Articles - Mathematics and its Applications

ACADEMY OF ROMANIAN SCIENTISTSANNALSSERIES ON MATHEMATICS AND ITS APPLICATIONSVOLUME 2 2010 NUMBER 2ISSN 2066 – 6594TOPICS: ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS OPTIMIZATION, OPTIMAL CONTROL AND DESIGN NUMERICAL ANALYSIS AND SCIENTIFIC COMPUTING ALGEBRAIC, TOPOLOGICAL AND DIFFERENTIALSTRUCTURES PROBABILITY AND STATISTICS ALGEBRAIC AND DIFFERENTIAL GEOMETRY MATHEMATICAL MODELLING IN MECHANICSENGINEERING SCIENCES MATHEMATICAL ECONOMY AND GAME THEORY MATHEMATICAL PHYSICS AND APPLICATIONSEDITURAACADEMIEI OAMENILOR DE ȘTIINȚĂ DIN ROMÂNIA

ACADEMY OF ROMANIAN SCIENTISTSANNALSSERIES ON MATHEMATICS AND ITS APPLICATIONSVOLUME 2 2010 NUMBER 2ISSN 2066 – 6594TOPICS: ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS OPTIMIZATION, OPTIMAL CONTROL AND DESIGN NUMERICAL ANALYSIS AND SCIENTIFIC COMPUTING ALGEBRAIC, TOPOLOGICAL AND DIFFERENTIALSTRUCTURES PROBABILITY AND STATISTICS ALGEBRAIC AND DIFFERENTIAL GEOMETRY MATHEMATICAL MODELLING IN MECHANICSENGINEERING SCIENCES MATHEMATICAL ECONOMY AND GAME THEORY MATHEMATICAL PHYSICS AND APPLICATIONSEDITURAACADEMIEI OAMENILOR DE ȘTIINȚĂ DIN ROMÂNIA

Annals of the Academyof Romanian ScientistsSeries on Mathematics and its ApplicationsFounding Editor-in-ChiefGen.(r) Prof. Dr. Vasile CândeaPresident of the Academy of Romanian ScientistsCo-EditorAcademician Aureliu SăndulescuPresident of the Section of MathematicsSeries EditorsFrederic Bonnans (Ecole Polytechnique, Paris), Frederic.Bonnans@inria.frDan Tiba (Institute of Mathematics, Bucharest), Dan.Tiba@imar.roEditorial BoardM. Altar (Bucharest), altarm@gmail.com, D. Andrica (Cluj), dorinandrica@yahoo.com,L. Badea (Bucharest), Lori.Badea@imar.ro, A.S. Carstea (Bucharest),carstas@yahoo.com, L. Gratie (Hong Kong), mcgratie@cityu.edu.hk, D. Jula(Bucharest), dorinjula@yahoo.fr, K. Kunisch (Graz), karl.kunisch@uni-graz.at,R. Litcanu (Iasi), litcanu@uaic.ro, M. Megan (Timisoara), megan@math.uvt.ro,M. Nicolae-Balan (Bucharest), mariana_prognoza@yahoo.com, C.P. Niculescu(Craiova), c.niculescu47@clicknet.ro, A. Perjan (Chisinau), perjan@usm.md,J.P. Raymond (Toulouse), raymond@mip.ups-tlse.fr, C. Scutaru (Bucharest),corneliascutaru@yahoo.com, J. Sprekels (Berlin), sprekels@wias-berlin.de, M. Sofonea(Perpignan), sofonea@univ-perp.fr, S. Solomon (Jerusalem), co3giacs@gmail.com,F. Troltzsch (Berlin), troeltzsch@math.tu-berlin.de, M. Tucsnak (Nancy),Tucsnak@iecn.u-nancy.fr, I.I. Vrabie (Iasi), ivrabie@uaic.ro, M. Yamamoto (Tokyo),myama@ms.u-tokyo.ac.jpSecretariate: stiintematematice@gmail.com© 2010, Editura Academiei Oamenilor de Ştiinţă, Bucureşti, sect. 5, str. SplaiulIndependenţei 54, 050094 ROMÂNIA

Annals of the Academy of Romanian ScientistsSeries on Mathematics and its ApplicationsISSN 2066 - 6594 Volume 2, Number 2 / 2010CONTENTSMihail MEGAN, Codruţa STOICAConcepts of dichotomy for skew-evolution semiflowsin Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Vasile DRAGAN, Toader MOROZANRobust stability and robust stabilization of discrete-time linearstochastic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141Peter PHILIPAnalysis, optimal control, and simulation of conductive-radiativeheat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171Nicolae CÎNDEA, Marius TUCSNAKInternal exact observability of a perturbed Euler-Bernoulli equationof arbitrary order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205Klaus KRUMBIEGEL, Ira NEITZEL, Arnd RÖSCHSufficient optimality conditions for the Moreau-Yosida-typeregularization concept applied to semilinear elliptic optimalcontrol problems with pointwise state constraints . . . . . . . . . . . . . . . . . . 222123

Annals of the Academy of Romanian ScientistsSeries on Mathematics and its ApplicationsISSN 2066 - 6594 Volume 2, Number 2 / 2010CONCEPTS OF DICHOTOMY FORSKEW-EVOLUTION SEMIFLOWS INBANACH SPACES ∗Mihail Megan † Codruța Stoica ‡AbstractIn this paper we investigate some dichotomy concepts for skewevolutionsemiflows in Banach spaces. Our main objective is to establishrelations between these concepts. We motivate our approach byillustrative examples.MSC: 34D05, 34D09keywords: skew-evolution semiflow, exponential dichotomy, polynomialdichotomy1 IntroductionIn the qualitative theory of evolution equations, the exponential dichotomyis one of the most important asymptotic properties, and in the last years itwas treated from various perspectives (see [1] –[16]).∗ Accepted for publication in revised form on May 2, 2010.† megan@math.uvt.ro Academy of Romanian Scientists, No.54, Independenței Str.,050094, Bucharest, Romania; Department of Mathematics, West University of Timişoara,No.4, Vasile Pârvan Blv., 300223, Timişoara, Romania‡ codruta.stoica@uav.ro Department of Mathematics and Computer Science, "AurelVlaicu" University of Arad, No.2 Elena Drăgoi Str., 310330, Arad, Romania125

126 Mihail Megan, Codruța StoicaThe notion of exponential dichotomy for linear differential equations wasintroduced by O. Perron in 1930. The classic paper [12] of Perron served asa starting point for many works on the stability theory.The property of exponential dichotomy for linear differential equationshas gained prominence since the appereance of two fundamental monographsdue to J.L. Daleckiĭ and M.G. Kreĭn (see [6]) and J.L. Massera and J.J.Schäffer (see [8]).The notion of linear skew-product semiflow arises naturally when oneconsiders the linearization along an invariant manifold of a dynamical systemgenerated by a nonlinear differential equation (see [14], Chapter 4).Diverse and important concepts of dichotomy for linear skew-productsemiflows were studied by C. Chicone and Y. Latushkin in [4], S.N. Chowand H. Leiva in [5], R.J. Sacker and G.R. Sell in [13].The particular cases of exponential stability and exponential instabilityfor linear skew-product semiflows have been considered in [9] and [10] .In this paper we consider the general case of skew-evolution semiflows(introduced in our paper [11]) as a natural generalization of skew-productsemiflows. The major difference consists in the fact that a skew-evolutionsemiflow depends on three variables t, t 0 and x, while the classic conceptof skew-product semiflow depends only on t and x, thus justifying a furtherstudy of asymptotic behaviors for skew-evolution semiflows in a more generalcase, the nonuniform setting (relative to the third variable t 0 ).The aim of this paper is to define and exemplify various concepts of dichotomiesas exponential dichotomy, Barreira-Valls exponential dichotomy,uniform exponential dichotomy, polynomial dichotomy, Barreira-Valls polynomialdichotomy and uniform polynomial dichotomy, and to emphasize connectionsbetween them. Thus we consider generalizations of some asymptoticproperties for differential equations studied by L. Barreira and C. Valls in[1], [2] and [3].Some results concerning the properties of stability and instability forskew-evolution semiflows were published by us in [11], in [15] and in [16].The obtained results clarify the difference between uniform dichotomiesand nonuniform dichotomies.

Concepts of dichotomy in Banach spaces 1272 Skew-evolution semiflowsLet us consider a metric space (X, d), a Banach space V and B(V ) thespace of all bounded linear operators from V into itself. I is the identityoperator on V . We denote Y = X × V and we consider the following sets∆ = { (t, t 0 ) ∈ R 2 + : t ≥ t 0}and T ={(t, s, t0 ) ∈ R 3 + : t ≥ s ≥ t 0 ≥ 0 } .Definition 1. A mapping ϕ : ∆ × X → X is called evolution semiflow onX if the following relations hold:(s 1 ) ϕ(t, t, x) = x, ∀(t, x) ∈ R + × X;(s 2 ) ϕ(t, s, ϕ(s, t 0 , x)) = ϕ(t, t 0 , x), ∀(t, s), (s, t 0 ) ∈ ∆, x ∈ X.Definition 2. A mapping Φ : ∆×X → B(V ) is called evolution cocycle overan evolution semiflow ϕ if:(c 1 ) Φ(t, t, x) = I, ∀(t, x) ∈ R + × X;(c 2 ) Φ(t, s, ϕ(s, t 0 , x))Φ(s, t 0 , x) = Φ(t, t 0 , x), ∀(t, s), (s, t 0 ) ∈ ∆, x ∈ X.Definition 3. The mapping C : ∆ × Y → Y defined by the relationC(t, s, x, v) = (Φ(t, s, x)v, ϕ(t, s, x)),where Φ is an evolution cocycle over an evolution semiflow ϕ, is called skewevolutionsemiflow on Y .Remark 1. The concept of skew-evolution semiflow generalizes the notionof skew-product semiflow, considered and studied by M. Megan, A.L. Sasuand B. Sasu in [9] and [10], where the mappings ϕ and Φ do not depend onthe variables t ≥ 0 and x ∈ X.Example 1. Let E : ∆ → B(V ) be an evolution operator on V . If thereexists P : X → B(V ) with the propertiesP (x) 2 = P (x) and P (x)E(t, s) = E(t, s)P (x),for all (t, s, x) ∈ ∆ × X, then C = (Φ, ϕ), whereis a linear skew-evolution semiflow.Φ(t, s, x) = P (x)E(t, s), ϕ(t, s, x) = x

128 Mihail Megan, Codruța StoicaExample 2. Let us consider a skew-evolution semiflow C = (Φ, ϕ) and aparameter λ ∈ R. We define the mappingΦ λ : ∆ × X → B(V ), Φ λ (t, t 0 , x) = e λ(t−t 0) Φ(t, t 0 , x).One can remark that C λ = (Φ λ , ϕ) also satisfies the conditions of Definition3, being called λ-shifted skew-evolution semiflow on Y .Let us consider on the Banach space V the Cauchy problem{ ˙v(t) = Av(t), t > 0v(0) = v 0where A is an operator which generates a C 0 -semigroup S = {S(t)} t≥0 .Then Φ(t, s, x)v = S(t − s)v, where t ≥ s ≥ 0, (x, v) ∈ Y , defines anevolution cocycle. Moreover, the mapping defined by Φ λ : ∆ × X → B(V ),Φ λ (t, s, x)v = S λ (t−s)v, where S λ = {S λ (t)} t≥0 is generated by the operatorA − λI, is also an evolution cocycle.Example 3. Let f : R + → R ∗ + be a decreasing function with the propertythat there exists lim f(t) = a > 0. We denote by C = C(R + , R + ) the sett→∞of all continuous functions x : R + → R + , endowed with the topology ofuniform convergence on compact subsets of R + , metrizable by means of thedistanced(x, y) =∞∑n=11 d n (x, y)2 n 1 + d n (x, y) , where d n(x, y) = sup |x(t) − y(t)|.t∈[0,n]If x ∈ C, then, for all t ∈ R + , we denote x t (s) = x(t + s), x t ∈ C. LetX be the closure in C of the set {f t , t ∈ R + }. It follows that (X, d) is ametric space. The mapping ϕ : ∆ × X → X, ϕ(t, s, x) = x t−s is an evolutionsemiflow on X.We consider V = R 2 , with the norm ‖v‖ = |v 1 | + |v 2 |, v = (v 1 , v 2 ) ∈ V .If u : R + → R ∗ +, then the mapping Φ u : ∆ × X → B(V ) defined byΦ u (t, s, x)v =( u(s)u(t) e− R ts x(τ−s)dτ v 1 , u(t)u(s) e R ts x(τ−s)dτ v 2),is an evolution cocycle over ϕ and C = (Φ u , ϕ) is a skew-evolution semiflow.

Concepts of dichotomy in Banach spaces 129Example 4. Let X be a metric space, ϕ an evolution semiflow on X andA : X → B(V ) a continuous mapping, where V is a Banach space. If Φ(t, s, x)is the solution of the Cauchy problem{ v ′ (t) = A(ϕ(t, s, x))v(t), t > sv(s) = x,then C = (Φ, ϕ) is a linear skew-evolution semiflow.Other examples of skew-evolution semiflows are given in [15].3 Exponential dichotomyIn this section we define three concepts of exponential dichotomy for skewevolutionsemiflows. We will establish connections between these notions andwe will emphasize that they are not equivalent.Let C : ∆ × Y → Y , C(t, s, x, v) = (Φ(t, s, x)v, ϕ(t, s, x)) be a skewevolutionsemiflow on Y .We recall that a mapping P : X → B(V ) with the propertyP (x) 2 = P (x), ∀x ∈ Xis called projections family on V .The mapping Q : X → B(V ) defined by Q(x) = I − P (x) is a projectionsfamily, which is called the complementary of P .Definition 4. A projections family P : X → B(V ) is said to be compatiblewith the skew-evolution semiflow C = (Φ, ϕ) iff:for all (t, s, x) ∈ ∆ × X.Φ(t, s, x)P (x) = P (ϕ(t, s, x))Φ(t, s, x),In what follows, if P is a given projections family, we will denoteΦ P (t, s, x) = Φ(t, s, x)P (x),for every (t, s, x) ∈ ∆ × X.We remark that(i) Φ P (t, t, x) = P (x), for all (t, x) ∈ R + × X;(ii) Φ P (t, s, ϕ(s, t 0 , x))Φ P (s, t 0 , x) = Φ P (t, t 0 , x), for all (t, s, t 0 , x) ∈ T × X.

130 Mihail Megan, Codruța StoicaDefinition 5. The skew-evolution semiflow C = (Φ, ϕ) is exponentially dichotomicrelative to the projections family P : X → B(V ) (and we denoteP.e.d.) iff there exist a constant α > 0 and a nondecreasing mappingN : R + → [1, ∞) such that:(ed 1 ) e α(t−s) ‖Φ P (t, t 0 , x 0 )v 0 ‖ ≤ N(s) ‖Φ P (s, t 0 , x 0 )v 0 ‖ ;(ed 2 ) e α(t−s) ‖Φ Q (s, t 0 , x 0 )v 0 ‖ ≤ N(t) ‖Φ Q (t, t 0 , x 0 )v 0 ‖ ,for all (t, s, t 0 , x 0 , v 0 ) ∈ T × Y , where Q is the complementary of P .Remark 2. The skew-evolution semiflow C = (Φ, ϕ) is P.e.d. if and only ifthere exist a constant α > 0 and a nondecreasing mapping N : R + → [1, ∞)such that:(ed ′ 1 ) eα(t−s) ‖Φ P (t, s, x)v‖ ≤ N(s) ‖P (x)v‖ ;(ed ′ 2 ) eα(t−s) ‖Q(x)v‖ ≤ N(t) ‖Φ Q (t, s, x)v‖ ,for all for all (t, s, x, v) ∈ ∆ × Y .A particular case of P.e.d. is given byDefinition 6. The skew-evolution semiflow C = (ϕ, Φ) is called Barreira-Valls exponentially dichotomic relative to the projections family P : X →B(V ) (and we denote P.B.V.e.d.) iff there exist N ≥ 1, α > 0 and β ≥ 0such that:(BV ed 1 ) e α(t−s) ‖Φ P (t, t 0 , x 0 )v 0 ‖ ≤ Ne βs ‖Φ P (s, t 0 , x 0 )v 0 ‖;(BV ed 2 ) e α(t−s) ‖Φ Q (s, t 0 , x 0 )v 0 ‖ ≤ Ne βt ‖Φ Q (t, t 0 , x 0 )v 0 ‖ ,for all (t, s, t 0 , x 0 , v 0 ) ∈ T × Y .Remark 3. The skew-evolution semiflow C = (Φ, ϕ) is P.B.V.e.d. if andonly if there exist N ≥ 1, α > 0 and β ≥ 0 such that:(BV ed ′ 1 ) eα(t−s) ‖Φ P (t, s, x)v‖ ≤ Ne βs ‖P (x)v‖(BV ed ′ 2 ) eα(t−s) ‖Q(x)v‖ ≤ Ne βt ‖Φ Q (t, s, x)v‖ ,for all for all (t, s, x, v) ∈ ∆ × Y .Remark 4. It is obvious that if C is P.B.V.e.d., then it is P.e.d.The converse is not true, fact illustrated byExample 5. We consider the metric space (X, d), the Banach space V andthe evolution semiflow ϕ defined as in Example 3. Let us consider thecomplementary projections families P, Q : X → B(V ), P (x)v = (v 1 , 0),Q(x)v = (0, v 2 ), for all x ∈ X and all v = (v 1 , v 2 ) ∈ V , compatible with C.

Concepts of dichotomy in Banach spaces 131Let g : R + → [1, ∞) be a continuous function with(g(n) = e n·22n and g n + 1 )2 2n = e 4 , for all n ∈ N.The mapping Φ : ∆ × X → B(V ), defined by( g(s)tΦ(t, s, x)v =g(t) e−(t−s)−R s x(τ−s)dτ v 1 , g(s))tg(t) et−s+R s x(τ−s)dτ v 2is an evolution cocycle over the evolution semiflow ϕ.We observe that for α = 1 + a we have thatande α(t−s) ‖Φ P (t, s, x)v‖ ≤ g(s) ‖P (x)v‖e α(t−s) ‖Q(x)v‖ ≤ g(s)e α(t−s) ‖Q(x)v‖ ≤ g(t) ‖Φ Q (t, s, x)v‖ ,for all (t, s, x, v) ∈ ∆ × Y. Thus, conditions (ed ′ 1 ) and (ed′ 2 ) are satisfied forα = 1 + a and N(t) = sup g(s)s∈[0,t]and, hence, C = (Φ, ϕ) is P.e.d.If we suppose that C is P.B.V.e.d., then there exist N ≥ 1, α > 0 andβ ≥ 0 such thatg(s)e αt ≤ Ng(t)e βs+t−s+R ts x(τ−s)dτ ,for all (t, s, x) ∈ ∆ × X.From here, for t = n + 1 and s = n, it follows that22n e n(22n +α−β) ≤ 81Ne 1−α+f(0)2 2n ,which, for n → ∞, implies a contradiction.Another particular case of P.e.d. is introduced byDefinition 7. The skew-evolution semiflow C = (Φ, ϕ) is uniformly exponentiallydichotomic relative to the projections family P : X → B(V ) (andwe denote P.u.e.d.) iff there exist some constants N ≥ 1 and α > 0 suchthat:(ued 1 ) e α(t−s) ‖Φ P (t, t 0 , x 0 )v 0 ‖ ≤ N ‖Φ P (s, t 0 , x 0 )v 0 ‖ ;(ued 2 ) e α(t−s) ‖Φ Q (s, t 0 , x 0 )v 0 ‖ ≤ N ‖Φ Q (t, t 0 , x 0 )v 0 ‖ ,for all (t, s, t 0 , x 0 , v 0 ) ∈ T × Y .

132 Mihail Megan, Codruța StoicaRemark 5. The skew-evolution semiflow C = (Φ, ϕ) is P.u.e.d. if and onlyif there exist some constants N ≥ 1 and α > 0 such that:(ued ′ 1 ) eα(t−s) ‖Φ P (t, s, x)v‖ ≤ N ‖P (x)v‖ ;(ued ′ 2 ) eα(t−s) ‖Q(x)v‖ ≤ N ‖Φ Q (t, s, x)v‖ ,for all for all (t, s, x, v) ∈ ∆ × Y .Remark 6. It is obvious that if C is P.u.e.d., then it is P.B.V.e.d.The following example shows that the converse implication is not valid.Example 6. We consider the metric space (X, d), the Banach space V andthe evolution semiflow ϕ defined as in Example 3. Let us consider thecomplementary projections families P, Q : X → B(V ), P (x)v = (v 1 , 0),Q(x)v = (0, v 2 ), for all x ∈ X and all v = (v 1 , v 2 ) ∈ V , compatible with C.The mapping Φ : ∆ × X → B(V ), defined by=Φ(t, s, x)v =(v 1 e t sin t−s sin s−2(t−s)−R ts x(τ−s)dτ , v 2 e 3(t−s)−2t cos t+2s cos s+R ts x(τ−s)dτ )is an evolution cocycle over the evolution semiflow ϕ.We observe that for α = 1 + a we have thate α(t−s) ‖Φ P (t, s, x)v‖ ≤ e α(t−s) e −(1+a)t e (3+a)s ‖P (x)v‖ ≤ e 2s ‖P (x)v‖ ,for all (t, s, x, v) ∈ ∆ × Y. Similarly,e α(t−s) ‖Q(x)v‖ ≤ e α(t−s) e −3t+3s+2t cos t−2s cos s−R ts x(τ−s)dτ ‖Φ Q (t, s, x)v‖≤ ‖Φ Q (t, s, x)v‖ ,for all (t, s, x, v) ∈ ∆×Y. Thus, conditions (BV ed ′ 1 ) and (BV ed′ 2 ) are satisfiedforα = 1 + a, N = 1 and β = min{0, 2}.This shows that C = (Φ, ϕ) is P.B.V.e.d.If we suppose that C is P.u.e.d., then there exist N > 1 and α > 0 suchthate α(t−s) e t sin t−s sin s−2t+2s−R ts x(τ−s)dτ ≤ N,

Concepts of dichotomy in Banach spaces 133for all (t, s) ∈ ∆. In particular, for t = 2nπ + π 2and s = 2nπ, we obtain2nπ + (α − 1) π 2 ≤ ln N ∫ 2nπ+π2= ln N∫ π202nπx(u)du ≤ fwhich, for n → ∞, leads to a contradiction.4 Polynomial dichotomyx(τ − 2nπ)dτ =( π2)ln N,Let C : ∆×Y → Y , C(t, s, x, v) = (Φ(t, s, x)v, ϕ(t, s, x)) be a skew-evolutionsemiflow on Y and let P : X → B(V ) be a projections family on V , compatiblewith C, and Q : X → B(V ) the complementary projections family ofP .Definition 8. The skew-evolution semiflow C = (Φ, ϕ) is polynomially dichotomicwith respect to P (and we denote P.p.d.) iff there exist α > 0,t 1 > 0 and a nondecreasing function N : R + → [1, ∞) such that:(pd 1 ) t α ‖Φ P (t, t 0 , x 0 )v 0 ‖ ≤ N(s)s α ‖Φ P (s, t 0 , x 0 )v 0 ‖ ;(pd 2 ) t α ‖Φ Q (s, t 0 , x 0 )v 0 ‖ ≤ N(t)s α ‖Φ Q (t, t 0 , x 0 )v 0 ‖ ,for all (t, s, t 0 , x 0 , v 0 ) ∈ T × Y with t 0 ≥ t 1 .Remark 7. The skew-evolution semiflow C = (Φ, ϕ) is P.p.d. if and onlyif there exist α > 0, t 0 > 0 and a nondecreasing function N : R + → [1, ∞)such that:(pd ′ 1 ) tα ‖Φ P (t, s, x)v‖ ≤ N(s)s α ‖P (x)v‖ ;(pd ′ 2 ) tα ‖Q(x)v‖ ≤ N(t)s α ‖Φ Q (t, s, x)v‖ ,for all for all (t, s, x, v) ∈ ∆ × Y with s ≥ t 0 .Example 7. We consider the metric space (X, d), the Banach space V andthe evolution semiflow ϕ defined as in Example 3. Let us consider thecomplementary projections families P, Q : X → B(V ), P (x)v = (v 1 , 0),Q(x)v = (0, v 2 ), for all x ∈ X and all v = (v 1 , v 2 ) ∈ V , compatible with C.The mapping Φ : ∆ × X → B(V ), defined by( s + 1R tΦ(t, s, x)v =t + 1 e− s x(τ−s)dτ v 1 , t + 1)Rs + 1 e ts x(τ−s)dτ v 2

134 Mihail Megan, Codruța Stoicais an evolution cocycle withandt a ‖Φ P (t, s, x)v‖ ≤ ta (s + 1)e −a(t−s)t + 1‖P (x)v‖ ≤ s a ‖P (x)v‖t a ‖Q(x)v‖ ≤ s a e a(t−s) ‖Q(x)v‖ ≤ sa (t + 1)s + 1 e R ts x(τ−s)dτ ≤ s a ‖Φ Q (t, s, x)v‖ ,for all t ≥ s ≥ 1 and all (x, v) ∈ Y. It follows that C = (Φ, ϕ) is P.p.d.Proposition 1. If C =(Φ, ϕ) is a P–exponentially dichotomic skew-evolutionsemiflow, then it is P –polynomially dichotomic.Proof. If C is P.e.d., then there exist α > 0 and N : R + → [1, ∞) such thatconditions (ed ′ 1 ) and (ed′ 2 ) are satisfied.We observe that the functionu : [1, ∞) → (0, ∞), u(t) = ettis nondecreasing on [1, ∞) and, hence,t αs α ‖Φ P (t, s, x)v‖ ≤ e α(t−s) ‖Φ P (t, s, x)v‖ ≤ N(s) ‖P (x)v‖andt αs α ‖Q(x)v‖ ≤ eα(t−s) ‖Q(x)v‖ ≤ N(t) ‖Φ Q (t, s, x)v‖ ,for all t ≥ s ≥ t 0 ≥ 1 and all (x, v) ∈ Y .Finally, it results that conditions (pd ′ 1 ) and (pd′ 2 ) are satisfied, whichproves that C is P.p.d.The converse of the preceding proposition is not valid. This fact is illustratedbyExample 8. Let X = R + and V = R 2 . The mapping ϕ : ∆ × X → X,defined by ϕ(t, s, x) = x is an evolution semiflow on R + .We define the evolution cocycle Φ : ∆ × X → B(V ) by( s + 1Φ(t, s, x)(v 1 , v 2 ) =t + 1 v 1, t + 1 )s + 1 v 2 ,

Concepts of dichotomy in Banach spaces 135with (t, s, x, v) ∈ ∆ × Y. Then P : X → B(V ), P (x)(v 1 , v 2 ) = (v 1 , 0) is aprojections family which is compatible with the skew-evolution semiflow C =(Φ, ϕ). Q denotes the complementary projections family of P . Furthermoreandt ‖Φ P (t, s, x)v‖ ≤ s 2 ‖P (x)v‖t ‖Q(x)v‖ ≤ ts ‖Φ Q (t, s, x)v‖for all (t, s, x, v) ∈ ∆ × Y.Hence, the conditions (pd ′ 1 ) and (pd′ 2 ) are satisfied forα = 1, t 0 = 1 and N(t) = t.Thus, C is P.p.d.If we suppose that C is P.e.d., then there exist α > 0 and a mappingN : R + → [1, ∞) such that(s + 1)e α(t−s) ≤ (t + 1)N(s),for all t ≥ s ≥ 0. From here, for s fixed and t → ∞, we obtain a contradiction.A particular case of polynomial dichotomy is introduced byDefinition 9. The skew-evolution semiflow C = (Φ, ϕ) is polynomially dichotomicin the sense Barreira-Valls with respect to the projections familyP : X → B(V ) (and we denote P.B.V.p.d.) iff there exist N ≥ 1, t 1 > 0,α > 0 and β ≥ 0 such that:(BV pd 1 ) t α ‖Φ P (t, t 0 , x 0 )v 0 ‖ ≤ Ns α+β ‖Φ P (s, t 0 , x 0 )v 0 ‖ ;(BV pd 2 ) t α ‖Φ Q (s, t 0 , x 0 )v 0 ‖ ≤ Ns α t β ‖Φ Q (t, t 0 , x 0 )v 0 ‖ ,for all (t, s, t 0 , x 0 , v 0 ) ∈ T × Y with t 0 ≥ t 1 .Remark 8. The skew-evolution semiflow C = (Φ, ϕ) is P.B.V.p.d. if andonly if there exist N ≥ 1, t 0 > 0, α > 0 and β ≥ 0 such that:(BV pd ′ 1 ) tα ‖Φ P (t, s, x)v‖ ≤ Ns α+β ‖P (x)v‖ ;(BV pd ′ 2 ) tα ‖Q(x)v‖ ≤ Ns α t β ‖Φ Q (t, s, x)v‖ ,for all for all (t, s, x, v) ∈ ∆ × Y with s ≥ t 0 .Remark 9. It is obvious that if C is P.B.V.p.d. then it is P.p.d.The following example shows that the converse is not true.

136 Mihail Megan, Codruța StoicaExample 9. We consider the skew-evolution semiflow C = (Φ, ϕ) given inExample 3 and the complementary projections families P, Q : X → B(V ),P (x)v = (v 1 , 0), Q(x)v = (0, v 2 ), for all x ∈ X and all v = (v 1 , v 2 ) ∈ V ,compatible with C. Because C is P.e.d., then it is also P.p.d.If we suppose that C is P.B.V.p.d., then there exist N ≥ 1, t 0 > 0, α > 0and β ≥ 0 such thatt α g(s) ≤ Ng(t)s α+β e t−s+R t−s0 x(u)du ,for all t ≥ s ≥ t 0 . From here, for t = n + 1 and s = n → ∞, we obtain a22n contradiction.Another particular case of polynomial dichotomy is given byDefinition 10. The skew-evolution semiflow C = (Φ, ϕ) is uniformly polynomiallydichotomic in rapport with the projections family P : X → B(V )(and we denote P.u.p.d.) iff there exist N ≥ 1, α > 0 and t 1 > 0 such that:(upd 1 ) t α ‖Φ P (t, t 0 , x 0 )v 0 ‖ ≤ Ns α ‖Φ P (s, t 0 , x 0 )v 0 ‖ ;(upd 2 ) t α ‖Φ Q (s, t 0 , x 0 )v 0 ‖ ≤ Ns α ‖Φ Q (t, t 0 , x 0 )v 0 ‖ ,for all (t, s, t 0 , x 0 , v 0 ) ∈ T × Y with t 0 ≥ t 1 .Remark 10. The skew-evolution semiflow C = (Φ, ϕ) is P.u.p.d. if and onlyif there exist N ≥ 1, α > 0 and t 0 > 0 such that:(upd ′ 1 ) tα ‖Φ P (t, s, x)v‖ ≤ Ns α ‖P (x)v‖ ;(upd ′ 2 ) tα ‖Q(x)v‖ ≤ Ns α ‖Φ Q (t, s, x)v‖ ,for all for all (t, s, x, v) ∈ ∆ × Y with s ≥ t 0 .Remark 11. If C is P.u.p.d. then it is P.B.V.p.d.The reciprocal is not valid, fact illustrated byExample 10. We consider the metric space (X, d), the Banach space Vand the evolution semiflow ϕ defined as in Example 3. Let us consider thecomplementary projections families P, Q : X → B(V ), P (x)v = (v 1 , 0),Q(x)v = (0, v 2 ), for all x ∈ X and all v = (v 1 , v 2 ) ∈ V , compatible with C.We consider the functiong : R + → R, g(t) =(t + 1) 3(t + 1) sin ln(t+1)

Concepts of dichotomy in Banach spaces 137and the evolution cocycle Φ : ∆ × X → B(V ) over ϕ defined by( g(s)Φ(t, s, x)v =g(t) v 1, g(t) )g(s) v 2 .Thent ‖Φ P (t, s, x)v‖ ≤t(s + 1)4(t + 1) 2 ‖P (x)v‖ ≤ s(s + 1)2 ‖P (x)v‖ ≤ 4s 3 ‖P (x)v‖andt ‖Q(x)v‖ ≤ t(t + 1) 2 ‖Q(x)v‖ ≤s(t + 1)4(s + 1) 2 ‖Q(x)v‖≤ s ‖Φ Q (t, s, x)v‖ ≤ 4st 2 ‖Φ Q (t, s, x)v‖ ,for all t ≥ s ≥ 1 and all (x, v) ∈ Y. Thus, the conditions (BV pd ′ 1 ) and(BV pd ′ 2 ) are satisfied forα = 1, β = 2, N = 4 and t 0 = 1.If we suppose that C is P.u.p.d., then there are N ≥ 1, α > 0 and t 0 > 0such thatt α (s + 1) 3 (t + 1) sin ln(t+1) ≤ Ns α (t + 1) 3 (s + 1) sin ln(s+1) ,for all t ≥ s ≥ t 0 . From here, for t = e 2nπ+ π 2 − 1 and s = e 2nπ− π 2 − 1 andn → ∞, we obtain a contradiction.Proposition 2. If the skew-evolution semiflow C = (Φ, ϕ) is uniformly exponentiallydichotomic with respect to the projections family P : X → B(V ),then C is uniformly polynomially dichotomic with respect to P .Proof. If C = (Φ, ϕ) is P.u.e.d., then there are N ≥ 1 and α > 0 suchthat the conditions (ued 1 ) and (ued 2 ) are satisfied. Using the inequalitieswe obtaint + 1 ≤ e t ,e ss ≤ ett and t ≤ t − s + 1, for t ≥ s ≥ 1,st α ‖Φ P (t, s, x)v‖ ≤ Nt α e −α(t−s) ‖P (x)v‖ ≤ Ntα ‖P (x)v‖(1 + t − s) α ≤ Nsα ‖P (x)v‖

138 Mihail Megan, Codruța Stoicaandt α‖Q(x)v‖ ≤ Nt α e −α(t−s) ‖Φ Q (t, s, x)v‖ ≤ Ns α ‖Φ Q (t, s, x)v‖ ,for all (t, s, x, v) ∈ ∆ × Y with s ≥ t 0 = 1.Finally, we obtain that C is P.u.p.d.Now, we give an example which shows that the converse of the precedingresult is not valid.Example 11. We consider the metric space (X, d), the Banach space Vand the evolution semiflow ϕ defined as in Example 3. Let us consider thecomplementary projections families P, Q : X → B(V ), P (x)v = (v 1 , 0),Q(x)v = (0, v 2 ), for all x ∈ X and all v = (v 1 , v 2 ) ∈ V , compatible with C.We consider the evolution cocycle Φ : ∆ × X → B(V ), defined by( s 2 + 1R )tΦ(t, s, x)v =t 2 + 1 e− s x(τ−s)dτ v 1 , t2 + 1Rs 2 + 1 e ts x(τ−s)dτ v 2 ,for (t, s, x) ∈ ∆ × X and v = (v 1 , v 2 ) ∈ V = R 2 . Using the inequalitieswe obtains 2 + 1t 2 + 1 ≤ s tesande t ≤ s , for t ≥ s ≥ 1,tt α ‖Φ P (t, s, x)v‖ ≤ tα (s 2 + 1)t 2 e −a(t−s) ‖P (x)v‖+ 1≤ tα · s( s) a= s α ‖P (x)v‖ ,t tfor all t ≥ s ≥ t 0 = 1 and (x, v) ∈ Y , where α = 1 + a.Similarly,t α ‖Q(x)v‖ = t · t a ≤ ts a e at e −as ‖Q(x)v‖ ≤ t s sα e a(t−s) ‖Q(x)v‖≤ sα (t 2 + 1)s 2 + 1 ea(t−s) ‖Q(x)v‖ ≤ s α ‖Φ Q (t, s, x)v‖ ,for all t ≥ s ≥ t 0 = 1 and (x, v) ∈ Y , with α = 1 + a. Thus, C is P.u.p.d.If we suppose that C is P.u.e.d., then there exist N ≥ 1, α > 0 and t 0 > 0such that(s 2 + 1)e α(t−s) ≤ N(t 2 + 1)e −a(t−s) ,for all t ≥ s ≥ t 0 . Then, for s = t 0 and t → ∞, we obtain a contradiction,which can be eliminated only if C is not P.u.e.d.

Concepts of dichotomy in Banach spaces 139Acknowledgement. This work was supported by CNCSIS–UEFISCSU,project number PN II–IDEI 1080/2008. The authors wish to express specialthanks to the reviewers for their helpful suggestions, which led to theimprovement of this paper.References[1] L. Barreira, C. Valls. Stability of Nonautonomous Differential Equations.Lect. Notes Math. 1926, 2008.[2] L. Barreira, C. Valls. Polynomial growth rates. Nonlinear Analysis.71:5208–5219, 2009.[3] L. Barreira, C. Valls. Existence of nonuniform exponential dichotomiesand a Fredholm alternative. Nonlinear Analysis. 71:5220–5228, 2009.[4] C. Chicone, Y. Latushkin. Evolution semigroups in dynamical systemsand differential equations. Mathematical Surveys and Monographs,Amer. Math. Soc., Providence, Rhode Island, 70, 1999.[5] S.N. Chow, H. Leiva. Existence and roughness of the exponential dichotomyfor linear skew-product semiflows in Banach spaces. J. DifferentialEquations. 120:429–477, 1995.[6] J.L. Daleckiĭ, M.G. Kreĭn. Stability of solutions of differential equationsin Banach space. Translations of Mathematical Monographs, Amer.Math. Soc., Providence, Rhode Island, 43, 1974.[7] N.T. Huy. Existence and robustness of exponential dichotomy for linearskew-product semiflows. J. Math. Anal. Appl. 33:731-752, 2007.[8] J.L. Massera, J.J. Schäffer. Linear Differential Equations and FunctionSpaces. Pure Appl. Math. 21 Academic Press, New York-London, 1966.[9] M. Megan, A.L. Sasu, B. Sasu. On uniform exponential stability of linearskew-product semiflows in Banach spaces. Bull. Belg. Math. Soc. SimonStevin. 9:143–154, 2002.

140 Mihail Megan, Codruța Stoica[10] M. Megan, A.L. Sasu, B. Sasu. Exponential stability and exponentialinstability for linear skew-product flows. Math. Bohem. 129, No. 3:225–243, 2004.[11] M. Megan, C. Stoica. Exponential instability of skew-evolution semiflowsin Banach spaces. Studia Univ. Babeş-Bolyai Math. LIII, No. 1:17–24,2008.[12] O. Perron. Die Stabilitätsfrage bei Differentialgleichungen. Math. Z.32:703–728, 1930.[13] R. J. Sacker, G. R. Sell. Dichotomies for linear evolutionary equationsin Banach spaces. J. Differential Equations. 113(1):17–67, 1994.[14] G. R. Sell, Y. You. Dynamics of evolutionary equations. Appl. Math.Sciences. 143, Springer Verlag, New–York, 2002.[15] C. Stoica, M. Megan. On uniform exponential stability for skewevolutionsemiflows on Banach spaces. Nonlinear Analysis. 72, Issues3–4:1305–1313, 2010.[16] C. Stoica, M.Megan. Nonuniform behaviors for skew-evolution semiflowsin Banach spaces. Operator Theory Live, Theta Ser. Adv. Math. 203–211, 2010.

Annals of the Academy of Romanian ScientistsSeries on Mathematics and its ApplicationsISSN 2066 - 6594 Volume 2, Number 2 / 2010ROBUST STABILITY AND ROBUSTSTABILIZATION OF DISCRETE-TIMELINEAR STOCHASTIC SYSTEMS ∗Vasile Dragan † Toader Morozan ‡AbstractIn this paper the problem of robust stabilization of a general class ofdiscrete-time linear stochastic systems subject to Markovian jumpingand independent random perturbations is investigated. A stochasticversion of the bounded real lemma is derived and the small gain theoremis proved. Finally, methodology for the designing of a stabilizingfeedback gain for discrete-time linear stochastic system with structuredparametric uncertainties is proposed.MSC: 93E15, 93E20, 93C55, 93B36keywords: discrete-time stochastic systems, input-output operators,bounded real lemma, small gain theorem, disturbance attenuation problem1 IntroductionIn many applications the mathematical model of the controlled process isnot completely known. Even if the multiplicative white noise perturbationsare introduced in order to model the stochastic environmental perturbationswhich are hard to quantify, it is also possible that some parametric uncertaintiesoccur in the coefficients of the stochastic system. Thus a robust∗ Accepted for publication in revised form on May 20, 2010.† Vasile.Dragan@imar.ro Institute of Mathematics of the Romanian Academy,P.O.Box. 1-764, RO-014700, Bucharest, Romania‡ Toader.Morozan@imar.ro Institute of Mathematics of the Romanian Academy,P.O.Box. 1-764, RO-014700, Bucharest, Romania141

142 Vasile Dragan, Toader Morozanstabilization problem, ask to construct a control law in a static or dynamicfeedback form which stabilizes all discrete-time linear stochastic systems intoa neighborhood of a given system often called the nominal system.To be more specific, let us consider the controlled system:x(t + 1) = (A 0 (η t ) + ∆ A (t, η t ) +r∑w k (t)A k (η t ))x(t) + B(η t )u(t) (1)k=1where A k (i), 0 ≤ k ≤ r, B(i), 1 ≤ i ≤ N, ar known matrices of appropriatedimensions, while ∆ A (t, i), t ≥ 0 are unknown matrices. A robust stabilizationproblem, via state feedback control law, ask to construct a controlu(t) = F (η t )x(t) such that the zero state equilibrium of the nominal systemx(t + 1) = (A 0 (η t ) + B(η t )F (η t ) +r∑w k (t)A k (η t ))x(t) (2)k=1and the zero state equilibrium of the perturbed systemx(t + 1) = (A 0 (η t ) + B(η t )F (η t ) + ∆ A (t, η t ) +r∑w k (t)A k (η t ))x(t) (3)are exponentially stable in mean square (ESMS) for all uncertainties ∆ A (t, i)in a neighborhood of the origin in R n×n .It is known that if the zero state equilibrium of the nominal system (2)is ESMS then the zero state equilibrium of the perturbed system (3) is stillESMS for some "small perturbations" ∆ A (t, i). In a robust stability problem,as well as in a robust stabilization problem, the goal is to preserve the stabilityof the nominal system for the perturbed systems in the case of the variationof the coefficients of the system which are not necessarily small.In this paper we shall investigate different aspects of the problem of robuststability and robust stabilization of discrete-time linear stochastic systems (1)with structured parametric uncertainties of the form:∆ A (t, η t ) = (G 0 (η t ) +k=1r∑w k (t)G k (η t ))∆(η t )C(η t )k=1where the matrices G k (i), 0 ≤ k ≤ r, C(i), 1 ≤ i ≤ N are assumed tobe known, and ∆(i), 1 ≤ i ≤ N are unknown matrices of appropriate dimensions.We shall see that in the definition of the set of the uncertainties

Robust stability of discrete-time linear stochastic systems 143∆ = (∆(1), ..., ∆(N)) for which the exponential stability in mean square ispreserved, an important role is played by the norm of linear operator adequatelychosen, named input-output operator.For this reason we shall start with the proof of the stochastic version ofthe Bounded Real Lemma. This result allows us to obtain information aboutthe norm of an input-output operator. Further we shall prove a stochasticversion of the Small Gain Theorem which is a powerful tool in the estimationof the stability radius of a perturbed system, with structured parametricuncertainties.Bounded Real Lemma and other H ∞ control problems for discrete-timelinear systems affected by independent random perturbations were consideredin [1, 4, 5, 11, 12, 13, 15, 17, 19] while in the Markovian case in [2, 3, 14, 16,18, 20, 21]. The proof of the Bounded Real Lemma in this paper follows theideas in [1, 16]. In fact, the result of this paper is the discrete-time counterpart of the ones developed in chapter 6 in [6].2 Input-output operatorsLet us consider the system (G) with the state space representation:r∑r∑x(t+1) = (A 0 (η t )+ w k (t)A k (η t ))x(t)+(B 0 (η t )+ w k (t)B k (η t ))v(t) (4)k=1z(t) = C(η t )x(t) + D(η t )v(t)where x(t) ∈ R n is the state of the system, v(t) ∈ R mv is the external inputand z(t) ∈ R nz is the output; {w(t)} t≥0 , ( w(t) = (w 1 (t), w 2 (t), ..., w r (t)) T )is a sequence of independent random vectors and the triple ({η t } t≥0 , P, D) isan homogeneous Markov chain, on a given probability space (Ω, F, P) withthe set of the states D = {1, 2, ..., N} and the transition probability matrixP = (p(i, j)) N i,j=1 .Concerning the processes {η t } t≥0 , {w(t)} t≥0 the following assumptionsare made:H 1 ) {w(t)} t≥0 is a sequence of independent random vectors with thefollowing properties:I r being the identity matrix of size r.k=1E[w(t)] = 0, E[w(t)w T (t)] = I r , t ≥ 0,

144 Vasile Dragan, Toader MorozanH 2 ) The stochastic processes {w(t)} t≥0 and {η(t)} t≥0 are independent.Throughout the paper we assume that together with the hypothesesH 1 ) − H 2 ), the Markov chain verifies the additional assumption:H 3 ) (i) The transition probability matrix P is a nondegenerate stochasticmatrix, that isN∑p(j, i) > 0, (∀) 1 ≤ i ≤ N.j=1(ii) π 0 (i) = P{η 0 = i} > 0, 1 ≤ i ≤ N.It is easy to verify by induction that the assumption H 3 ) holds iff π t (i) =P{η t = i} > 0 for all t ∈ Z + and i ∈ D.In (4) A k (i), B k (i), 0≤k ≤r, C(i), D(i), 1 ≤ i ≤ N are given matrices ofappropriate dimensions.For each t ≥ 0 we denote F t =σ(w(s); 0≤s≤t) and G t = σ(η s ; 0≤s≤t).Let H t = F t ∨ G t , t ∈ Z + . ˜Ht = F t−1 ∨ G t if t ≥ 1 and ˜H 0 = σ(η 0 ).In the following l 2˜H{0, τ; R m } stands for the space of all finite sequences{v(t)} 0≤t≤τ of m-dimensional random vectors with the properties that for all0 ≤ t ≤ τ, v(t) is ˜H t -measurable and E[|v(t)| 2 ] < ∞. Also l 2˜H{0, ∞; R m } isthe space of all sequences {v(t)} t≥0 of m-dimensional random vectors with theproperties that for all t ≥ 0, v(t) is ˜H t -measurable and ∑ ∞t=0 E[|v(t)|2 ] < ∞.In this paper, the inputs v = {v(t)} t≥0 are stochastic processes eitherin l 2˜H{0, τ; R mv } for τ > 0 or in l 2˜H{0, ∞; R mv }. Both l 2˜H{0, τ; R mv } andl 2˜H{0, ∞; R mv } are real Hilbert spaces.The norms induced by the usual inner product on each of this Hilbertspace are:τ∑||v|| l 2˜H{0,τ;R mv } = ( E[|v(t)| 2 ]) 1 2for all v ∈ l 2˜H{0, τ; R mv } andt=0∞||v|| l 2˜H{0,∞;R mv } = ( ∑E[|v(t)| 2 ]) 1 2respectively, for all v ∈ l 2˜H{0, ∞; R mv }.t=0

Robust stability of discrete-time linear stochastic systems 145Let x(t, 0, v) be the solution of the system (4) corresponding to the inputv = {v(t)} t≥0 with the initial condition x(0, 0, v) = 0. Letz(t, 0, v) = C(η t )x(t, 0, v) + D(η t )v(t) (5)the corresponding output. One can see that if v ∈ l 2˜H{0, τ; R mv } for someτ ≥ 1, then x(t, 0, v) is H t−1 -measurable and E[|x(t, 0, v)| 2 ] < ∞.Hence from (5) it follows that {z(t, 0, v)} 0≤t≤τ ∈ l 2˜H{0, τ; R nz }.Consider the linear system:r∑x(t + 1) = (A 0 (η t ) + w k (t)A k (η t ))x(t) (6)k=1Definition 2.1 We say that the zero state equilibrium of the system(6) is exponentially stable in mean square (ESMS) if there exist β ≥ 1 andq ∈ (0, 1) such thatE[|x(t, 0, x 0 )| 2 ≤ βq t |x 0 | 2for all t ∈ Z + , x 0 ∈ R n , where x(t, 0, x 0 ) is the solution of (6) starting fromx 0 at time t = 0.Applying Lemma 4.3 in [9] we deduce that if the zero state equilibriumof (6) is ESMS, then there exists γ > 0 such that∞∑∑∞E[|z(t, 0, v)| 2 ] ≤ γ 2 E[|v(t)| 2 ] (7)t=0for all v ∈ l 2˜H{0, ∞; R mv }.It can be remarked that in the absence of the property of the exponentialstability in mean square of the linear system (6) one can prove that for eachτ ≥ 1 there exists γ(τ) > 0 such thatτ∑τ∑E[|z(t, 0, v)| 2 ] ≤ γ 2 (τ) E[|v(t)| 2 ] (8)t=0for all v ∈ l 2˜H{0, τ; R mv }.Since v → z(t, 0, v) is a linear dependence, we deduce that if the stateequilibrium of (6) is ESMS, we may define a linear operator T : l 2˜H{0, ∞; R mv }→ l 2˜H{0, ∞; R nz } by:for all v ∈ l 2˜H{0, ∞; R mv }.t=0t=0(T v)(t) = z(t, 0, v) (9)

146 Vasile Dragan, Toader MorozanIn the absence of the assumption of exponential stability for each τ ≥ 1,the equality (9) defines a linear operator T τ : l 2˜H{0, τ; R mv } → l 2˜H{0, τ; R nz }.From (7) and (8) one obtains that T and T τ are bounded operators.The linear operator T introduced by (9) will be called input-output operatordefined by the system (4) while, the system (4) is known as a state spacerepresentation of the operator T . From the definition of the input-outputoperator one sees that a such operator maps only finite-energy disturbancesignal v into the corresponding finite energy output signal z of the consideredsystem.To obtain an estimate of a robustness radius of the stabilization achievedby a control law, an important role is played by the norm of an input-outputoperator. It is well known, from the deterministic context, that the normof an input-output operator cannot be explicitly computed as in the case ofH 2 -norms. That is why, we are looking for necessary and sufficient conditionswhich guarantee the fact that the norm of an input-output operator is smallerthan a prescribed level γ > 0.Such conditions are provided by the well known Bounded Real Lemma.In the last part of this section we present several auxiliary results usefulin the developments of the next sections.Firstly, we remark that it is easy to prove the next inequality:for all τ ≥ 1.‖T τ ‖ ≤ ‖T ‖ (10)Let γ > 0, 0 < τ ∈ Z ∪ {∞} and x 0 ∈ R n be arbitrary but fixed. Weconsider the following cost functionalsi ∈ D andJ γ (τ, x 0 , i, v) =τ∑E[|z(t, x 0 , v)| 2 − γ 2 |v(t)| 2 |η 0 = i] (11)t=0˜J γ (τ, x 0 , v) =τ∑E[|z(t, x 0 , v)| 2 − γ 2 |v(t)| 2 ] (12)t=0for all v = {v(t)} 0≤t≤τ ∈ l 2˜H{0, τ; R mv }.It should be noted that if (11) and (12) are written for τ = +∞ weassume tacitly that the zero state equilibrium of the system (6) is ESMS. It

Robust stability of discrete-time linear stochastic systems 147is clear that ‖T τ ‖ ≤ γ if and only if ˜J γ (τ, 0, v) ≤ 0 for all v ∈ l 2˜H{0, τ; R mv }and ‖T ‖ ≤ γ if and only if ˜J γ (∞, 0, v) ≤ 0 for all v ∈ l 2˜H{0, ∞; R mv }.Throughout this paper SnN = S n ⊕ S n ⊕ .... ⊕ S n , S n being the Hilbertspace of n × n symmetric matrices. If X(t) = (X(t, 1), ..., X(t, N)) ∈ SnN weshall use the notations:( )Π1i X(t + 1) Π[ΠX(t + 1)](i) =2i X(t + 1)(Π 2i X(t + 1)) T = (13)Π 3i X(t + 1)r∑ (Ak (i) B k (i) ) T Ei (X(t+1)) ( A k (i) B k (i) )k=0∑with E i (X(t + 1)) = N p(i, j)X(t + 1, j), 1 ≤ i ≤ N.j=1Let F (t) = (F (t, 1), ..., F (t, N)), F (t, i) ∈ R mv×n , 0 ≤ t ≤ τ, τ ≥ 1.Let X γ F (t) = (Xγ F (t, 1), ..., Xγ F(t, N)) be the solution of the following problemwith the given final valueX(t, i) =r∑(A k (i) + B k (i)F (t, i)) T E i (X(t + 1))(A k (i) + (14)k=0X(τ + 1, i) = 0, 1 ≤ i ≤ N.+B k (i)F (t, i)) + (C(i) + D(i)F (t, i)) T (C(i) ++D(i)F (t, i)) − γ 2 F T (t, i)F (t, i)Let x F = {x F (t)} 0≤t≤τ+1 be the solution of the following problem withthe initial given value:r∑x(t + 1) = [A 0 (η t ) + B 0 (η t )F (t, η t ) + w k (t)(A k (η t ) + (15)x(0) = x 0 .k=1+B k (η t )F (t, η t ))]x(t) + [B 0 (η t ) +r∑+ w k (t)B k (η t )]v(t)k=1

148 Vasile Dragan, Toader MorozanApplying Lemma 3.2 in [9] we obtain:Lemma 2.1. Let F = {F (t)} 0≤t≤τ , F (t) = (F (t, 1), ..., F (t, N)), F (t, i) ∈R mv×n be a sequence of gain matrices. If {X γ F (t)} 0≤t≤τ+1 is the solution ofthe problem (14), then we have:J γ (τ, x 0 , i, v + F x F ) = x T 0 X γ F (0, i)x 0+τ∑E[v T (t)H γ (X γ F (t + 1), η t)v(t) + 2v T (t)N(X γ F (t + 1), η t)x F (t)|η 0 = i]t=0for all i ∈ D, x 0 ∈ R n , v ∈ l 2˜H{0, τ; R mv }, x F (t) being the solution of theproblem (15) corresponding to the input v andH γ (X γ F (t + 1), i) = Π 3iX γ F (t + 1) + DT (i)D(i) − γ 2 I mv (16)N(X γ F (t+1), i)=(Π 2iX γ F (t+1)+CT (i)D(i)) T +H γ (X γ F(t+1), i)F (t, i). (17)Proof may be done by direct calculations. It is omitted for shortness.Now we prove:Proposition 2.2. If for an integer τ ≥ 1 and a real number γ > 0,‖T τ ‖ < γ, thenr∑Bk T (i)E i(X γ F (t + 1))B k(i) + D T (i)D(i) − γ 2 I mv ≤ −ε 0 I mv (18)k=0for all 0 ≤ t ≤ τ, with ε 0 ∈ (0, γ 2 − ‖T τ ‖ 2 ).Proof. Let us remark that (18) can be rewrittenH γ (X γ F (t + 1), i) ≤ −ε 0I mv , 0 ≤ t ≤ τ. (19)We prove (19) in two steps. First we show thatfor all 0 ≤ t ≤ τ, i ∈ D.H γ (X γ F(t + 1), i) ≤ 0 (20)

150 Vasile Dragan, Toader MorozanWe deduce recursively thatThereforeX ˜γ F (t, i) ≥ Xγ F(t, i), 0 ≤ t ≤ τ, i ∈ D. (25)H˜γ (X γ F (t + 1), i) ≤ H˜γ(X ˜γ F(t + 1), i) ≤ 0.Having in mind the definition of ˜γ we obtain thatH γ (X γ F (t + 1), i) ≤ −ε 0I mv , 0 ≤ t ≤ τ, i ∈ Dwhich completes the proof.Let X γ (t) = (X γ (t, 1), ..., X γ (t, N)) be the solution of the problem (14) inthe special case F (t) = 0. One obtains recursively for t ∈ {τ +1, τ, ..., 0}, i ∈D that X γ (t, i) ≥ 0. Applying Proposition 2.2 for X γ (t) instead of X γ F (t)one obtains:Corollary 2.3 If there exists an integer τ ≥ 1 such that ‖T τ ‖ < γ, thenγ 2 I mv − D T (i)D(i) > 0, i ∈ D.3 Stochastic version of Bounded Real LemmaIn the developments of this section an important role is played by the followingbackward discrete-time stochastic generalized Riccati equations (DTS-GRE):X(t, i) =r∑A T k (i)E i(X(t + 1))A k (i) + C T (i)C(i) −k=0−(r∑A T k (i)E i(X(t + 1))B k (i) +k=0+C T (i)D(i))(r∑Bk T (i)E i(X(t + 1))B k (i) + (26)k=0+D T (i)D(i) − γ 2 I mv ) −1 × (+D T (i)C(i)), 1 ≤ i ≤ N.r∑Bk T (i)E i(X(t + 1))A k (i)k=0

Robust stability of discrete-time linear stochastic systems 151Using the notation introduced in (13) we may rewrite (26) in the followingcompact form:X(t)=Π 1 X(t+1)+M−(Π 2 X(t+1)+L)(Π 3 X(t+1)+R) −1 (Π 2 X(t+1)+L) T (27)whereM = (M(1), M(2), ..., M(N)) ∈ S N n ,L = (L(1), L(2), ..., L(N)) ∈ M N n,m v,M(i) = C T (i)C(i),L(i) = C T (i)D(i),R = (R(1), R(2), ..., R(N)) ∈ S N m v, R(i) = D T (i)D(i) − γ 2 I mv .For each integer τ ≥ 1, let X τ (t) = (X τ (t, 1), ..., X τ (t, N)) be the solutionof DTSGRE (26) with the final valueX τ (τ + 1, i) = 0, i ∈ D. (28)By using Proposition 2.2 we can prove by induction the next result:Lemma 3.1 If for an integer τ ≥ 1 and a real number γ > 0 we have‖T τ ‖ < γ, then the solution X τ (t) of the problem (26)-(28) is well definedfor all 0 ≤ t ≤ τ and it has the properties:X τ (t, i) ≥ 0 andr∑Bk T (i)E i(X τ (t + 1))B k (i) + D T (i)D(i) − γ 2 I mv ≤ −ε 0 I mv (29)k=0for all 0 ≤ t ≤ τ, i ∈ D, where ε 0 ∈ (0, γ 2 − ‖T τ ‖ 2 ).Lemma 3.2 Assume: a) the zero state equilibrium of the system (6) isESMS,b) the input-output operator T associated to the system (4) satisfies‖T ‖ < γ.Then there exists ρ > 0 such that ˜J γ (∞, x 0 , v) ≤ ρ|x 0 | 2 for all x 0 ∈ R n andv ∈ l 2˜H{0, ∞; R mv }.Proof. Under the assumption a) and Theorem 3.5 in [7] it follows thatthe linear equationZ(i) =r∑A T k (i)E i(Z)A k (i) + C T (i)C(i), 1 ≤ i ≤ N. (30)k=0

152 Vasile Dragan, Toader Morozanhas a unique solution Z = (Z(1), Z(2), ..., Z(N)) ∈ S N+n . We recall thatunder the assumption a), if v ∈ l 2˜H{0, ∞; R mv } then, from Lemma 4.3 in [9],we have limt→∞E[|x(t, x 0 , v)| 2 ] = 0.Applying Lemma 2.1 in the special case F (t, i) = 0, X(t, i) = Z(i) andtaking the limit for τ → ∞, one gets:˜J γ (∞, x 0 , v) =N∑∞∑π 0 (i)x T 0 Z(i)x 0 + E[v T (t)H γ (Z, η t )v(t) +i=1t=0+2x T (t, x 0 , v)N T (Z, η t )v(t)] (31)for all v ∈ l 2˜H{0, ∞; R mv } and all x 0 ∈ R n , where N(Z, i) and H γ (Z, i) areas in (16) and (17) with Z(i) instead of X γ F(t, i).Let ε be such that ‖T ‖ 2 < γ 2 − ε 2 . Thus we may write:˜J γ (∞, 0, v) = ‖T v‖ 2 l 2˜H{0,∞;R nz } − γ2 ‖v‖ 2 l 2˜H{0,∞;R mv } ≤ −ε2 ‖v‖ 2 l 2˜H{0,∞;R mv } .Therefore˜J γ (∞; x 0 , v) ≤orN∑∞∑π 0 (i)x T 0 Z(i)x 0 + E[2x T (t, x 0 , 0)N T (Z, η t )v(t)−ε 2 |v(t)| 2 ]i=1t=0˜J γ (∞; x 0 , v) ≤N∑λ max (Z(i))|x 0 | 2 + 1 ε 2i=1∞ ∑t=0E[|N(Z, η t )x(t, x 0 , 0)| 2 ]∞∑− E[|εv(t) − 1 ε N(Z, η t)x(t, x 0 , 0)| 2 ].t=0Let ν > 0 such that max|N(Z, i)| ≤ ν. Thus we have˜J γ (∞; x 0 , v) ≤N∑i=1λ max (Z(i))|x 0 | 2 + ν2ε 2∞ ∑t=0E[|x(t, x 0 , 0)| 2 ]. (32)From the assumption a) we deduce that there exists ρ 1 > 0 not dependingupon x 0 such that ∞ E[|x(t, x 0 , 0)| 2 ] ≤ ρ 1 |x 0 | 2 . Introducing the∑lastt=0

Robust stability of discrete-time linear stochastic systems 153inequality in (32) one obtains the inequality from the statement with ρ =N∑νλ max Z(i) + ρ 21 . Thus the proof is complete.ε 2i=1If X τ (t), 0 ≤ t ≤ τ + 1 is the solution of the problem with given finalvalue (26)-(28) we define K(t) = (K(t, 1), ..., K(t, N)) byK(t, i) = X τ (τ + 1 − t, i). (33)We see that K(0, i) = X τ (τ + 1, i) = 0, 1 ≤ i ≤ N. Also, by direct calculationone obtains that K = {K(t)} t≥0 solves the following forward nonlinearequation on S N n :K(t + 1, i) = Π 1i K(t) + C T (i)C(i) − (Π 2i K(t) + C T (i)D(i))(Π 3i K(t) (34)+D T (i)D(i) − γ 2 I mv ) −1 (Π 2i K(t) + C T (i)D(i)) T .Let us denote K 0 (t) = (K 0 (t, 1), ..., K 0 (t, N)) the solution of (34) with giveninitial value K 0 (0, i) = 0, 1 ≤ i ≤ N.Several properties of the solution K 0 (t) are summarized in the next result:Proposition 3.3 Assume: a) the zero state equilibrium of (6) is ESMS.b) ‖T ‖ < γ.Then the solution K 0 (t) of the forward equation (34) with the given initialvalue K 0 (0, i) = 0 is defined for all t ≥ 0. It has the properties:(i)r∑Bk T (i)E i(K 0 (t))B k (i) + D T (i)D(i) − γ 2 I mv ≤ −ε 0 I mv (35)k=0where ε 0 ∈ (0, γ 2 − ‖T ‖ 2 ).(ii) 0 ≤ K 0 (τ, i) ≤ K 0 (τ + 1, i) ≤ cI n , (∀) t, i ∈ Z + × D, where c > 0 isa constant not depending upon t, i.Proof. Based on (10) we obtain that ‖T τ ‖ ≤ ‖T ‖ < γ for all τ ≥ 1.Therefore, we deduce, via Lemma 3.1, that for any integer τ ≥ 1 the solutionX τ (t) of the problem with given final value (26), (28) is well defined for0 ≤ t ≤ τ + 1 and it verifies (29). Thus we deduce via (33) that K 0 (t) is welldefined for all t ≥ 0. If 0 < ε 0 < γ 2 − ‖T ‖ 2 it follows that ε 0 < γ 2 − ‖T τ ‖ 2for all τ ≥ 1.Hence in (29) we may choose ε 0 independent of τ. Writing (29) for t = 0and taking into account that K 0 (τ, i) = X τ (1, i) we obtain that (i) is fulfilled.Further, from (35) and (34) we deduce that K 0 (t, i) ≥ 0 for all (t, i) ∈ Z + ×D.

154 Vasile Dragan, Toader MorozanLet X τ (t) and X τ+1 (t) be the solutions of the DTSGRE (26) with thefinal value X τ (τ + 1) = 0 and X τ+1 (τ + 2) = 0 in S N n . Under the consideredassumptions we know that these two solutions are well defined for 0 ≤ t ≤τ + 1 and 0 ≤ t ≤ τ + 2, respectively.Let Z τ (t, i) = X τ+1 (t, i) − X τ (t, i), 0 ≤ t ≤ τ + 1, 1 ≤ i ≤ N.We can deduce recursively that Z τ (t, i) ≥ 0 for 0 ≤ t ≤ τ + 1.This means that X τ (t, i) ≤ X τ+1 (t, i), 0 ≤ t ≤ τ + 1, i ∈ D. ParticularlyX τ (1, i) ≤ X τ+1 (1, i), i ∈ D.Using (33) we see that the above inequality is equivalent to K 0 (τ, i) ≤K 0 (τ + 1, i), i ∈ D, τ ≥ 1. Further we consider v τ = {v τ (t)} 0≤t≤τ definedby v τ (t) = F τ (t, η t )x τ (t) where x τ (t) is the solution of (4) corresponding tov τ (t) and F τ is defined byF τ (t, i) = −(r∑Bk T (i)E i(X τ (t + 1))B k (i) + D T (i)D(i) − γ 2 I mv ) −1 (36)k=0×(r∑Bk T (i)E i(X τ (t + 1))A k (i) + D T (i)C(i)), i ∈ D.k=0Let v τ = {v τ (t)} t≥0 ∈ l 2 H {0, ∞; Rmv } be the natural extension of v τ takingv τ (t) = 0 for t ≥ τ + 1.Applying Lemma 3.2 from above and Lemma 3.2 in [9] we may writesuccessivelyπ 0 (i)x T 0 X τ (0, i)x 0 ≤E[x T 0 X τ (0, η 0 )x 0 ]= ˜J γ (τ, x 0 , v τ ) ≤ ˜J γ (∞; x 0 , v τ )≤ρ|x 0 | 2for all x 0 ∈ R n , i ∈ D. Henceπ 0 (i)x T 0 X τ (0, i)x 0 ≤ ρ|x 0 | 2 (37)for all x 0 ∈ R n , i ∈ D and for all initial distribution π 0 = (π 0 (1), ..., π 0 (N))with π 0 (i) > 0. Particulary, (37) is valid for the special case π 0 (i) = 1 N .This leads to x T 0 X τ (0, i)x 0 ≤Nρ|x 0 | 2 for all i ∈ D. Thus x T 0 K 0(τ+1, i)x 0 ≤c|x 0 | 2 (∀) τ ≥ 1, i ∈ D, x 0 ∈ R n where c = Nρ. Thus the proof is complete.

Robust stability of discrete-time linear stochastic systems 155Let us consider the following system of discrete-time coupled algebraicRiccati equations (DTSARE):r∑X(i) = A T k (i)E i(X)A k (i) + C T (i)C(i) −k=0−(×((r∑A T k (i)E i(X)B k (i) + C T (i)D(i))k=0r∑Bk T (i)E i(X)B k (i) + D T (i)D(i) − γ 2 I mv ) −1k=0r∑Bk T (i)E i(X)A k (i) + D T (i)C(i)). (38)k=0We have:Corollary 3.4 Under the assumptions of the Proposition 3.3 the DT-SARE (38) has a solution ˜X = ( ˜X(1), ..., ˜X(N)) ∈ S N +n with the additionalproperty:r∑Bk T (i)E i(X)B k (i) + D T (i)D(i) − γ 2 I mv < 0, 1 ≤ i ≤ N. (39)k=0Proof. From Proposition 3.3 one obtains that the sequences {K 0 (τ, i)} τ≥1 ,1 ≤ i ≤ N are convergent. Let ˜X(i) = lim K 0(τ, i). Taking the limit forτ→∞t → ∞ in (34) one obtains that ˜X = ( ˜X(1), ..., ˜X(N)) is a solution of DT-SARE (38). Finally, taking the limit for t → ∞ in (35) we deduce that (39)is fulfilled. The proof ends.We say that a solution X s = (X s (1), ..., X s (N)) of the DTSARE (38) isa stabilizing solution if the zero state equilibrium of the closed-loop systemx s (t + 1) = [A 0 (η t ) + B 0 (η t )F s (η t ) +is ESMS, where1 ≤ i ≤ N.F s (i) = −(r ∑k=1w k (t)(A k (η t ) + B k (η t )F s (η t ))]x s (t)r∑Bk T (i)E i(X s )B k (i) + D T (i)D(i) − γ 2 I mv ) −1k=0×(r∑Bk T (i)E i(X s )A k (i) + D T (i)C(i)) (40)k=0

156 Vasile Dragan, Toader MorozanBefore to prove the main result of this section we recall several definitionsand results from [10].Consider the discrete-time general Riccati equationX = Π 1 X + M − (L + Π 2 X)(R + Π 3 X) −1 (L + Π 2 X) T (41)whereX → ΠX =(Π1 X Π 2 X(Π 2 X) T Π 3 X)(42)is a linear and positive operator defined on Sn N taking values in Sn+m N vand( ) M LQ =L T ∈ Sn+m RN v. To the pair Σ = (Π, Q) (which defines theequation (41)) we associate the so called dissipation operator D Σ : SnN →Sn+m N vby: D Σ X = (D Σ 1 X, ..., DΣ NX) where( )D Σ Π1i X − X(i) + M(i) L(i) + Πi X =2i X(L(i) + Π 2i X) T(43)R(i) + Π 3i Xfor all X = (X(1), ..., X(N)) ∈ Sn N .If Π : Sn N → Sn+m N vis a linear operator and F = (F (1), F (2), ..., F (N)),F (i)∈R mv×n then we denote Π F X =((Π F X)(1), (Π F X)(2), ..., (Π F X)(N))with(Π F X)(i) = ( I n F T (i) ) ( Π 1i X Π 2i X(Π 2i X) T Π 3i X) (InF (i)). (44)Definition 3.1 We say that a linear and positive operator Π : SnN →Sn+m N vis stabilizable if there exists F = (F (1), F (2), ..., F (N)), F (i) ∈ R mv×nwith the property that the eigenvalues of the operator Π F are located in theinside of the disk |λ| < 1.It should be remarked that in the special case of Π introduced by (13)the concept of stabilizability introduced in Definition 3.1 is equivalent to theconcept of stochastic stabilizability introduced in [8].Definition 3.2 A solution X s = (X s (1), ..., X s (N)) of (41) is a stabilizingsolution if the eigenvalues of the operator Π Fs are in the insideof the disk |λ| < 1, where Π Fs is defined as in (44) with F replaced byF s = (F s (1), ..., F s (N)),F s (i) = −(R(i) + Π 3i X s ) −1 (L(i) + Π 2i X s ) T . (45)

Robust stability of discrete-time linear stochastic systems 157The next result provides a set of necessary and sufficient conditions forthe existence of a stabilizing solution of (41).Theorem 3.5 ([10]) With the considered notations, the following areequivalent:(i) the linear and positive operator Π is stabilizable and there exists ˆX ∈S N n ,ˆX = ( ˆX(1), ˆX(2), ..., ˆX(N)) such thatD Σ iˆX > 0, (∀) i ∈ {1, 2, ..., N}; (46)(ii) the algebraic Riccati equation (41) has a stabilizing solution X s = (X s (1),X s (2), ..., X s (N)) which satisfiesR(i) + Π 3i X s > 0, 1 ≤ i ≤ N. (47)The main result of this section is:Theorem 3.6 (Bounded Real Lemma) Under the considered assumptions,for a given scalar γ > 0, the following are equivalent:(i) the zero state equilibrium of (6) is ESMS and the input-output operatorT defined by the system (4) satisfies ‖T ‖ < γ.(ii) there exists X = (X(1), ..., X(N)) ∈ Sn N , X(i) > 0, 1 ≤ i ≤ N,which solves the following system of LMI’s:(Π1i X − X(i) + C T (i)C(i) Π 2i X + C T )(i)D(i)(Π 2i X + C T (i)D(i)) T Π 3i X + D T (i)D(i) − γ 2 < 0,I mv1 ≤ i ≤ N, (48)where the operators Π li are introduced by (13);(iii) the DTSARE (38) has a stabilizing solution ˜X = ( ˜X(1), ..., ˜X(N)) ∈Sn N with ˜X(i) ≥ 0, 1 ≤ i ≤ N which satisfies (39);(iv) there exists Y = (Y (1), Y (2), ..., Y (N)) ∈ Sn N , Y (i) > 0, 1 ≤ i ≤ N,which solves the following system of LMIs⎛−Y (i) Ψ 0i (Y ) Ψ 1i (Y ) ... Ψ ri (Y ) Y (i)C T ⎞(i)Ψ0i T (Y ) G 00(i)−Y G 01 (i) ... G 0r (i) G 0r+1 (i)Ψ1i T (Y ) GT 01 (i) G 11(i)−Y ... G 1r (i) G 1r+1 (i)⎜ ... ... ... ... ... ...< 0(49)⎝ Ψri T (Y ) GT 0r (i) GT 1r (i) ... G ⎟rr(i) − Y G rr+1 (i) ⎠C(i)Y (i) G T 0r+1 (i) GT 1r+1 (i) ... GT rr+1 (i) D(i)DT (i)−γ 2 I nz

158 Vasile Dragan, Toader MorozanwhereΨ ki (Y ) = ( √ p(i, 1)Y (i)A T k (i) √p(i, 2)Y (i)ATk(i) ... √ p(i, N)Y (i)A T k (i)) ,Y = diag(Y (1), ..., Y (N)) ∈ S nNG lk (i) = I T (i)B l (i)Bk T (i)I(i), 0 ≤ l ≤ k ≤ r,G lr+1 (i) = I T (i)B l (i)D T (i)andI(i) = ( √ p(i, 1)I n√p(i, 2)In ...√p(i, N)In).Proof. Let us assume that (i) holds. If δ >0 denote T δ :l 2˜H{0, ∞; R mv } →l 2˜H{0, ∞; R n+nz } the linear operator defined by v→(T δ v)(t)=C δ (η t )x(t, 0, v)+D δ (η t )v(t) where x(t, 0, v) is the zero(initial)value solution(of (4))correspondingto the input v and C δ (i) = , D C(i)D(i)δI δ (i) = . Based onn 0(7) we deduce that for δ > 0 sufficiently small we have ‖T δ ‖ < γ. ApplyingCorollary 3.4 we deduce that there exists X δ = (X δ (1), ..., X δ (N)), X δ (i) ≥ 0solving the DTSARE:X δ (i) = Π 1i X δ −(Π 2i X δ + C T (i)D(i))(Π 3i X δ + D T (i)D(i)−γ 2 I mv ) −1 (50)with additional property(Π 2i X δ + C T (i)D(i)) T + C T (i)C(i) + δ 2 I n , 1 ≤ 1 ≤ N,Π 3i X δ + D T (i)D(i) − γ 2 I mv < 0, 1 ≤ 1 ≤ N. (51)Since the right hand side of (50) is positive definite it follows that X δ (i) >0, 1 ≤ i ≤ N.Also (50) impliesΠ 1i X δ − X δ (i) + C T (i)C(i) − (Π 2i X δ ++C T (i)D(i))(Π 3i X δ + D T (i)D(i) − γ 2 I mv ) −1 (52)(Π 2i X δ + C T (i)D(i)) T < 0, 1 ≤ 1 ≤ N.By a Schur complement technique one obtains that (51) and (52) areequivalent to (48) and thus the proof of the implication (i) → (ii) is complete.

Robust stability of discrete-time linear stochastic systems 159To prove the converse implication, (ii) → (i) we remark that if (ii) is fulfilledthen the (1;1) block of (48) is negative definite. Thus we obtained thatthere exists X = (X(1), ..., X(N)) ∈ Sn N with X(i) > 0, such that X(i) >r∑A T k (i)E i(X)A k (i), 1 ≤ i ≤ N. Applying Corollary 4.8 in [8] we deducek=0that the zero state equilibrium of the system (6) is ESMS. Further, applyingCorollary 3.3 in[9] for X(t, i) = X(i), 0 ≤ t ≤ τ, τ ≥ 1, 1 ≤ i ≤ N, andtaking the limit for τ → ∞ we have:˜J γ (∞; 0, v) =∞∑( x(t, 0, v)E[v(t)t=0) T ( x(t, 0, v)Q(X, η t )v(t))] (53)where Q(X, i) is the left hand side of (48). If X = (X(1), ..., X(N)) verifies(48) then for ε > 0 small enough we haveQ(X, i) ≤ −ε 2 I n+mv , 1 ≤ 1 ≤ N. (54)Combining (53) and (54) we deduceor equivalently∑˜J ∞ γ (∞; 0, v) ≤ −ε 2 E[|x(t, 0, v)| 2 ] + E[|v(t)| 2 ]t=0∑∞˜J˜γ (∞; 0, v) ≤ −ε 2 E[|x(t, 0, v)| 2 ] < 0t=0for all v ∈ l 2˜H{0, ∞; R mv } where ˜γ = (γ 2 − ε 2 ) 1 2 .The last inequality may be written:‖T v‖ 2 l 2˜H{0,∞;R nz } ≤ ˜γ2 ‖v‖ 2 l 2˜H{0,∞;R mv }for all v ∈ l 2˜H{0, ∞; R mv }. This leads to ‖T ‖ 2 ≤ γ 2 − ε 2 and thus theimplication (ii) → (i) is proved.To prove the equivalence (ii) ↔ (iii) let us consider the DTSGRE:whereX = Π 1 X + ˆM − (Π 2 X + ˆL)(Π 3 X + ˆR) −1 (Π 2 X + ˆL) T (55)ˆM(i) = −C T (i)C(i), ˆL(i) = −C T (i)D(i), ˆR(i) = γ 2 I mv − D T (i)D(i),

160 Vasile Dragan, Toader Morozan1 ≤ i ≤ N. One can sees that (55) is a nonlinear equation of type (41) defined( ) ˆMby the pair Σ = (Π, ˆQ) with ˆQˆL=ˆL T ∈ Sn+m ˆRN v. One can checkthat if X = (X(1), X(2), ..., X(N)) solves (48) then ˆX = ( ˆX(1), ..., ˆX(N))with ˆX(i) = −X(i), i ∈ D solves the corresponding LMIs (46).Also if (ii) is fulfilled then from (1,1) block of (48) one obtains thatΠ 1i X − X(i) < 0, 1 ≤ i ≤ N. Using the implication (vii) → (i) of Theorem3.4 in [7] in the special case of the positive operator Π 1 we deduce that theeigenvalues of this operator are located in the inside(of the disc |λ| < 1.)Π1 X ΠThis means that the operator Π defined by ΠX =2 X(Π 2 X) T Π 3 Xis stabilizable (in the sense of Definition 3.1 from above). Thus we obtainthat if (ii) is fulfilled then in the case of DTSGRE (55) the assertion (i)in Theorem 3.5 is fulfilled. Hence, (55) has a stabilizing solution X s =(X s (1), X s (2), ..., X s (N)) which satisfiesΠ 3i X s − R(i) > 0, 1 ≤ i ≤ N.x (56)A simple computation shows that ˜X = ( ˜X(1), ..., ˜X(N)) defined by ˜X(i) =−X s (i) is the stabilizing solution of DTSARE (38) which satisfies (39). Sincethe eigenvalues of the positive operator Π 1 are located in the inside of thedisk |λ| < 1 from Theorem 3.5 in [7] it follows that ˜X(i) ≥ 0, i ∈ D and then(ii) → (iii) is true.Conversely, let ˜X = ( ˜X(1), ..., ˜X(N)) be the stabilizing solution of theDTSARE (38) which satisfies (39). If X s (i) = − ˜X(i), 1 ≤ i ≤ N, thenX s = (X s (1), ..., X s (N)) is the stabilizing solution of (55) which satisfies thecondition (56). Applying Theorem 3.5 in the case of (55) one deduces thatthere exists ˆX = ( ˆX(1), ..., ˆX(N)) which solves( )Π1i ˆX − ˆX(i) + ˆM(i) Π2i ˆX + ˆL(i)(Π 2i ˆX + ˆL(i))T> 0, 1 ≤ i ≤ N. (57)Π 3i ˆX + ˆR(i)On the other hand from Proposition 5.1 in [10] we deduce that X s coincideswith the maximal solution of (55). Therefore, ˆX(i) ≤ Xs (i) = − ˜X(i) ≤0, 1 ≤ i ≤ N.Let ∆ i ( ˆX) be defined by ∆ i ( ˆX) = Π 1i ˆX − ˆX(i) + ˆM(i). Since ∆i ( ˆX)is the (1,1) block of the matrix from the left hand side of (57) we have∆ i ( ˆX) > 0, 1 ≤ i ≤ N. Writing ˆX(i) = Π 1i ˆX + ˆM(i) − ∆i ( ˆX) and taking

Robust stability of discrete-time linear stochastic systems 161into account that ˆM(i) ≤ 0, we conclude that ˆX(i) < 0, 1 ≤ i ≤ N. TakingX(i) = − ˆX(i) one sees that X = (X(1), X(2), ..., X(N)) solves (48) andX(i) > 0, 1 ≤ i ≤ N.This completes the proof of the implication (iii) → (ii).The equivalence (ii) ↔ (iv) follows immediately by a Schur complementtechnique. This completes the proof of the theorem.If the system (4) is either in the case N = 1 or N ≥ 2, with A k (i) = 0,B k (i) = 0, 1 ≤ k ≤ r, i ∈ D, the result proved in Theorem 3.6 recover as specialcases the stochastic version of the Bounded Real Lemma for discrete-timelinear stochastic systems perturbed by independent random perturbationsand the discrete-time linear stochastic systems with Markovian switching,respectively.Let us remark that if the zero state equilibrium of (6) is ESMS fromTheorem 3.6, it follows that‖T ‖ = inf{γ > 0, for which it exists X ∈ S N n , X > 0,such that (48) holds} = inf{γ > 0,DTSARE (38) has a positive semidefinite solution verifying (39)}4 The small gain theorem and robust stabilityOne of the important consequence of the Bounded Real Lemma is the socalled Small Gain Theorem. It is known that this result is a powerful toolin the derivation of some estimates of the stability radius with respect toseveral classes of parametric uncertainties. We start with an auxiliary resultwhich is interesting in itself:Theorem 4.1 Regarding the system (4) we assume that the followingassumptions are fulfilled:a) the number of inputs equals the number of outputs (i.e. m v = n z = m);b) the zero state equilibrium of the corresponding linear system (6) isESMS;c) the input-output operator T associated to the system (4) satisfies‖T ‖

162 Vasile Dragan, Toader Morozan(ii) the zero state equilibrium of the systemx(t + 1) = (A(η t ) +r∑w k (t)A k (η t ))x(t) (58)k=1is ESMS, where either A k (i) = A k (i) − B k (i)(I m + D(i)) −1 C(i) or A k (i) =A k (i) + B k (i)(I m − D(i)) −1 C(i).Proof. Based on (10) and assumption c) we deduce that ‖T τ ‖ < 1for any integer τ ≥ 1. Thus applying Corollary 2.3 one obtains that I m −D T (i)D(i) > 0, i ∈ {1, 2..., N}. Therefore for each i the eigenvalues of thematrix D(i) are located in the inside of the disk |λ| < 1. Hence det(I m ±D(i)) ≠ 0, 1 ≤ i ≤ N. Thus we obtain that (i) is true. To prove (ii) we usethe implication (i) → (ii) of Theorem 3.6. Thus if the assumptions b) andc) are fulfilled, then there exist X = (X(1), ..., X(N)) ∈ S N n , X(i) > 0 suchthat (48) hold with γ = 1.TakingF (i) = ±(I m ∓ D(i)) −1 C(i) (59)by direct calculation one obtains via (13) and (48) thatr∑[A k (i) + B k (i)F (i)] T E i (X)[A k (i) + B k (i)F (i)] − (60)k=0X(i) + (C(i) + D(i)F (i)) T (C(i) + D(i)F (i)) − F T (i)F (i) < 01 ≤ i ≤ N.If we take into account (59) we obtain C(i)+D(i)F (i)=(I m ∓D(i)) −1 C(i).Thus we have (C(i) + D(i)F (i)) T (C(i) + D(i)F (i)) − F T (i)F (i) = 0. Hence(60) becomes:r∑A T k (i)E i (X)A k (i) − X(i) < 0, X(i) > 0, 1 ≤ i ≤ N. (61)k=0Applying Corollary 4.8 in [8] one deduces that the zero state equilibrium ofthe system (58) is ESMS. This completes the proof.

Robust stability of discrete-time linear stochastic systems 163Consider the systemx(t + 1) = [A 0 (η t ) ++r∑w k (t)A k (η t )]x(t) + [B 0 (η t ) +k=1r∑w k (t)B k (η t )]u(t) (62)k=1z(t) = C(η t )x(t)with the input u(t) ∈ R m and the output z(t) ∈ R p .Let ˆD = ( ˆD(1), ..., ˆD(N)), ˆD(i)∈R m×p . By definition | ˆD|=max{| ˆD(i)|,1 ≤ i ≤ N}.Theorem 4.2 ( The small gain theorem). Assume:a) The zero state equilibrium of the system (6) is ESMS.b) ‖ ˜T ‖ < γ where ˜T : l 2˜H{0, ∞; R m } → l 2˜H{0, ∞; R p } is the input-outputoperator defined by the system (62).c) | ˆD| < γ −1 .Under these conditions the zero state equilibrium of the systemx(t + 1) = [A 0 (η t ) + B 0 (η t ) ˆD(η t )C(η t ) +r∑w k (t)(A k (η t ) + B k (η t ) ˆD(η t )C(η t ))]x(t) (63)k=1is ESMS.Proof.Definethelinearbounded operator ˆT :l 2˜H{0, ∞,R p }→l 2˜H{0, ∞,R m }by( ˆT v)(t) = D(η t )v(t)with v(t) ∈ l 2˜H{0, ∞, R p }. Reasoning as in the proof of Proposition 13 in [6]page 234 one can prove that ‖ ˆT ‖ = | ˆD|.Let us consider the system:ˆx(t + 1) = [A 0 (η t ) ++r∑w k (t)A k (η t )]ˆx(t) + [B 0 (η t ) +k=1r∑w k (t)B k (η t )] ˆD(η t )v(t) (64)k=1z(t) = C(η t )ˆx(t).

164 Vasile Dragan, Toader MorozanWe observe that ˜T ˆT is the input output operator associated with the system(64). Since ‖ ˜T ˆT ‖ < 1, the conclusion follows via Theorem 4.1. Thus theproof is complete.In this section the problem of the robust stability is investigated for aclass of discrete-time linear stochastic systems subject to linear parametricuncertainties.Let us consider the discrete-time linear stochastic system described by:+x(t + 1) = [A 0 (η t ) + B 0 (η t )∆(η t )C(η t ) +r∑w k (t)(A k (η t ) + B k (η t )∆(η t )C(η t ))]x(t) (65)k=1where A k (i) ∈ R n×n , B k (i) ∈ R n×m , 0 ≤ k ≤ r, C(i) ∈ R p×n are assumedto be known matrices, ∆(i) ∈ R m×p are unknown matrices. The system (65)is a perturbed model of the nominal system (6).The matrices B k (i), C(i) occurring in (65) determine the structure of theparametric uncertainties presented in the perturbed model.If the zero state equilibrium of the nominal system (6) is ESMS we willanalyze if the zero state equilibrium of the perturbed model (65) remainsESMS for some ∆(i) ≠ 0. This would be, in few words the formulation ofthe problem of the robust stability. For a more precise formulation of therobust stability problem we introduced a norm in the set of the uncertainties.If ∆ = (∆(1), ∆(2), ..., ∆(N)) ∈ M N m,p i.e. ∆(i) are m × p real matrices,we set|∆| = max |∆(i)| = max (λ max(∆ T (i)∆(i))) 1 2 . (66)i∈D i∈DAs a measure of the robustness of the stability we introduce the conceptof stability radius.Definition 4.1 The stability radius of the nominal system (6), or equivalently,the stability radius of the pair (A, P ) with respect to the structuredparametric uncertainties with the structure determined by the pair (B, C)is the number ρ L [A, P |B, C] = inf{ρ > 0|(∃)∆ = (∆(1), ..., ∆(N)) ∈ M N m,pwith |∆| ≤ ρ that the zero state equilibrium of the corresponding system(4.8) is not ESMS}.The next result provides a lower bound of the stability radius introducedin the above definition. To this end, let us consider the fictitious

Robust stability of discrete-time linear stochastic systems 165system constructed based on the known matrices occurring in the perturbedmodel (65):x(t + 1) = (A 0 (η t ) ++r∑w k (t)A k (η t ))x(t) + (B 0 (η t ) +k=1r∑w k (t)B k (η t ))v(t); z(t) = C(η t )x(t) (67)k=1Theorem 4.3 Assume that the zero state equilibrium of the nominal system(6) is ESMS. Let T : l 2˜H{0, ∞; R m } → l 2˜H{0, ∞; R p } be the input outputoperator defined by the fictitious system (67). Then we have:ρ L [A, P |B, C] ≥ ‖T ‖ −1 (68)Proof. Let ρ < ‖T ‖ −1 be arbitrary but fixed. We show that for any perturbation∆ = (∆(1), ∆(2), ..., ∆(N)) ∈ M N m,p with |∆| < ρ, the zero stateequilibrium of the perturbed system (65) is ESMS. Let ∆ ∈ M N m,p be a perturbationwith |∆| < ρ. Setting γ = ρ −1 , we have ‖T ‖ < γ and |∆| < γ −1 .Hence the fictitious system (67) and the perturbation ∆ are in the conditionsof Theorem 4.2. Thus the prooof is complete.5 The disturbance attenuation problem and the robuststabilizationConsider the control system:If we takex(t + 1) = A 0 (η t )x(t) + G 0 (η t )v(t) + B 0 (η t )u(t) +r∑w k (t)[A k (η t )x(t) + G k (η t )v(t) + B k (η t )u(t)]k=1z(t) = C z (η t )x(t) + D zv (η t )v(t) + D zu (η t )u(t).y(t) = x(t) (69)u(t) = F (η t )x(t). (70)

166 Vasile Dragan, Toader Morozanthe closed-loop system obtained when coupling (70) and (69) is:r∑x(t + 1) = [A 0 (η t ) + B 0 (η t )F (η t )+ w k (t)(A k (η t ) + B k (η t )F (η t ))]x(t) +(G 0 (η t ) +k=1r∑w k (t)G k (η t ))v(t) (71)k=1z(t) = (C z (η t ) + D zu (η t )F (η t ))x(t) + D zv (η t )v(t).If F = (F (1), F (2), ..., F (N)) is a stabilizing feedback gain, that is (71)with v(t) = 0 is ESMS, then the system (71) defines an input output operator,T F : l 2˜H{0, ∞; R mv } → l 2˜H{0, ∞; R nz } by (T F v)(t) = (C z (η t ) +D zu (η t )F (η t ))x(t, 0, v) + D zv (η t )v(t), t ∈ Z + .The disturbance attenuation problem with level of attenuation γ > 0 asksfor constructing a stabilizing feedback gain F , such that ‖T F ‖ < γ.Remark 5.1 The disturbance attenuation problem (DAP) stated beforeextends to this general framework the H ∞ control problem from the deterministiccontext. Therefore, this problem will be often named stochasticH ∞ -problem.The solution of the above problem is given in the next result:Theorem 5.1 For the system (69) and a given scalar γ > 0, the followingare equivalent:(i) there exists a control law u(t) = F (η t )x(t) such that the zero state equilibriumof the linear system x(t+1) = [A 0 (η t )+B 0 (η t )F (η t )+ r w k (t)(A k (η t∑)+B k (η t )F (η t ))]x(t) is ESMS and ‖T F ‖ < γ.(ii) there exist Y = (Y (1), Y (2), ..., Y (N)) ∈ S N n and Γ = (Γ (1), Γ (2), ...,Γ (N)) ∈ M N mn, Y (i) > 0, 1 ≤ i ≤ N, which solve the following system ofLMIs:⎛−Y (i) W 0i (Y, Γ ) W 1i (Y, Γ ) ... W ri (Y, Γ ) Y(i)C z T (i)+Γ T (i)D T ⎞zu(i)W0i T (Y, Γ ) G 00−Y G 01 (i) ... G 0r (i) G 0r+1 (i)W1i T (Y, Γ ) GT 01 (i) G 11(i)−Y ... G 1r (i) G 1r+1 (i)⎜ ... ... ... ... ... ...< 0(72)⎝ Wri T (Y, Γ ) GT 0r (i) GT 1r (i) ... G ⎟rr(i)−Y G rr+1 (i) ⎠C z (i)Y(i)+D zu (i)Γ(i) G T 0r+1 (i) GT 1r+1 (i) ... GT rr+1 (i) DT zv(i)D zv(i)−γ T 2 I nzk=1

Robust stability of discrete-time linear stochastic systems 167where W ki (Y, Γ ) = (Y (i)A T k (i) + Γ T (i)Bk T (i))I(i), 0 ≤ k ≤ r,I(i) = ( √ √ √ )p(i, 1)I n p(i, 2)In ... p(i, N)InG lk (i) = I T (i)G l (i)G T k (i)I(i), 0 ≤ l ≤ k ≤ r, (73)G lr+1 (i) = I T (i)G l (i)Dzv(i), T 0 ≤ l ≤ rY = diag(Y (1), Y (2), ..., Y (N)).Moreover, if (Y, Γ ) is a solution of the above LMI (72), then a solution ofthe disturbance attenuation problem under consideration is given by F =(F (1), F (2), ..., F (N)), F (i) = Γ (i)Y −1 (i), 1 ≤ i ≤ N.Proof. It follows immediately via the equivalence (i) ↔ (iv) in Theorem3.6 specialized in the case of the system (71) and taking Γ (i) = F (i)Y (i).We shall apply Theorem 5.1 in order to solve a robust stabilization problem.Consider the system described by:+x(t + 1) = [A 0 (η t ) + Ĝ0(η t )∆ 1 (η t )Ĉ(η t)]x(t) ++[B 0 (η t ) + ˆB 0 (η t )∆ 2 (η t ) ˆD(η t )]u(t)r∑w k (t){[A k (η t ) + Ĝk(η t )∆ 1 (η t )Ĉ(η t)]x(t) +k=1+[B k (η t )∆ 2 (η t ) ˆD(η t )]u(t)} (74)where A k (η t ), Ĝk(i), B k (i), Ĉ(i), ˆD(i), 0 ≤ k ≤ r, i ∈ D are known matricesof appropriate dimensions and ∆ 1 = (∆ 1 (1), ..., ∆ 1 (N)) and ∆ 2 =(∆ 2 (1), ..., ∆ 2 (N)) are unknown matrices and they describe the magnitudeof the uncertainties of the system (74). It is assumed that the whole statevector is accessible for measurements.The robust stabilization problem considered here can be stated as follows:For a given ρ > 0 find a control u(t) = F (η t )x(t) stabilizing (74) for any∆ 1 and ∆ 2 such that max(|∆ 1 |, |∆ 2 |) < ρ.The closed-loop system obtained with u(t) = F (η t )x(t) is given byx(t+1) = {A 0 (η t )+B 0 (η t )F (η t )+G 0 (η t )∆(η t )[C(η t )+D(η t )F (η t )]}x(t)r∑+ w k (t){A k (η t )+B k (η t )F (η t )+G k (η t )∆(η t )[C(η t )+D(η t )F (η t )]}x(t) (75)k=1

168 Vasile Dragan, Toader Morozan)where G k (i) =(Ĝk (i) ˆBk (i) , C(i) =( )∆1 (i) 0.(Ĉ(i)0) ( 0, D(i) =ˆD(i)), ∆(i) =0 ∆ 2 (i)If the zero state equilibrium of the linear system obtained from (75)taking∆ = 0 is ESMS, then from Theorem 4.3 it follows that the zero state equilibriumof (75) is ESMS for all ∆ with |∆| < ρ, |∆| = max(|∆ 1 |, |∆ 2 |),if the input-output operator T F associated to the system (71) with z(t) =[C(η t ) + D(η t )F (η t )]x(t) satisfies the condition ‖T F ‖ < 1 ρ .Therefore, F is a robust stabilizing feedback with the robustness radiusρ if it is a solution of the DAP with level of attenuation γ = 1 ρfor the system(69) with z(t) = C(η t )x(t) + D(η t )u(t) where the matrices C(i) and D(i)were defined above.The next result follows directly from Theorem 5.1:Theorem 5.2 Suppose that there exist Y ∈ Sn N and Γ ∈ M N mn , Y >0 verifying the system of LMIs (72) where C z (i) = C(i), D zu (i) = D(i),D zv (i) = 0, γ = 1 ρ . Then the state feedback gain F (i) = Γ (i)Y −1 (i) is asolution of the robust stabilization problem.References[1] A. El Bouhtouri, D. Hinrichsen, A.J. Pritchard, H ∞ type control fordiscrete-time stochastic systems, Int. J. Robust Nonlinear Control, 9,(1999), 923-948.[2] O.L.V. Costa, R.P. Marques, Mixed H 2 /H ∞ control of discrete-timeMarkovian jump linear systems, IEEE Trans. Autmatic Control, AC-43,1, (1998), 95-100.[3] O.L.V. Costa, M.D.Fragoso, R.P.Marques, Discrete-time Markov jumplinear systems, Ed. Springer, 2005.[4] J.C.Doyle, K. Glover, P.P.Khargonekaar, B.A.Francis, State space solutionsto standard H 2 and H ∞ control pproblems, IEEE Trans. AutomaticControl, 34, 8, (1989), 831-847.[5] V.Dragan, A.Stoica, A robust stabilization problem for discrete timetime- varying stochastic systems with multiplicative noise, Math. Reports,2(52), 3, (2000), 275-293.

Robust stability of discrete-time linear stochastic systems 169[6] V. Dragan, T. Morozan, A.M. Stoica, Mathematical methods in robustcontrol of linear stochastic systems, Mathematical Concepts and Methodsin Science and Engineering, series Editor Angelo Miele, Volume 50,Springer, (2006).[7] V. Dragan, T. Morozan, Exponential stability for discrete time linearequations defined by positive operators. Integral Equ. Oper. Theory, 54,(2006), 465-493.[8] V. Dragan, T.Morozan, Mean square exponential stability for somestochastic linear discrete time systems, European Journal of Control,12, (2006), 373-395.[9] V. Dragan, T. Morozan, The linear quadratic optimization problem fora class of discrete-time stochastic linear systems, Int. J. of InnovativeComputing, Inform. and Control, 4, 9, (2008), 2127-2137.[10] V. Dragan, T. Morozan, A class of discrete-time generalized Riccatiequations, Journal of Difference equations and Applications, 16, 4(2010), 291-320[11] E.Ghershon, U.Shaked, I.Yaesh, H ∞ control and filtering of discretetimestochastic systems with multiplicative noise, Automatica, 37,(2001), 409-417.[12] E. Ghershon, U. Shaked, I. Yaesh, H ∞ Control and Estimation of State-Multiplicative Linear Systems, Lecture Notes in Control and InformationSciences, 318,(2005), Springer-Verlag London Limited.[13] M. Green, D.J.N.Limebeer, Linear robust control, Prentice Hall, New-York, 1995.[14] L. Hu, P. Shi, P. Frank, Robust sampled data control for Markovianjump linear systems, Automatica, 42, (2006), 2025-2030.[15] T. Morozan, Parametrized Riccati equations and input output operatorsfor discrete-time systems with state dependent noise, Stochastic Anal.and Appl., 16,5, (1998), 915-931.[16] T. Morozan, Parametrized Riccati equations for linear discrete-time systemswith Markov perturbations, Rev. Roum. Math. Pures et Appl.,43,7-8, (1998), 761-777.

170 Vasile Dragan, Toader Morozan[17] C.E. de Souza, L.Xie, On the discrete-time bounded real lemma with applicationin the characterization of static state feedback H ∞ controllers,System and Control Letters, 18, (1992), 61-71.[18] E. Yaz, X. Niu, New results on the robustness of discrete-time systemswith stochastic perturbations, Proceedings of American Control conference,(1991), Boston MA, 2698-2699.[19] X. Yu, C.S. Hsu, Reduced order H ∞ filter design for discrete-time invariantsystems, International Journal of Robust and Nonlinear Control,7, (1997), 797-809.[20] L. Zhang, P. Shi, E. K. Boukas, C. Wang, H ∞ Model Reduction forUncertain Switched Linear Discrete-Time Systems, Automatica, 44, 11,(2008), 2944-2949.[21] L. Zhang, P. Shi, E. K. Boukas, H ∞ model reduction for discrete-timeMarkov jump linear systems with partially known transition, Int. J. ofControl, 82,2, (2009), 243-351.

Annals of the Academy of Romanian ScientistsSeries on Mathematics and its ApplicationsISSN 2066 - 6594 Volume 2, Number 2 / 2010ANALYSIS, OPTIMAL CONTROL,AND SIMULATION OFCONDUCTIVE-RADIATIVEHEAT TRANSFER ∗Peter Philip †AbstractThis article surveys recent results regarding the existence of weaksolutions to quasilinear partial dierential equations (PDE) couplednonlocally by the integral operator of the radiosity equation, modelingconductive-radiative heat transfer. Both the stationary and the transientcase are considered. For the stationary case, an optimal controlproblem with control constraints is presented with rst-order necessaryoptimality conditions, where recent results on the solution theoryof the linearized state equation allow to close a previous gap. A -nite volume scheme for the discretization of the stationary system isdescribed and, based on this scheme, a numerical computation of thetemperature eld (solution of the state equation) is shown as well asthe numerical solution to a realistic control problem in the context ofindustrial applications in crystal growth.MSC: 49K20, 35J60, 35D30, 35K05, 35K55, 45P05, 49J20, 65C20,80M15, 80M50.∗Accepted for publication in revised form on June 13, 2010.†philip@math.lmu.de Department of Mathematics, Ludwig-Maximilians University(LMU), Theresienstrasse 39, 80333 Munich, Germany.171

172 Peter Philipkeywords: nonlinear elliptic equation, nonlinear parabolic equation,heat equation, nonlocal boundary condition, diuse-gray radiation, radiosityequation, weak solution, optimal control, nite volume method, numericalsimulation1 IntroductionModeling and numerical simulation of conductive-radiative heat transfer hasbecome a standard tool to support and improve numerous industrial processessuch as crystal growth by the Czochralski method [1] and by the physicalvapor transport method [2] to mention just two examples. Moreover, inthe context of industrial applications, one is often not only interested in computingstationary or transient temperature elds as they arise from models ofconductive-radiative heat transfer, but one also needs to optimize and controlsuch temperature elds according to objective functionals arising fromthe application [3, 4].Models including diuse-gray radiative interactions between cavity surfacesconsist of nonlinear elliptic (stationary) or parabolic (transient) PDE(heat equations), where a nonlocal coupling occurs due to the integral operatorof the radiosity equation. There have been numerous papers, where themathematical theory of existence and uniqueness of weak solutions has beendeveloped in recent years (see [5, 6, 7, 8, 9, 10, 11] and references therein).Distributed over several papers, there have been important recent advancesreducing regularity requirements on the data [8, 9] as well as improving regularityresults for the solution [12].In the present survey, we compile such results for both the stationaryand the transient model. For the stationary model, it is then described, howthese results together with advances from [12] regarding the linearized modelcan be applied to obtain rst-order necessary optimality conditions for theoptimal control problem considered in [4], closing a gap left open in [4].We then proceed to the description of a nite volume scheme that hasbeen successfully applied for the numerical solution of conductive-radiativeheat transfer models. Numerical solutions based on the nite volume schemeare presented, where the stationary system has been solved for a complexcrystal growth arrangement. Finally, numerical results of a related appliedoptimal control problem are depicted and briey discussed.The paper is organized as follows: The model of conductive-radiative

Conductive-Radiative Heat Transfer 173heat transfer is reviewed in Sec. 2.1, surveying some recent improvementsregarding the solution of the radiosity equation, which is a key buildingblock of the model. Section 2.2 treats the existence theory of weak solutions,with applications on optimal control in Sec. 3. The nite volume scheme isdescribed in Sec. 4, and we conclude with the numerical results in Sec. 5.2 Modeling Conductive-Radiative Heat Transfer2.1 The ModelThe space domain Ω ⊆ R 3 is assumed to consist of two parts Ω s and Ω g ,where Ω s represents an opaque solid and Ω g represents a transparent gas.More precisely, we assume:(A-1) Ω = Ω s ∪ Ω g , Ω s ∩ Ω g = ∅, and each of the sets Ω, Ω s , Ω g , is anonvoid, bounded, open subset of R 3 such that the interface surfaceΣ := Ω s ∩ Ω g is Lipschitz and piecewise C 1 , i.e. Σ can be partitionedinto nitely many C 1 -surfaces.(A-2) Ω g is enclosed by Ω s , i.e. Σ = ∂Ω g (see Fig. 1).Heat conduction is considered throughout Ω. Nonlocal radiative heattransport is considered between points on the surface Σ of Ω g .Stationary heat conduction is described by− div ( κ(x, θ) ∇ θ ) = f(x) in Ω, (1)where θ(x) ∈ R + represents absolute temperature, depending on the spacecoordinate x; κ > 0 represents the thermal conductivity, and f ≥ 0 is aheat source due to some heating mechanism. In practice, for many heatingmechanisms such as induction or resistance heating, one has f = 0 in Ω g .One can also allow f ≤ 0 to model heat sinks due to cooling.While (1) models the thermal equilibrium, for example, at the end of aheating process, it is often also desirable to model transient heat conduction,for instance, to model crystal growth apparatus during the heating phase, importantin situations, where the growth process and possible defect creationis already initiated during the heating phase [13]. Transient heat conductionis described by∂ε(x, θ)∂t− div ( κ(x, θ) ∇ θ ) = f(t, x) in ]0, T [×Ω, (2)

174 Peter PhilipΩ g,3Ω sΩ sΩ g,2Ω g,1Ω g = Ω g,1 ∪ Ω g,2 ∪ Ω g,2Figure 1: Possible shape of a 2-dimensional section through the 3-dimensionaldomain Ω = Ω s ∪ Ω g . Here, Ω g has the 3 connected components Ω g,1 , Ω g,2 ,Ω g,3 . Note that, according to (A-2), Ω g is engulfed by Ω s , which can not beseen in the 2-dimensional section.where T > 0 represents the nal time, θ and f now depend on the timevariable t as well as on the space variable x, and ε > 0 represents internalenergy.Remark 1. If θ is to represent absolute temperature, then it must be alwayspositive. However, it is also mathematically interesting to study equations(1) and (2) in situations, where the solution θ can be negative. For example,negative solutions can occur if the right-hand side function f is allowed to benegative. For that reason, it is often desirable to keep the problem formulationsuciently exible, such that it makes sense even if θ ≥ 0 can not beguaranteed (considering κ to be dened on Ω × R rather than on Ω × R + isan example, also cf. Rem. 2 below).On the interface Σ between solid and gas, one needs to account for radiosityR and for irradiation J, resulting in a jump in the normal heat ux(κ(x, θ) ∇ θ)· ⃗ng according to the following interface condition for (1) (thesame works for (2) after replacing the space domains by the correspondingtime-space cylinders):(κ(x, θ) ∇ θ)↾Ωg · ⃗n g + R(θ) − J(θ) = ( κ(x, θ) ∇ θ ) ↾ Ωs · ⃗n g on Σ. (3)

Conductive-Radiative Heat Transfer 175Here, ⃗n g denotes the unit normal vector pointing from gas to solid and ↾denotes restriction (or trace). Thus, eectively, (1) consists of two equations,one on Ω s and one on Ω g , coupled via (3) (and analogously for (2)).It is assumed that the solid is opaque such that R(θ) and J(θ) are computedaccording to the net radiation model for diuse-gray surfaces, i.e. re-ection and emittance are taken to be independent of the angle of incidenceand independent of the wavelength. At each point of the surface Σ, the radiosityis the sum of the emitted radiation E(θ) and of the reected radiationJ r (θ):R = E + J r on Σ. (4)According to the Stefan-Boltzmann law,E(θ) = σ ɛ |θ| 3 θ on Σ, (5)where σ ∈ R + represents the Boltzmann radiation constant, and ɛ representsthe potentially material-dependent emissivity of the solid surface. It isassumed that:(A-3) ɛ ∈ L ∞ (Σ) with values in [0, 1] is such that, for each connected componentΩ g,k of Ω g (cf. Fig. 1), there exists M k ⊆ Σ k := ∂Ω g,k suchthat M k has positive surface measure and ɛ > 0 on M k .Remark 2. The physically inclined reader might expect to read θ 4 on theright-hand side of (5) rather than |θ| 3 θ. However, whenever θ > 0 does,indeed, represent an absolute temperature, both terms are identical, while,due to its monotonicity properties, |θ| 3 θ is more suitable for the mathematicaltheory in situations, where θ can become negative (cf. Rem. 1 above). Thisremark should be born in mind for each subsequent occurrence of |θ| 3 θ.Using the presumed opaqueness together with Kirchho's law yieldsJ r = (1 − ɛ) J. (6)Due to diuseness, the irradiation can be calculated asJ(θ) = K(R(θ)), (7)

176 Peter Philipusing the nonlocal integral radiation operator K dened by∫K(ρ)(x) :=ω(x, y) :=V (x, y) :=ΣV (x, y) ω(x, y) ρ(y) dy for a.e. x ∈ Σ, (8)(⃗ns (y) · (x − y) ) ( ⃗n s (x) · (y − x) )π ( (y − x) · (y − x) ) 2for a.e. (x, y) ∈ Σ × Σ, (9){0 if Σ ∩ ]x, y[≠ ∅,1 if Σ ∩ ]x, y[= ∅for each (x, y) ∈ Σ × Σ, (10)where ω is called view factor, V is called visibility factor (being 1 if, and onlyif, x and y are mutually visible), and ⃗n s denotes the outer unit normal tothe solid domain Ω s , existing almost everywhere on the Lipschitz interfaceΣ. The following Th. 3 summarizes properties of ω, V , and K, relevant toour considerations.Theorem 3. Assume (A-1) and (A-2).(a) The kernel V ω of K is almost everywhere nonnegative (actually positivefor V (x, y) = 1), symmetric, and V (x, ·) ω(x, ·) is in L 1 (Σ) with∫ΣV (x, y) ω(x, y) dy = 1 for a.e. x ∈ Σ. (11a)Moreover, if Ω s and Ω g are polyhedral, then∫ΣV (x, y) ω(x, y) dy > 0 for every x ∈ Σ, (11b)where one can choose each of the nitely many possible values of ⃗n(x) ifx belongs to more than one face of Σ.(b) For each 1 ≤ p ≤ ∞, the operator K : L p (Σ) −→ L p (Σ) given by (8) iswell-dened, linear, bounded, and positive with ‖K‖ = 1.Proof. See [14, Lem. 1] and [15, Lem. 2]. For (11b), let Σ z := {y ∈ Σ :V (z, y) = 1}, and note that, if Σ is a polyhedral enclosure, then meas(Σ z ) > 0for each z ∈ Σ. Since ω(z, y) > 0 for each y ∈ Σ z , (11b) holds.Remark 4. It is noted that, for Σ being polyhedral, K is noncompact onL p (Σ) for each p ∈ [1, ∞], as shown in [16]. In settings where the geometry

Conductive-Radiative Heat Transfer 177of the domains is such that Σ is at least C 1,α , α > 0, K is known to becompact on L p (Σ) [8, 15]. However, it is also shown in [16] that, for p < ∞,K can never be compact when reinterpreted as a linear bounded operatorK : L p (0, T, L p (Σ)) −→ L p (0, T, L p (Σ)) in a transient setting (regardless ofthe regularity of Σ).Combining (4) through (7) provides the so-called radiosity equation forR:( )Id −(1 − ɛ)K (R) = ɛσ|θ| 3 θ, (12)where Id denotes the identity operator. The following Th. 5 allows to solve(12) for R. Its hypothesis involves the technical condition introduced as(A-4).(A-4) K is compact (Σ being C 1,α , α > 0, is sucient, cf. Rem. 4) or thereexists r 0 > 0 such thatess supx∈Σ∫B r0 (x)V (x, y) ω(x, y) dy < 1, (13)where B r0 (x) := {y ∈ Σ : ‖x − y‖ 2 < r 0 } (Σ being polyhedral issucient for (13) to hold, see [16, Lem. 6]).Theorem 5. Let p ∈ [1, ∞], and assume (A-1) (A-4). Then the operatorId −(1−ɛ)K has an inverse in the Banach space L(L p (Σ), L p (Σ)) of boundedlinear operators, and the operatorG := (Id −K) ( Id −(1 − ɛ)K ) −1 ɛ (14)is an element of L(L p (Σ), L p (Σ)).Proof. For compact K, see [6, Lem. 2]; for the case that (13) holds, see [16,Th. 5].Corollary 6. Under the hypotheses of Th. 5, given θ ∈ L 4 (Σ), the radiosityequation (12) has the unique solution R(θ) = ( Id −(1 − ɛ)K ) −1 (ɛσ|θ| 3 θ) ∈L 1 (Σ) (recall σ > 0 and ɛ ∈ L ∞ (Σ)).Combining (4) (7) yieldsR(θ) − J(θ) = −ɛ ( K(R(θ)) − σ |θ| 3 θ ) (15a)(12)= (Id −K) ( Id −(1 − ɛ)K ) −1 (ɛσ|θ| 3 θ) (15b)(14)= G(σ|θ| 3 θ) on Σ, (15c)

178 Peter Philipsuch that (3) becomes( )κ(x, θ) ∇ θ ↾Ωg · ⃗n g + G(σ|θ| 3 θ) = ( κ(x, θ) ∇ θ ) ↾ Ωs · ⃗n g on Σ. (16)Assuming the domain Ω is exposed to a black body environment (e.g. alarge isothermal room) radiating at θ ext (some given absolute temperature),the Stefan-Boltzmann law provides the outer boundary conditionκ(x, θ) ∇ θ · ⃗n s − σ ɛ (θ 4 ext − |θ| 3 θ) = 0 on ∂Ω. (17)The outer boundary condition (17) does not allow for nonlocal radiativeinteractions between open cavities and the outer environment. It is physicallyreasonable provided the considered domain does not have any open cavities,i.e. under the simplifying assumption:(A-5) Ω is convex.One can also omit Assumption (A-5) to include nonlocal radiative interactionsbetween open cavities and the outer environment [17]. Here, (A-5) isused for the sake of simpler notation and briefness.The summarized stationary model reads− div ( κ(x, θ) ∇ θ ) = f(x) in Ω, (18a)(κ(x, θ) ∇ θ)↾Ωg · ⃗n g + G(σ|θ| 3 θ) = ( κ(x, θ) ∇ θ ) ↾ Ωs · ⃗n g on Σ, (18b)κ(x, θ) ∇ θ · ⃗n s − σ ɛ (θ 4 ext − |θ| 3 θ) = 0 on ∂Ω, (18c)an integro-dierential boundary value problem for the unknown θ : Ω −→ R.The summarized transient model is similar, employing time-dependentvariants of the interface and boundary condition, respectively, as well asinitial condition (19d):∂ε(x, θ)( )∂tκ(x, θ) ∇ θ ↾]0,T[×Ωg · ⃗n g + G(σ|θ| 3 θ) = ( κ(x, θ) ∇ θ ) ↾ ]0,T [×Ωs · ⃗n gon ]0, T [×Σ,− div ( κ(x, θ) ∇ θ ) = f(t, x) in ]0, T [×Ω, (19a)(19b)κ(x, θ) ∇ θ · ⃗n s + σ ɛ |θ| 3 θ = σ ɛ θext 4 on ]0, T [×∂Ω, (19c)θ(0, x) = θ init (x) on Ω, (19d)an integro-dierential initial-boundary value problem for the unknown θ :[0, T ] × Ω −→ R.

Conductive-Radiative Heat Transfer 1792.2 Existence of Weak SolutionsThe assumptions introduced in the previous section were sucient for theformulation and solution of the radiosity equation (12). For the existencetheory of (18) and (19), we need to introduce further conditions on thematerial and data functions and on the domain.(A-6) κ : Ω×R −→ R + is piecewise continuous in the following sense: Thereexist nitely many open sets with Lipschitz boundary Ω 1 , . . . Ω M ⊆ Ωsuch thatΩ =M⋃i=1Ω i , Ω i ∩ Ω j = ∅ for i ≠ j, Ω i ⊆ Ω g or Ω i ⊆ Ω s for each i,(20)and there exist continuous functions κ 1 , . . . , κ M : R −→ R + such thatκ(x, θ) = κ i (θ) for each x ∈ Ω i .(A-7) There exist κ l , κ u ∈ R + such that 0 < κ l ≤ κ ≤ κ u .(A-8) There exists ɛ l ∈]0, 1[ such that 0 < ɛ l ≤ ɛ ≤ 1.(A-9) θ ext ∈ L 4 (∂Ω) for (18) and θ ext = θ init = const. for (19).(A-10) f ∈ L 1 (Ω) for (18) and f ∈ L 1( ]0, T [×Ω ) for (19).(A-11) dist(Σ, ∂Ω) > 0.We rst consider the stationary case (18), surveying recent results from[12, 18] that extend earlier results from [4, 6] (see [19] for some correctionsto [18]). Related results for a wavelength-dependent emissivity can be foundin [11].Notation 7. For p, q ∈ [1, ∞], letV p,q (Ω) := { u ∈ W 1,p (Ω) : u ∈ L q (Σ ∪ ∂Ω) } . (21)Here, as elsewhere in the paper, we simply write u instead of tr(u) whenconsidering u on Σ ∪ ∂Ω, suppressing the trace operator tr.

180 Peter PhilipDenition 8. Assume (A-1) (A-4) and (A-6) (A-11). Following [18,Def. 1.2], we dene θ ∈ V s,4 (Ω) for some s ∈ [1, ∞] to be a weak solution to(18) if, and only if,∫Ω∫κ(·, θ) ∇ θ · ∇ ψ +∂Ω∫∫ ∫σɛ|θ| 3 θψ + G(σ|θ| 3 θ)ψ = fψ + σɛθextψ4ΣΩ ∂Ω(22)for each ψ ∈ V s′ ,∞ (Ω), where s ′ ∈ [1, ∞] is the conjugate exponent to s, i.e.1s + 1 s= 1. ′Theorem 9. Assume (A-1) (A-4) and (A-6) (A-11).(a) If f ∈ L p (Ω), where p > 9 7or just p > 1 under the additional assumptionthat Σ is C 1,α , α > 0, then (18) has a weak solution θ. If f ≥ 0, thenθ ≥ ess inf θ ext . Moreover, regarding the regularity of θ, if p ≥ 3 2andθ ext ∈ L 8 (∂Ω), then |θ| r ∈ W 1,2 (Ω) for each r ∈ [1, ∞[. If p ∈] 9 7 , 3 2 [ andθ ext ∈ L 8p/(3−p) (∂Ω), then θ ∈ V 2,2p/(3−2p) (Ω) with 2p/(3 − 2p) > 6.If Σ is C 1,α , α > 0, p ∈ [ 6 5 , 9 7 ] and θ ext ∈ L 8p/(3−p) (∂Ω), then θ ∈V 2,(9−5p)/(3−2p) (Ω) with 5 ≤ (9 − 5p)/(3 − 2p) ≤ 6. If Σ is C 1,α , α > 0,p ∈]1, 6 5 [ and θ ext ∈ L 8p/(3−p) (∂Ω), then θ ∈ V 3p/(3−p),(9−5p)/(3−2p) (Ω)with 3 2< 3p/(3 − p) < 2.(b) If f ∈ L 1 (Ω), Σ is C 1,α , α > 0, and ɛ < 1, then (18) has a weak solutionθ ∈ ⋂ s∈[1, 3 2 [ V s,4 (Ω).(c) If θ ext ∈ L ∞ (∂Ω), f ∈ L p (Ω) with p > 3 2, and all ∂Ω i are C 1 , then (18)has a weak solution θ ∈ W 1,q with q := 2p > 3 (in particular, the solutionis Hölder continuous, θ ∈ C γ (Ω), γ > 0). This solution is uniqueprovided the functions κ 1 , . . . , κ M of (A-6) are Lipschitz continuous.Proof. For (a) see [18, Th. 5.1], for (b) see [18, Th. 6.1], and for (c) see [12,Lem. 3.6] and its proof.It remains to discuss the transient case. Here, we survey recent resultsfrom [9] that extend earlier results from [6]. For the sake of clarity andbriefness, the results in [9] were stated for (19) with a Dirichlet conditioninstead of (19c). However, the proofs in [9] do carry over to the situation of(19), and Def. 11 and Th. 12 below are formulated in this spirit. A similartransient problem, where the transparent region Ω g was not enclosed by theopaque region Ω s was rst solved in [5]. Once again, related results for a

Conductive-Radiative Heat Transfer 181wavelength-dependent emissivity can be found in [10]. All the mentionedpapers make the additional simplifying assumption that the internal energyfunction is trivial:(A-12) ε : Ω × R −→ R, ε(x, θ) = θ.Notation 10. We introduce the abbreviationsQ :=]0, T [×Ω, S :=]0, T [×Σ, C :=]0, T [×∂Ω, (23)and, for p, q ∈ [1, ∞], the spacesWp 1,0 (Q) := { u ∈ L p (Q) : ∂ xi ∈ L p (Q) for i = 1, 2, 3 } , (24a)Wp 1 (Q) := { u ∈ Wp 1,0 (Q) : ∂ t ∈ L p (Q) } , (24b)V p,q0 (Q) := { u ∈ Wp 1,0 (Q) : u ∈ L q (S ∪ C) } , (24c)V p,q (Q) := { u ∈ Wp 1 (Q) : u ∈ L q (S ∪ C) } . (24d)Denition 11. Assume (A-1) (A-4) and (A-6) (A-12). Following [9,Def. 1.1], we dene θ ∈ V s,40 (Q) for some s ∈ [1, ∞] to be a weak solution to(19) if, and only if,∫−∫=QΩθ ∂ψ∂t∫Q+ κ(·, θ) ∇ θ · ∇ ψ +∫ ∫θ init ψ(0) + fψ +Q C∫C∫σɛ|θ| 3 θψ + G(σ|θ| 3 θ)ψSσɛθ 4 extψ (25)for each ψ ∈ V s′ ,∞ (Q) with ψ(T, ·) = 0 a.e. in Ω, where, as before, s ′ ∈ [1, ∞]is the conjugate exponent to s.The formulations in [9, Ths. 2.1, 2.2] and [9, Lem. 3.1] provide strongerresults than the formulation of Th. 12(a) below, which has been simpliedfor the sake of clarity and briefness.Theorem 12. Assume (A-1) (A-4) and (A-6) (A-12).(a) If f ∈ L 2 (Q), then (19) has a weak solution θ ∈ V s,50 (Q)∩C(0, T ; L2 (Ω))and ∂ t θ exists in a distributional sense. This solution is unique providedthe functions κ 1 , . . . , κ M of (A-6) are Lipschitz continuous. If f ∈ L s (Q)for s > 2, 5 then θ ∈ L ∞ (Q) and there is c > 0 such that‖θ‖ L ∞ (Q) ≤ max{|θ ext |, |θ init |} + c ‖f‖ L s (Q). (26)

182 Peter Philip(b) If f ∈ L 1 (Ω), Σ is C 1,α , α > 0, and ɛ < 1, then (19) has a weak solutionV p,40 (Q) ∩ (Q), whereL∞,1θ ∈ ⋂ 1≤p< 5 4L ∞,1 (Q) :={u ∈ L 1 (Q) : ess supt∈]0,T [∫Ω|u| < ∞}. (27)Proof. (a) has been shown in Th. 2.1 and Lemmas 3.1, 3.2, 3.3 of [9], alsosee [9, Rem. 3.4]; (b) has been the subject of [9, Th. 4.1].3 Optimal ControlWhen modeling heat transfer for industrial applications such as crystal growth,one is usually not merely interested in determining the temperature distributionθ, but one aims at optimizing θ according to a suitable objectivefunctional. For example, during sublimation growth of silicon carbide, smallhorizontal temperature gradients in the gas domain Ω g are desirable to avoiddefects of the growing crystal, while suciently large vertical temperaturegradients are required to guarantee a material transport from the siliconsource to the seed crystal [20, 21]. This background led to the optimal controlproblem considered in [4]:∫minimize J(θ, u) := 1 ‖ ∇ θ − z‖ 2 2 + ν u 2 (28a)2 Ω g2 Ω s{usubject to system (18) withon Ω s ,f =(28b)0 on Ω g ,and 0 < u a ≤ u ≤ u b in Ω s , (28c)where z : Ω g −→ R 3 is a given desired distribution for the temperature gradient.Here, (28b) imposes the condition of no heat sources in the gas regionΩ g , which is the case for the motivating application of induction heating. Thecontrol constraints (28c) reect the fact that only heating (and no cooling)is considered, and they take into account that, due to technical limitations,an actual heating device can not produce heat sources of arbitrarily largevalues. Precisely stated, the assumptions on ν, z, u a , u b are:(A-13) ν > 0, z ∈ L 2 (Ω g , R 3 ).∫

Conductive-Radiative Heat Transfer 183(A-14) u a , u b ∈ L ∞ (Ω s ), 0 < u a ≤ u b .In view of Th. 9(c), in (28a), we consider the objective functionalJ : W 1,2 (Ω) × L 2 (Ω s ) −→ R + 0 . (29)Actually, from the point of view of the application, a control problemlike (28), where the heat sources f are controlled directly, is only the rststep. In practice, the heat sources are generated by a heating mechanismsuch as induction heating, i.e. f itself is again the solution to some equation.A control problem, where f is obtained as a solution to Maxwell's equationsdescribing induction heating, has been considered in [12]. An even morerealistic situation was used for the numerical results of Sec. 5 below.Denition 13. Under the assumptions of Th. 9(c), dene the control-tostateoperator S : L 2 (Ω s ) −→ W 1,q (Ω) ⊆ C γ (Ω) ⊆ L ∞ (Ω) (q > 3 as in Th.9(c), γ > 0), u{↦→ θ, assigning to u ∈ L 2 (Ω s ) the unique weak solution θ ofu(18) withon Ω s ,f := provided by Th. 9(c).0 on Ω g ,Denition 14. Employing the control-to-state operator of Def. 13, and lettingU ad := { u ∈ L ∞ }(Ω s ) : u a ≤ u ≤ u b , (30)(¯θ, ū) ∈ W 1,q (Ω) × U ad (q > 3 as in Th. 9(c)) is called an optimal controlfor (28) if, and only if, ¯θ = S(ū), and ū minimizes the reduced objectivefunctionalj : L 2 (Ω s ) −→ R + 0 , j(u) := J( S(u), u ) (31)on U ad .The following theorem provides the existence of an optimal control for(28) under the simplifying assumption of θ-independent κ:Theorem 15. Under the assumptions of Th. 9(c) plus (A-13), (A-14), κ i =const., and ess inf θ ext > 0, there exists an optimal control (¯θ, ū) for (28).Proof. See [4, Th. 5.2].Theorem 18 below provides the dierentiability of the control-to-stateoperator as well as rst-order necessary optimality conditions for (28), which

184 Peter Philipare related to weak solutions to the linearized form of (18) as dened inDef. 16 below. For the situation of Th. 15, variants of Ths. 17 and 18below had already been considered in [4], where the Fredholm alternativewas employed to show the linearized form of (18) admits a unique solution,provided the homogeneous version admits only 0 as its solution. However,the latter question remained open in [4]. This gap has now been closed dueto the availability of [12, Th. 4.4].While the existence of an optimal control for (28) has only been provedfor θ-independent κ, the following theory of rst-order necessary optimalityconditions merely requires the much milder condition:(A-15) Each of the functions κ 1 , . . . , κ M of (A-6) are Lipschitz continuous andcontinuously dierentiable (i.e. C 1 with bounded derivative).Denition 16. Under the assumptions of Th. 9(c) plus ess inf θ ext > 0 and(A-15), let ū ∈ L 2 (Ω s ), ū ≥ 0, ¯θ := S(ū) ∈ W 1,q (Ω) with q > 3 as in Th.9(c). Given F in the dual of W 1,q′ (Ω) (q ′ the conjugate exponent to q), afunction θ ∈ W 1,q (Ω) is called a weak solution to the linearized form of (18)(or (22)) with right-hand side F if, and only if,∫∫κ(·, ¯θ) ∇ θ · ∇ ψ +Ω∫∫+ 4 σɛ|¯θ| 3 θψ + 4∂ΩΩΣ∂κ∂θ (·, ¯θ)θ ∇ ¯θ · ∇ ψG(σ|¯θ| 3 θ)ψ = F(ψ) (32)for each ψ ∈ W 1,q′ (Ω) (recall G : L ∞ (Σ) −→ L ∞ (Σ) according to Th. 5).Theorem 17. In the situation of Def. 16, there exists a unique weak solutionto the linearized form of (18), i.e. θ ∈ W 1,q (Ω) such that (32) holds for eachψ ∈ W 1,q′ (Ω). Moreover, there exists c > 0 such that‖θ‖ W 1,q (Ω) ≤ c ‖F‖. (33)Proof. See [12, Th. 4.4]. The proof is based on the Fredholm alternative,which yields that (32) admits a unique solution θ ∈ W 1,q (Ω) if, and only if,its homogeneous version (F = 0) has 0 as its unique solution. The latter isestablished in the proof of [12, Th. 4.4] via a comparison principle.Theorem 18. Under the assumptions of Th. 9(c) plus ess inf θ ext > 0 and(A-15), the control-to-state operator of Def. 13 is Fréchet dierentiable on

Conductive-Radiative Heat Transfer 185L 2 +(Ω s ) := {ū ∈ L 2 (Ω s ) : ū > 0}. Moreover, for ū ∈ L 2 +(Ω s ), ¯θ = S(ū), andu ∈ L 2 (Ω s ), one has θ := S ′ (ū)(u) given by the weak solution to the linearizedform of (18) with right-hand side F(ψ) := F u (ψ) := ∫ L 2 (Ω s ) uψ.If (A-13) and (A-14) hold, then (¯θ, ū) ∈ W 1,q (Ω) × U ad (q > 3 as in Th.9(c)) being an optimal control for (28) implies the necessary conditionj ′ (ū)(u−ū) = 〈∇ ¯θ−z, ∇ θ〉 L 2 (Ω g )+ν〈ū, (u−ū)〉 L 2 (Ω s ) ≥ 0 for each u ∈ U ad ,(34)with j as in (31), ¯θ = S(ū), and θ = S ′ (ū)(u − ū).Proof. Using Th. 17, the proof is based on the implicit function theorem andcan be conducted as in [4, Th. 7.1].Second-order sucient optimality conditions for the situation of Th. 15have been proved in [22], and a similar problem with constraints on θ (i.e.state constraints) has been treated in [23].4 Finite Volume Discretization4.1 SettingThe numerical simulation results presented in Sec. 5 below are based on anite volume discretization of (18), which is described in the present section.As for the considerations on optimal control, we will restrict ourselves to thestationary setting. For transient simulation results solving (19), we refer to[24, 25, 26]. Descriptions of nite volume schemes suitable for the transientsituation can be found in [17, 24].The described nite volume scheme was designed for polyhedral domains.In consequence, within the present section, assume:(A-16) Ω g and Ω s are polyhedral.For the sake of more readable notation, we use the following simplied versionof (A-6):(A-17) There are precisely two materials, i.e. (A-6) holds with M = 2, Ω 1 =Ω g , Ω 2 = Ω s .We also impose more regularity on the emissivity, the external temperature,and the heat sources:

186 Peter Philip(A-18) ɛ : Σ −→ [0, 1] and θ ext : ∂Ω −→ R + are continuous. Moreover, thereare continuous functions f m : Ω m −→ R such that f m ↾ Ωm = f ↾ Ωm ,m ∈ {s, g}.4.2 Discretization of the Local TermsAn admissible discretization of Ω is given by a nite family T := (ω i ) i∈I ofsubsets of Ω satisfying a number of assumptions, subsequently denoted by(DA-∗).(DA-1) T = (ω i ) i∈I forms a nite partition of Ω (i.e. Ω = ⋃ i∈I ω i), and, foreach i ∈ I, ω i is a nonvoid, polyhedral, connected, and open subsetof Ω.From T , one can dene discretizations of Ω s and Ω g : For m ∈ {s, g} andi ∈ I, letω m,i := ω i ∩ Ω m , I m := { j ∈ I : ω m,j ≠ ∅ } , T m := (ω m,i ) i∈Im . (35)To allow the incorporation of the interface condition (18b) into the scheme, itis assumed that, if some ω i has a 2-dimensional intersection with the interfaceΣ, then it lies on both sides of the intersection. More precisely:(DA-2) For each i ∈ I: ∂ reg ω s,i ∩ Σ = ∂ reg ω g,i ∩ Σ, where ∂ reg denotes theregular boundary of a polyhedral set, i.e. the parts of the boundary,where a unique outer unit normal vector exists (see Fig. 2), ∂ reg ∅ :=∅.Integrating (18a) over ω m,i and applying the Gauss-Green integrationtheorem yields ∫∫− κ m (θ) ∇ θ · ⃗n ωm,i = f, (36)∂ω m,i ω m,iwhere ⃗n ωm,i denotes the outer unit normal vector to ω m,i .The nite volume scheme is furnished by incorporating the interface andboundary conditions (18b) and (18c) followed by an approximation of theintegrals by quadrature formulas. To approximate θ by a nite number ofdiscrete unknowns θ i , i ∈ I, precisely one value θ i is associated with eachcontrol volume ω i . Introducing a discretization point x i ∈ ω i for each controlvolume ω i , the θ i can be interpreted as θ(x i ). Moreover, the discretization

Conductive-Radiative Heat Transfer 187ω g,1ω 3ω s,1Ω gω 2ΣΩ sΩ sFigure 2: Illustration of condition (DA-2): Ω s consists of the outer wall ofthe box as well as of the region to the right of the vertical plane in the middleof the box, which is contained in Σ; Ω g consists of the region to the left ofthat plane and engulfed by the wall. Both ω 1 and ω 2 satisfy (DA-2) (since∂ reg ω s,2 ∩ Σ = ∅ and ∂ reg ω g,2 ∩ Σ = ∂ reg ∅ ∩ Σ = ∅), however ω 3 does notsatisfy (DA-2) (since ∂ reg ω s,3 ∩ Σ ≠ ∅ and ∂ reg ω g,3 ∩ Σ = ∂ reg ∅ ∩ Σ = ∅).makes use of regularity assumptions concerning the partition (ω i ) i∈I that canbe expressed in terms of the x i (see (DA-3), (DA-4), and (DA-5) below).The boundary of each control volume ω m,i can be decomposed accordingto (see Fig. 3)∂ω m,i = ( ∂ω m,i ∩ Ω m)∪(∂ωm,i ∩ ∂Ω ) ∪ ( ∂ω m,i ∩ Σ ) . (37)To guarantee that there is a discretization point x i in each of the integrationdomains occurring in (37), it is assumed that the discretization Trespects interfaces and outer boundaries in the following sense:(DA-3) For each m ∈ {s, g}, i ∈ I m : x i ∈ ω m,i . In particular, if ω s,i ≠ ∅ andω g,i ≠ ∅, then x i ∈ ω s,i ∩ ω g,i .(DA-4) For each i ∈ I, the following holds: If meas(ω i ∩ ∂Ω) ≠ 0, thenx i ∈ ω i ∩ ∂Ω (cf. Fig. 4).

188 Peter Philipω s,1 ω s,1ω s,1∂ω m,i ∩ Ω ω m s,2 ∂ω m,i ∩ ∂Ω ω s,2ω g,2 ω g,2 ω g,2∂ω m,i ∩ Σω s,2ω s,2ω 1 = ω s,1ω g,3ω g,2Ω sΩ g,3Ω sΩ g,2Ω g,1ω g,3ω s,3Ω g = Ω g,1 ∪ Ω g,2 ∪ Ω g,2ω g,3ω g,3ω s,3 ω s,3 ω s,3∂ω m,i ∩ Ω m∂ω m,i ∩ ∂Ω∂ω m,i ∩ ΣFigure 3: Illustration of the decomposition of the boundary of control volumesω m,i according to (37). The lower control volume ω 3 is not admissible,as it has 2-dimensional intersections with both Σ and ∂Ω (see Rem. 19).Remark 19. Suppose a control volume ω i has a 2-dimensional intersectionwith both ∂Ω and Σ. Then, by (DA-2), ω s,i ≠ ∅ and ω g,i ≠ ∅. Thus, by (DA-3), x i ∈ Σ. On the other hand, by (DA-4), x i ∈ ∂Ω, which means that (A-11)is violated. It is thus shown that ω i cannot have 2-dimensional intersectionswith both ∂Ω and Σ. In particular, the lower control volume ω 3 in Fig. 3 isnot admissible.

Conductive-Radiative Heat Transfer 189x 7 x 1ω m,1 ∩ ω m,7ω m,7 ω m,1 ω m,1 ∩ ω m,4ω m,1 ∩ ω m,2x 6ω m,6x 2ω m,4ω m,2x 4x 3ω m,3 ∩ ω m,2ω m,3 ∩ ω m,4ω m,5ω m,3ω m,3 ∩ ω m,5x 5Figure 4: Illustration of conditions (DA-4) and (DA-5) as well as of thepartition of ∂ω m,i ∩ Ω m according to (40). One has nb m (1) = {2, 4, 7} andnb m (3) = {2, 4, 5}.∫−Using the boundary condition (18c) leads to the following approximation:κ s (θ) ∇ θ · ⃗n ωs,i ≈ −σ ɛ(x i ) (θext(x 4 i ) − |θ i | 3 θ i ) meas ( ∂ω s,i ∩ ∂Ω ) .∂ω s,i ∩∂Ω(38)The nonlocal interface condition (18b) with G(σ|θ| 3 θ) = −ɛ ( K(R(θ)) −σ |θ| 3 θ ) according to (15) yields−∑m∈{s,g}∫∂ω m,i ∩Σ∫κ m (θ) ∇ θ · ⃗n ωm,i = −ω i ∩Σɛ ( K(R(θ)) − σ |θ| 3 θ ) . (39)

190 Peter PhilipThe approximation of the nonlocal term K(R(θ)) is more involved and willbe considered in detail in Sec. 4.3 below. First, to approximate the integralsover ∂ω m,i ∩ Ω m , this set is partitioned further (see Fig. 4):∂ω m,i ∩ Ω m =⋃j∈nb m (i)∂ω m,i ∩ ∂ω m,j , (40)where nb m (i) := {j ∈ I m \ {i} : meas(∂ω m,i ∩ ∂ω m,j ) ≠ 0} is the set ofm-neighbors of i. Moreover, it is assumed that:(DA-5) For each i ∈ I, j ∈ nb(i) := {j ∈ I \ {i} : meas(∂ω i ∩ ∂ω j ) ≠0}: x i ≠ x j and x j−x i‖x i −x j ‖= ⃗n ωi ↾ ∂ωi ∩∂ω j 2, where ⃗n ωi ↾ ∂ωi ∩∂ω jis therestriction of the normal vector ⃗n ωi to the interface ∂ω i ∩∂ω j . Thus,the line segment joining neighboring vertices x i and x j is alwaysperpendicular to ∂ω i ∩ ∂ω j (see Fig. 4, where the vertices x i arechosen such that (DA-5) is satised).The approximation of the integrals over ∂ω m,i ∩ Ω m , is now provided byreplacing the normal gradient of θ on ∂ω i ∩ ∂ω j by the corresponding dierencequotient and by approximating κ m (θ) on ∂ω m,i ∩∂ω m,j by the arithmeticmean of κ m (θ i ) and κ m (θ j ):∫κ m (θ) ∇ θ · ⃗n ωm,i∂ω m,i ∩ Ω m≈∑ κ m (θ i ) + κ m (θ j )2j∈nb m(i)Finally, for the approximation of the source term,∫θ j − θ i‖x i − x j ‖ 2meas ( ∂ω m,i ∩ ∂ω m,j). (41)ω m,if ≈ f m (x i ) meas(ω m,i ). (42)4.3 Discretization of the Nonlocal Radiation TermsSimilarly to the nite volume approximation of the local terms, the discretizationof K(R(θ)) proceeds by partitioning the surface Σ into 2-dimensionalpolyhedral control volumes (so-called boundary elements).(DA-6) (ζ α ) α∈IΣ is a nite partition of Σ, where for each α ∈ I Σ , the boundaryelement ζ α is a nonvoid, polyhedral, connected, and (relative)open subset of Σ, lying in a 2-dimensional ane subspace of R 3 .

Conductive-Radiative Heat Transfer 191On Σ, the boundary elements are supposed to be compatible with thecontrol volumes ω i :(DA-7) For each α ∈ I Σ , there is a unique i(α) ∈ I such that ζ α ⊆ ∂ω s,i(α) ∩Σ.Moreover, for each α ∈ I Σ : x i(α) ∈ ζ α (see Fig. 5).Denition and Remark 20. For each i ∈ I, dene J Σ,i := {α ∈ I Σ :meas(ζ α ∩ ∂ω s,i ) ≠ 0}. It then follows from (DA-1), (DA-6), and (DA-7),that (ζ α ∩ ∂ω s,i ) α∈JΣ,i is a partition of ∂ω s,i ∩ Σ = ω i ∩ Σ (see Fig. 5).i(1) = i(10) = 5, i(2) = 6, i(3) = i(4) = 1,i(5) = i(6) = 2, i(7) = 3, i(8) = i(9) = 4.x 6ζ 1 ζ 2ω s,49ζ 7ζ 5ζ 10Ω gζ 4ζ 8ζ 6ω s,3J Σ,1 = {3, 4}, J Σ,2 = {5, 6},J Σ,3 = {7}, J Σ,4 = {8, 9}, J Σ,5 = {1, 10}, J Σ,6 = {2}.x 3x 1ω s,1ω s,2x 2Figure 5: Illustration of the partitioning of Σ into the ζ α , and of the compatibilitycondition (DA-7) as well as of Def. and Rem. 20. Note that, in orderto satisfy (DA-2), each ω i must extend into Ω g (i.e. ω g,i ≠ ∅). However, onlythe parts ω s,i are drawn in the gure.The radiosity R(θ) is approximated as constant on each boundary elementζ α , α ∈ I Σ . The approximated value is denoted by R α (⃗u), depending on the

192 Peter Philipvector ⃗u := (θ i(α) ) α∈IΣ . From (8), one obtains∫whereζ αK(R(θ)) ≈ ∑ β∈I ΣR β (⃗u) V α,β for each α ∈ I Σ , (43)∫V α,β :=V ω for each (α, β) ∈ I Σ × I Σ . (44)ζ α ×ζ βThe V α,β are nonnegative since V ω is nonnegative according to Th. 3(a).The forms of V and ω imply the symmetry conditionV α,β = V β,α for each (α, β) ∈ I Σ × I Σ , (45)and (11a) implies∑V α,β = meas(ζ α ) for each α ∈ I Σ . (46)β∈I ΣUsing (43) and approximating ɛ as constant on ζ α allows to write the radiosityequation in the integrated and discretized formR α (⃗u) meas(ζ α ) − ( 1 − ɛ(x i(α) ) ) ∑R β (⃗u) V α,ββ∈I Σ= σ ɛ(x i(α) ) |θ i(α) | 3 θ i(α) meas(ζ α ) for each α ∈ I Σ . (47)If the vector ⃗u = (θ i(α) ) α∈IΣ is known, then (47) constitutes a linear systemfor the determination of the vector (R α (⃗u)) α∈IΣ .In matrix form, (47) readsG ⃗ R(⃗u) = ⃗ E(⃗u), (48)with vector-valued functions⃗R : R I Σ−→ R I Σ,(R(⃗u) ⃗ = Rα (⃗u) ) ,α∈I Σ(49a)⃗E : R I Σ−→ R I Σ, ⃗ E(⃗u) =(Eα (⃗u) ) α∈I Σ,and the matrixG = (G α,β ) (α,β)∈I 2Σ, G α,β :=E α (⃗u) :=σ ɛ(x i(α) ) |u α | 3 u α meas(ζ α ),(49b){meas(ζ α ) − ( 1 − ɛ(x i(α) ) ) V α,β for α = β,− ( 1 − ɛ(x i(α) ) ) V α,β for α ≠ β.(49c)

Conductive-Radiative Heat Transfer 193Lemma 21. (a) For each α ∈ I Σ : ∑ β∈I Σ \{α} |G α,β| ≤ (1−ɛ(x i(α) )) G α,α ≤G α,α . In particular, G is weakly diagonally dominant.(b) G is an M-matrix, i.e. G is invertible, G −1 is nonnegative, and G α,β ≤ 0for each (α, β) ∈ I 2 Σ, α ≠ β.Proof. See [17, Lem. 3.4 and Rem. 3.5].Now, Lemma 21(b) allows to give a precise denition of ⃗ R by completing(49a) with⃗R(⃗u) := G −1 ( ⃗ E(⃗u)). (50)Remark 22. The denition of ⃗ R in (50) implies that (47) and (48) holdwith ⃗u = (θ i(α) ) α∈IΣ replaced by a general vector ⃗u = (u α ) α∈IΣ ∈ R I Σ.Finally, introducing the vector-valued function⃗V : R I Σ−→ R I Σ, ⃗ V (⃗u) =(Vα (⃗u) ) α∈I Σ,V α (⃗u) := ɛ(x i(α) ) ∑ β∈I ΣR β (⃗u) V α,β ,(51)(43) provides the desired approximation of the nonlocal term in (39):∫ζ αɛ K(R(θ)) ≈ ɛ(x i(α) ) ∑ β∈I ΣR β (⃗u) V α,β = V α (⃗u). (52)The following Lem. 23 states some useful properties of the function ⃗ V . Weintroduce the following notation for ⃗u = (u i ) i∈IΣ ∈ R I Σ:min (⃗u) := min{u i : i ∈ I Σ }, max (⃗u) := max{u i : i ∈ I Σ }. (53)Lemma 23. (a) For each ⃗u ∈ (R + 0 )I Σ: ⃗ R(⃗u) ≥ 0 and ⃗ V (⃗u) ≥ 0.(b) For each ⃗u ∈ (R + 0 )I Σ, α ∈ I Σ :σ ɛ(x i(α) ) min (⃗u) 4 meas(ζ α ) ≤ V α (⃗u) ≤ σ ɛ(x i(α) ) max (⃗u) 4 meas(ζ α ).Proof. See [17, Lem. 3.7].

194 Peter Philip4.4 The Finite Volume SchemeFor ⃗u = (u i ) i∈I , dene⃗u↾ IΣ := (u i(α) ) α∈IΣ . (54)At this point, all preparations are in place to state the nite volume schemein (55) below. The terms in (55) arise from (36) after summing over m ∈{s, g} and employing the approximations (38), (39), (41), (42), and (52),respectively. One is seeking a solution ⃗u = (u i ) i∈I , to0 = − ∑∑m∈{s,g} j∈nb m(i)κ m (u i ) + κ m (u j )2u j − u imeas ( )∂ω m,i ∩ ∂ω m,j‖x i − x j ‖ 2(55a)+ σ ɛ(x i ) ( u 4 i − θ 4 ext(x i ) ) meas(∂ω s,i ∩ ∂Ω) (55b)+ σ ɛ(x i ) u 4 i meas ( ω i ∩ Σ ) − ∑−∑m∈{s,g}f m (x i ) meas(ω m,i ).α∈J Σ,iV α (⃗u↾ IΣ )(55c)(55d)5 Numerical SimulationAs discussed before, the modeling of conductive-radiative heat transfer ismotivated by industrial applications such as crystal growth. We now presentsimulation results, where the solution θ to (18) has been computed and optimizednumerically in the context of such applications for axisymmetricgeometries. Here, the heat sources f are due to induction heating, generatedby nitely many coil rings located outside the domain Ω. The heat sourcesare numerically computed according to the following model, where all materialsin Ω s are considered as potential conductors, whereas Ω g is treatedas a perfect insulator (see [2, Sec. 2.6] for details; due to the axisymmetry,cylindrical coordinates (r, z) are used):j =|j(r, z)|2f(r, z) =2 σ c (r, z) , (56)in the k-th coil ring,−iω σ c φ in Ω s ,(57){−iω σ c φ + σ c v k2πr

Conductive-Radiative Heat Transfer 195where σ c denotes the electrical conductivity, v k is the voltage imposed in thekth coil ring, ω is the common angular frequency of the imposed voltages,and i is the imaginary unit. The potential φ is determined from the followingsystem of elliptic partial dierential equations:− ν div ∇(rφ)− ν div ∇(rφ)r 2− ν div ∇(rφ)r 2r 2 = 0 in Ω g , (58a)+ i ωσ cφr+ i ωσ cφr= σ c v k2πr 2 in the k-th coil ring, (58b)= 0 in Ω s , (58c)where ν denotes the magnetic reluctivity. The system (58) is completed bythe interface conditions( ) ( )ν↾Ωiν↾Ωjr 2 ∇(rφ)↾ Ωi• ⃗n Ωi =r 2 ∇(rφ)↾ Ωj• ⃗n Ωi , (59)where Ω i and Ω j can be either Ω g or subsets of Ω s , representing dierent solidmaterials, as the magnetic reluctivity ν can be discontinuous at interfacesbetween such dierent solid materials. Moreover, the assumption φ = 0 isused both on the symmetry axis r = 0 and suciently far from the growthapparatus (imposed as Dirichlet boundary condition).A nite volume discretization as described in Sec. 4 above was used tocompute the solution θ to (18), where Newton's method was used to solve(55). The computation of the nonlocal radiation terms involves the calculationof visibility and view factors. The method used is based on [1] andis described in [27, Sec. 4]. The discrete scheme was implemented as partof the software WIAS-HiTNIHS 1 which is based on the program packagepdelib [28]. In particular, pdelib uses the grid generator Triangle [29] to produceconstrained Delaunay triangulations of the domains, and it uses thesparse matrix solver PARDISO [30, 31] to solve the linear system arisingfrom the linearization of the nite volume scheme via Newton's method.Figure 6 depicts a numerical solution for the temperature eld θ inside acomplex Czochralski crystal growth apparatus (for crystal growth from melt[32]). As described above, the model equations (18) and (56) (59) havebeen used for the computation.1 High Temperature Numerical Induction Heating Simulator; pronunciation: ∼hitnice.

196 Peter PhilipGeometry Temperature350Figure 6: Numerical solution of (18) for the temperature eld θ inside aCzochralski crystal growth arrangement computed by WIAS-HiTNIHS. Thegure shows detail of an enlarged and rotated section, not according to scale see [32] for the entire apparatus and the precise dimensions.OuterpressurechamberInsulationInnerpressurechamber1500165015501511CrystalseedMelt155015004001200Insulationtemp. [K]Holesbehindbus-bars170015001300Bus-bars

Conductive-Radiative Heat Transfer 197For the following numerical results of an optimal control problem, a simplerdomain was used, schematically depicted in Fig. 7 (see [3, Figs. 1,3]for the precise dimensions). This domain represents an apparatus for siliconcarbide single crystal growth via sublimation by physical vapor transport.As discussed in Sec. 3, during sublimation growth of silicon carbide, smallhorizontal temperature gradients in the gas domain Ω g (more precisely inthe part of Ω g close to the surface of the growing crystal) are desirable toavoid defects of the growing crystal, while suciently large vertical temperaturegradients are required to guarantee a material transport from the siliconsource to the seed crystal [20, 21].z = z rimcopper induction coil ringsgasblind hole(forcoolingof seed)SiC seedcrystalpolycrystallineSiC powdergraphiteinsulationz = 0Figure 7: Schematic picture of apparatus for silicon carbide single crystalgrowth by physical vapor transport. For the precise dimensions of the domainused for the temperature eld optimization, see [3, Figs. 1,3].In the following, numerical optimization results from [3] are presented,where a problem similar to (28) was solved. However, for the numericaloptimization of the temperature θ in [3], the heat sources were not controlled

198 Peter Philipdirectly as in (28), but they were computed according to (56) (59), whereasthe quantities heating power P, vertical upper rim z rim of the induction coil(cf. Fig. 7), and the frequency f = ω/(2π) of the heating voltage were usedas control parameters, which is more realistic from the point of view of theconsidered crystal growth application. The control parameters, thus, resultin a temperature distribution θ(P, z rim , f) via (56) (59) and (18) (see [3]for details).(a): θ(P = 10.0 kW, z rim = 24.0 cm, f = 10.0 kHz)SiC crystalSiC powder3002 K3007 K3012 K3022 K(b): θ(P = 7.98 kW, z rim = 22.7 cm, f = 165 kHz)Nelder-Mead result minimizing F r (θ)SiC powder2304 K2314 K(c): θ(P 1 = 10.3 kW, z rim = 12.9 cm, f = 84.9 kHz),Nelder-Mead result minimizing F r(θ)−F z (θ)2 F r(θ) − 1 2 F z(θ), F z (θ) :=ASiC crystal3042 KFigure 8: Stationary solution for the temperature eld θ(P, z rim , f) in gas regionA between SiC powder and crystal for the generic, unoptimized situationP = 10 kW, z rim = 24 cm, f = 10 kHz. Isotherms spaced at 5 K.While Fig. 8 depicts SiCthecrystaltemperature eld for a generic, unoptimizedsituation as a reference, the objective functional minimized in Fig. 9 is(∫F r (θ) :=Γ2334 K2π r ∂ r θ(r, z) 2 dr) 1/2, (60)aiming at minimizing the radial temperature gradient on the lower surface Γof the growing SiC crystal. The objective functional minimized in Fig. 10 is(∫2π r ∂ z θ(r, z) 2 d(r, z)) 1/2, (61)2aiming at minimizing the radial temperature 2299 K gradient on Γ, while simultaneouslymaximizing the vertical temperature gradient inside the region A2304 Kbetween the SiC crystal and the SiC powder, to guarantee material transportfrom the powder to the crystal. 2314 K2364 KThe optimization is subject to a number2324 Kof state constraints on θ: (a) Themaximal temperature in the apparatus must not surpass a prescribed boundSiC powderθ max ; (b) the temperature at the crystal surface Γ needs to stay within a

SiC powder3022 KConductive-Radiative Heat Transfer 199(b): θ(P = 7.98 kW, z rim = 22.7 cm, f = 165 kHz)Nelder-Mead result minimizing F r (θ)(a): θ(P = 10.0 kW, SiCz rim crystal = 24.0 cm, f = 10.0 kHz)SiC crystalSiC powder3002 2304K3007 K23143012 K3022 KSiC powder(c): θ(P = 10.3 kW, z rim = 12.9 cm, f = 84.9 kHz),Nelder-Mead result minimizing F r(θ)−F z (θ)(b): θ(P = 7.98 kW, z rim = 22.7 cm, f =2165 kHz)Nelder-Mead result minimizing 2299 KF r (θ)SiC crystalSiC crystal2304 K2324 K2314 KSiC θpowdermin,Γ ≤ min(θ) ≤ maxΓSiC powdermax2334 K3042 KFigure 9: Temperature eld θ(P, z rim , f) in gas region A between SiC powderand crystal according to Nelder-Mead minimization of F r (θ), resulting inP = 7.98 kW, z rim = 22.7 cm, f = 165 kHz. Isotherms spaced at 5 K.prescribed range [θ min,Γ , θ max,Γ ]; (c) the temperature gradient between sourceand seed must be negative, and must not surpass a prescribed value ∆ max

200 Peter Philiptimized solutions shown in Figures 9,10 is the gained homogeneity of thetemperature inside the SiC crystal in the optimized solutions (favorable withrespect to low thermal stress and few crystal defects) as well as the isothermsbelow the crystal's surface becoming more parallel to that surface (as intendedby the minimization of F r (θ)). As expected, in Fig. 10, the maximizationof F z (θ) leads to an increased number of isotherms between thecrystal and the source powder. Summarizing the results, the radial and thevertical gradient can be eectively tuned simultaneously.References[1] F. Dupret, P. Nicodéme, Y. Ryckmans, P. Wouters, and M.J. Crochet.Global modelling of heat transfer in crystal growth furnaces. Intern.J. Heat Mass Transfer, 33(9):18491871, 1990.[2] O. Klein, P. Philip, and J. Sprekels. Modeling and simulation of sublimationgrowth of SiC bulk single crystals. Interfaces and Free Boundaries,6:295314, 2004.[3] C. Meyer and P. Philip. Optimizing the temperature prole duringsublimation growth of SiC single crystals: Control of heating power,frequency, and coil position. Crystal Growth & Design, 5(3):11451156,2005.[4] C. Meyer, P. Philip, and F. Tröltzsch. Optimal control of a semilinearPDE with nonlocal radiation interface conditions. SIAM J. ControlOptim., 45:699721, 2006.[5] M. Metzger. Existence for a time-dependent heat equation with nonlocalradiation terms. Math. Meth. Appl. Sci., 22:11011119, 1999.[6] M. Laitinen and T. Tiihonen. Conductive-radiative heat transfer in greymaterials. Quart. Appl. Math., 59(4):737768, 2001.[7] J. Monnier and J.P. Vila. Convective and radiative thermal transferwith multiple reections. Analysis and approximation by a nite elementmethod. Math. Models Methods Appl. Sci., 11(2):229262, 2001.[8] P.-É. Druet. Analysis of a coupled system of partial dierential equationsmodeling the interaction between melt ow, global heat transfer and

Conductive-Radiative Heat Transfer 201applied magnetic elds in crystal growth. PhD thesis, Department ofMathematics, Humboldt University of Berlin, Germany, 2008. Availablein pdf format athttp://edoc.hu-berlin.de/dissertationen/druet-pierre-etienne-2009-02-05/PDF/druet.pdf.[9] P.-É. Druet. Weak solutions to a time-dependent heat equation withnonlocal radiation boundary condition and arbitrary p-summable righthandside. Applications of Mathematics, 55(2):111149, 2010.[10] A.A. Amosov. Nonstationary nonlinear nonlocal problem of radiativeconductiveheat transfer in a system of opaque bodies with propertiesdepending on the radiation frequency. Journal of Mathematical Sciences,165(3):141, 2010.[11] A.A. Amosov. Stationary nonlinear nonlocal problem of radiativeconductiveheat transfer in a system of opaque bodies with propertiesdepending on the radiation frequency. Journal of Mathematical Sciences,164(3):309344, 2010.[12] P.-É. Druet, O. Klein, J. Sprekels, F. Tröltzsch, and I. Yousept. Optimalcontrol of 3D state-constrained induction heating problems withnonlocal radiation eects. Preprint No. 1422, Weierstrass Institute forApplied Analysis and Stochastics (WIAS), Berlin, 2009. Submitted toSIAM J. Control Optim. Available in pdf format athttp://www.wias-berlin.de/preprint/1422/wias_preprints_1422.pdf.[13] D. Schulz, M. Lechner, H.-J. Rost, D. Siche, and J. Wollweber. Onthe early stages of sublimation growth of 4H-SiC using 8 ◦ o-orientedsubstrates. Mater. Sci. Forum, 433436:1720, 2003. Proceedings of4th European Conference on Silicon Carbide and Related Materials,September 25, 2002, Linköping, Sweden.[14] T. Tiihonen. A nonlocal problem arising from heat radiation on nonconvexsurfaces. Eur. J. App. Math., 8(4):403416, 1997.[15] T. Tiihonen. Stefan-boltzmann radiation on non-convex surfaces. Math.Meth. in Appl. Sci., 20(1):4757, 1997.

202 Peter Philip[16] P.-É. Druet and P. Philip. Noncompactness of integral operators modelingdiuse-gray radiation in polyhedral and transient settings. Integr.Equ. Oper. Theory, in press. Published online July 14, 2010, DOI:10.1007/s00020-010-1821-8.[17] O. Klein and P. Philip. Transient conductive-radiative heat transfer:Discrete existence and uniqueness for a nite volume scheme. Math.Mod. Meth. Appl. Sci., 15(2):227258, 2005.[18] P.-É. Druet. Weak solutions to a stationary heat equation with nonlocalradiation boundary condition and right-hand side in L p (p ≥ 1). Math.Meth. Appl. Sci., 32(2):135166, 2009.[19] P.-É. Druet. Weak solutions to a stationary heat equation with nonlocalradiation boundary condition and right-hand side in L p (p ≥ 1). PreprintNo. 1240, Weierstrass Institute for Applied Analysis and Stochastics(WIAS), Berlin, 2007, revised 2009. Available in pdf format athttp://www.wias-berlin.de/preprint/1240/wias_preprints_1240.pdf.[20] M. Selder, L. Kadinski, Yu. Makarov, F. Durst, P. Wellmann, T. Straubinger,D. Homann, S. Karpov, and M. Ramm. Global numericalsimulation of heat and mass transfer for SiC bulk crystal growth byPVT. J. Crystal Growth, 211:333338, 2000.[21] R. Ma, H. Zhang, V. Prasad, and M. Dudley. Growth kinetics andthermal stress in the sublimation growth of silicon carbide. CrystalGrowth & Design, 2(3):213220, 2002.[22] C. Meyer. Second-order sucient optimality conditions for a semilinearoptimal control problem with nonlocal radiation interface conditions.ESAIM: Optim. Calc. Var., 13:750775, 2007.[23] C. Meyer and I. Yousept. State-constrained optimal control of semilinearelliptic equations with nonlocal radiation interface conditions. SIAMJ. Control Optim., 48:734755, 2009.[24] P. Philip. Transient Numerical Simulation of Sublimation Growth of SiCBulk Single Crystals. Modeling, Finite Volume Method, Results. PhDthesis, Department of Mathematics, Humboldt University of Berlin, Germany,2003. Report No. 22, Weierstrass Institute for Applied Analysis

Conductive-Radiative Heat Transfer 203and Stochastics (WIAS), Berlin. Available in pdf format athttp://www.wias-berlin.de/report/22/wias_reports_22.pdf.[25] O. Klein and P. Philip. Transient numerical investigation of inductionheating during sublimation growth of silicon carbide single crystals.J. Crystal Growth, 247(12):219235, 2003.[26] J. Geiser, O. Klein, and P. Philip. Transient numerical study of temperaturegradients during sublimation growth of SiC: Dependence onapparatus design. J. Crystal Growth, 297:2032, 2006.[27] O. Klein, P. Philip, J. Sprekels, and K. Wilma«ski. Radiation- andconvection-driven transient heat transfer during sublimation growth ofsilicon carbide single crystals. J. Crystal Growth, 222(4):832851, 2001.[28] J. Fuhrmann, Th. Koprucki, and H. Langmach. pdelib: An open modulartool box for the numerical solution of partial dierential equations.Design patterns. In W. Hackbusch and G. Wittum, editors. Proceedingsof the 14th GAMM Seminar on Concepts of Numerical Software, Kiel,January 2325, 1998, University of Kiel, Germany, 2001.[29] J.R. Shewchuk. Triangle: Engineering a 2D Quality Mesh Generator andDelaunay Triangulator. In M.C. Lin and D. Manocha, editors. AppliedComputational Geometry: Towards Geometric Engineering, Vol. 1148of Lecture Notes in Computer Science, pp. 203-222, Springer-Verlag,Berlin, 1996.[30] O. Schenk, K. Gärtner, and W. Fichtner. Scalable parallel sparse factorizationwith left-strategy on shared memory multiprocessor. BIT,40(1):158176, 2000.[31] O. Schenk and K. Gärtner. Solving unsymmetric sparse systems of linearequations with PARDISO. Journal of Future Generation ComputerSystems, 20(3):475487, 2004.[32] O. Klein, Ch. Lechner, P.-E. Druet, P. Philip, J. Sprekels, Ch. Frank-Rotsch, F.-M. Kieÿling, W. Miller, U. Rehse, and P. Rudolph. Numericalsimulation of Czochralski crystal growth under the inuence of atraveling magnetic eld generated by an internal heater-magnet module(HMM). J. Crystal Growth, 310:15231532, 2008. Special Issue:

204 Peter PhilipProceedings of the 15th International Conference on Crystal Growth(ICCG-15) in conjunction with the International Conference on VaporGrowth and Epitaxy and the US Biennial Workshop on OrganometallicVapor Phase Epitaxy, Salt Lake City, USA, August 12-17, 2007.[33] M. Pons, M. Anikin, K. Chourou, J.M. Dedulle, R. Madar, E. Blanquet,A. Pisch, C. Bernard, P. Grosse, C. Faure, G. Basset, andY. Grange. State of the art in the modelling of SiC sublimation growth.Mater. Sci. Eng. B, 61-62:1828, 1999.

Annals of the Academy of Romanian ScientistsSeries on Mathematics and its ApplicationsISSN 2066 - 6594 Volume 2, Number 2 / 2010INTERNAL EXACT OBSERVABILITYOF A PERTURBED EULER-BERNOULLIEQUATION ∗Nicolae Cîndea † Marius Tucsnak ‡AbstractIn this work we prove that the exact internal observability for theEuler-Bernoulli equation is robust with respect to a class of linear perturbations.Our results yield, in particular, that for rectangular domainswe have the exact observability in an arbitrarily small time andwith an arbitrarily small observation region. The usual method of tacklinglower order terms, using Carleman estimates, cannot be appliedin this context. More precisely, it is not known if Carleman estimateshold for the evolution Euler-Bernoulli equation with arbitrarily smallobservation region. Therefore we use a method combining frequencydomain techniques, a compactness-uniqueness argument and a Carlemanestimate for elliptic problems.MSC: 35B37, 93B05, 93B07.keywords: exact observability, Euler-Bernoulli equation, compactnessuniqueness∗ Accepted for publication on July 27, 2010.† Nicolae.Cindea@iecn.u-nancy.fr Institut Elie Cartan, Nancy Université / CNRS /INRIA, BP 70239, 54506 Vandoeuvre-lès-Nancy, France‡ Marius.Tucsnak@iecn.u-nancy.fr Institut Elie Cartan, Nancy Université / CNRS /INRIA, BP 70239, 54506 Vandoeuvre-lès-Nancy, France205

206 Nicolae Cîndea, Marius Tucsnak1 Introduction and main resultsThe internal exact observability of the Euler-Bernoulli plate equation, modelingthe vibrations of elastic plates, is a subject which has been widely tackledin the literature. One of the features differentiating this problem with respectto the corresponding system for the wave equation is that the observabilitytime is arbitrarily small, as it has been first shown in the Appendix 1 of Lions[9]. Much later it has been shown in Miller [10] and Tucsnak and Weiss[13, Section 6.7] that if the wave equation with a given observation region isexactly observable then the Euler-Bernoulli equation with the same region isexactly observable in arbitrarily small time. This holds, in particular, if theobservation region satisfies the geometric optics condition of Bardos, Lebeauand Rauch [1] (see Lebeau [8] for a derivation of this result with no explicitreference to the wave equation). However, this condition is not necessaryfor the exact observability of the Euler-Bernoulli equation. In particular, forrectangular domains, it has been shown in Jaffard [6] and Komornik [7] thatthe exact observability holds for arbitrary open observation domains and inany time.The aim of this work is to study the robustness of the above mentioned observabilityproperties with respect to lower order perturbations of the Euler-Bernoulli equation. These perturbations may contain derivatives of order upto two and coefficients depending on the space variable. In the case in whicha strong version of the geometric optics condition holds such perturbationcan be studied using Carleman estimates for the evolution Euler-Bernoulliequation (see Wang [14]), which are quite appropriate to absorb the lowerorderterms. These Carleman estimates are not available for arbitrarily smallobservation regions so they cannot be used to generalize the results from [6]and [7] to the perturbed plate. Therefore we develop here a general perturbationargument showing that any internal observability result for theEuler-Bernoulli equation is robust with respect to the considered class ofperturbations. This implies, in particular, that for rectangular domains, wehave exact observability with arbitrarily small observation regions.Let Ω ⊂ R n (n ∈ N ∗ ) be an open and nonempty set with a C 2 boundaryor let Ω be a rectangle. We consider the following initial and boundary valueproblem :ẅ(x, t) + ∆ 2 w(x, t) − a∆w(x, t) + b(x) · ∇w(x, t) + c(x)w(x, t) = 0,for (x, t) ∈ Ω × (0, ∞)(1)

Internal exact observability of a perturbed Euler-Bernoulli equation 207w(x, t) = ∆w(x, t) = 0, for (x, t) ∈ ∂Ω × (0, ∞) (2)w(x, 0) = w 0 (x), ẇ(x, 0) = w 1 (x), for x ∈ Ω, (3)where a > 0, b ∈ (L ∞ (Ω)) n , c ∈ L ∞ (Ω), w 0 ∈ H 2 (Ω)∩H 1 0 (Ω) and w 1 ∈ L 2 (Ω).We consider the output given byy(t) = ẇ(·, t)| O , (4)where O is an open and nonempty subset of Ω and a dot denotes differentiationwith respect to the time t:ẇ = ∂w∂t ,ẅ = ∂2 w∂t 2 .For n = 2 the equations (1)-(3) model the vibration of a perturbed Euler-Bernoulli plate with a hinged boundary.The main result of this work is the following theorem :Theorem 1. Let O ⊂ Ω be an open and nonempty subset of Ω such that (1)-(4), with a = 0, b = 0, c = 0, is exactly observable in any time τ > 0. Then,for a = 0 and b = 0 the system (1)-(4) is exactly observable in arbitrarilysmall time for every c ∈ L ∞ (Ω).Moreover, (1)-(4) is exactly observable for every a > 0 and b, c realanalytic functions.Note that, in the case a > 0, the above result gives no information onthe observability time. For rectangular domains we have the following, moreprecise, result.Theorem 2. Assume that n = 2, Ω is a rectangle and let O be an open andnonempty subset of Ω. If b = 0 then (1) − (4) is exactly observable in anytime τ > 0 for every a > 0, c ∈ L ∞ (Ω). Moreover, if b ≠ 0 is an analyticfunction then (1)-(4) is exactly observable in any time τ > 0 for every a > 0and c analytic.To prove the above two theorems, we consider an abstract formulationof our exact observability problem. More precisely, in Section 3 we prove anexact observability result for a linear abstract perturbed system.In Section 5 we prove the Theorem 1 and Theorem 2, applying theabstract results from Section 3. A unique continuation result for the bi-Laplacian is proved in Section 4.

208 Nicolae Cîndea, Marius Tucsnak2 Background on exact observabilityIn this section we recall the definition of the exact observability of an infinitedimensional system and we give a perturbation result for the exact observabilityof a second order infinite dimensional system. In this purpose we needsome notation.Let X and Y be two complex Hilbert spaces which are identified withtheir duals, and let T = (T t ) t≥0 be a strongly continuous semigroup on X,with the generator A : D(A) → X.We consider the following infinite dimensional systemż(t) = Az(t), z(0) = z 0 , (5)y(t) = Cz(t), (6)where C ∈ L(X, Y ) is a bounded linear observation operator.We recall the classical definition of the exact observability.Definition 1. The pair (A, C) is exactly observable in time τ > 0 if thereexists a constant k τ > 0 such that any solution of (5)-(6) satisfies∫ τ0‖Cz(t)‖ 2 Y dt ≥ k 2 τ ‖z 0 ‖ 2 X, (z 0 ∈ X). (7)Let H be a Hilbert space equipped with the norm ‖·‖ H , let A 0 :D(A 0 )→Hbe a self-adjoint, positive and boundedly invertible operator, with compactresolvents and let C 0 ∈ L(H, Y ) be a bounded linear operator. For such anoperator A 0 we denote H α the Hilbert space defined by H α = D(A α 0 ) for anyα ≥ 0 and H −α is the dual space of H α with respect to the pivot space H.We consider the following second-order abstract system :with the output functionẅ(t) + A 2 0w(t) = 0, (8)w(0) = w 0 , ẇ(0) = w 1 , (9)y(t) = C 0 ẇ(t). (10)The system (8)-(10) can be described by a first order system. Indeed, ifwe denotes X = H 1 × H, D(A) = H 2 × H 1 and[ [ ] ([ ] )f g fA : D(A) → X, A =g]−A 2 0 f ∈ D(A) , (11)g

Internal exact observability of a perturbed Euler-Bernoulli equation 209we can write (8)-(9) asż(t) = Az(t), z(0) = z 0 ,[ ] [ ]w(t) w0where z(t) = , zẇ(t) 0 = . The operator A defined above is a skewadjointoperator and, therefore, generates a strongly continuous semigroupw 1(T t ) t on X.Let C ∈ L(X, Y ) be the operator defined by C = [0 C 0 ]. We say that(8)-(10) is exactly observable if the pair (A, C) is exactly observable in thesense of Definition 1.In our recent work [3], we have shown that if (8)-(10) is exactly observablethen the following initial value problemẅ(t) + A 2 0w(t) + aA 0 w(t) = 0 (12)w(0) = w 0 , ẇ(0) = w 1 , (13)is exactly observable with respect to the same output, for every a > 0. Moreprecisely, we proved the following result :Theorem 3. Assume that the system (8)-(10) is exactly observable. Thenthe system (12)-(13) is exactly observable in rapport with the observation(10), i.e., there exist a time τ > 0 and a constant k τ > 0 such that everysolution w of (12)-(13) satisfies∫ τ([ ] )‖C 0 ẇ(t)‖ 2 Y dt ≥ kτ2 (‖w0 ‖ 2 H 1+ ‖w 1 ‖ 2 w0H), ∈ Hw 2 × H 1 . (14)10Moreover, if Ω is a rectangle the observability time τ > 0 can be arbitrarilysmall.3 An exact observability result for second-order perturbedsystemsIn this section we study the exact observability of (12)-(13) with the output(10), perturbed with a term of the form P 0 w, where P 0 ∈ L(H 1−ε , H) andε ∈ (0, 1]. More precisely, we consider the following second-order system :¨v(t) + A 2 av(t) + P 0 v(t) = 0, t > 0 (15)

210 Nicolae Cîndea, Marius Tucsnakv(0) = v 0 , ˙v(0) = v 1 , (16)with the output functiony(t) = C 0 ˙v(t), (17)seen as a perturbation of (12)-(13), where we denote A a : H 1 → H the operatordefined by A a = (A 2 0 + aA 0) 1 2 . It is easy to see that A a is a self-adjoint,strictly positive, boundedly invertible operator, with compact resolvents. Remarkthat (12) can be written, using this notation, asẅ(t) + A 2 aw(t) = 0, t > 0.Let Ãa : H 2 × H 1 → H 1 × H be the operator defined byand denote P =Ã a =[ 0] I−A 2 a 0[ ] 0 0, C = [0 C−P 0 00 ] ∈ L(H 1 × H, Y ). We can considerP ∈ L(H 1 × H). Then A P : H 2 × H 1 → H 1 × H, with A P = Ãa + P , is welldefined. Hence, according to Theorem 1.1 from Pazy [11, p.76], A P is thegenerator of a strongly continuous semigroup in H 1 × H, denoted ( T P )t t≥0 .We denote{ [ ]}w0N (T ) = W 0 = ∈ Hw 1 × H | CT P t W 0 = 0, for any t ∈ [0, T ] . (18)1The aim of this section is to prove that the exact observability of (12)-(13) with the observation (10) implies the exact observability of (15)-(17).The main result of this section is the following theorem :Theorem 4. With the above notations, we assume that (12)-(13), with theobservation (10), is exactly observable in time τ > 0. We assume, moreover,that N (T ) = {0}. If C 0 φ ≠ 0 for every eigenvector φ of A 2 a + P 0 then (15)-(17) is exactly observable in any time T > τ, i.e., there exists a constantk T > 0 such that any solution v of (15)-(16) satisfies∫ T0‖C 0 ˙v(t)‖ 2 Ydt ≥ k 2 T([ ] )(‖v0 ‖ 2 H 1+ ‖v 1 ‖ 2 v0H), ∈ Hv 1 × H .1

Internal exact observability of a perturbed Euler-Bernoulli equation 211Lemma 1. Let G ∈ L(H 1−ε × H −ε , H 1 × H) be the compact operator definedby[ ] A−εa 0G =0 A −ε . (19)aThen, with the assumptions of Theorem 4, there exists a positive constantC T such that∫ T[ ]∥‖C 0 ˙ψ(t)‖ 2 Y dt ≤ CT2 ∥ G w0 ∥∥∥2 ([ ] )w0,∈ Hw 1 w 1 × H , (20)10where ψ is the solution ofH 1 ×H¨ψ(t) + A 2 aψ(t) + P 0 ψ(t) = −P 0 w(t), t ∈ (0, ∞) (21)and w is the solution of (12)-(13).Proof.ψ(0) = ˙ψ(0) = 0 (22)Since C 0 ∈ L(H, Y ) we have the following estimate:‖C 0 ˙ψ‖ C([0,T ];Y ) ≤ ‖C 0 ‖ L(H,Y ) ‖ ˙ψ‖ C([0,T ];H) ≤ ‖C 0 ‖ L(H,Y ) ‖Ψ‖ C([0,T ];H1 ×H),[ ] ψ(t)where Ψ(t) = is the solution of the following initial value problem˙ψ(t)[ ] w(t)˙Ψ(t) = A P Ψ(t) + P , Ψ(0) = 0. (23)ẇ(t)Then, there exists a constant ˜C T > 0 such that∥ [ ]∥ ∥∥∥‖Ψ‖ C([0,T ];H1 ×H) ≤ ˜C ẇ ∥∥∥LT P=w˜C T ‖P 0 w‖ L 1 ([0,T ];H).1 ([0,T ];H 1 ×H)Recall that P 0 ∈ L(H 1−ε , H). Combining this fact with the above two inequalities,we obtain :‖C 0 ˙ψ‖ C([0,T ];Y ) ≤ ˜C T ‖C 0 ‖ L(H,Y ) ‖P 0 ‖ L(H1−ε ,H)‖w‖ L 1 ([0,T ];H 1−ε ). (24)We recall that w is the solution of (12)-(13) and, therefore, we can boundits L 1 norm by the norm of the initial data. We have that[ ]∥‖w‖ L 1 ([0,T ];H 1−ε ) ≤ C w0 ∥∥∥H1−εT ∥. (25)w 1 ×H −ε

212 Nicolae Cîndea, Marius TucsnakUsing the fact that the operator G is an isomorphism from H 1−ε × H −εonto H 1 × H, we can conclude that there exists a constant C T > 0 such that‖C 0 ˙ψ‖ C([0,T ];Y ) ≤ C T∥ ∥∥∥G[w0w 1]∥ ∥∥∥H1×Hand the proof of the lemma is complete.□Proof of Theorem 4. The solution v of (15)-(16) can be written asv = w + ψ, where w is the solution of (12)-(13) and ψ is the solution of(21)-(22). Then we have∫ T∫ T‖C 0 ˙v(t)‖ 2 Y dt + ‖C 0 ˙ψ(t)‖ 2 Y dt ≥ 1 ∫ T‖C 0 ẇ(t)‖ 2 Y dt. (26)002 0Using the exact observability of (12)-(13), we obtain from (26) that thereexists a constant k T > 0 such that∫ T0‖C 0 ˙v(t)‖ 2 Y dt +∫ T0‖C 0 ˙ψ(t)‖ 2 YFrom (27) and Lemma 1 we obtain[ ]∥∫ T0 ‖C 0 ˙v(t)‖ 2 Y dt + C2 T ∥ G w0 ∥∥∥2w 1H 1 ×Hdt ≥ k2 T2 (‖w 0‖ 2 H 1+ ‖w 1 ‖ 2 H). (27)≥ 1 2 k2 T (‖w 0‖ 2 H 1+ ‖w 1 ‖ 2 H ),([ ] )w0for any ∈ Hw 1 × H ,1(28)where G is the operator defined by (19).The idea of the proof is to show that in (28), from Lemma 1, we canremove the term C T ‖GW 0 ‖ 2 H 1 ×Hand thus we obtain the requested observabilityinequality∫ T0‖CT P t W 0 ‖ 2 Y dt ≥ 1 2 k2 T ‖W 0 ‖ 2 H 1 ×H, (W 0 ∈ H 1 × H). (29)We assume that there exists a sequence (W n 0 ) n ∈ H 1 × H such that‖W n 0 ‖ H1 ×H = 1and∫ T0‖CT P t W n 0 ‖ 2 Y dt → 0, when n → ∞,

Internal exact observability of a perturbed Euler-Bernoulli equation 213which contradicts (29). Since the operator G provided by Lemma 1, is compact,we can extract a subsequence of (W n 0 ) n, denoted with the same notation,such thatGW n 0 → W 0 ∈ H 1 × H, when n → ∞,Passing to the limit in (28), we obtainthat isand so, W 0 ≠ 0. Recall thatwhich impliesC T ‖W 0 ‖ 2 H 1 ×H ≥ 1 2 k2 T‖W 0 ‖ 2 H 1 ×H ≥ k2 T2C T> 0,∫ T0‖CT P t W 0 ‖ 2 Y dt = 0,CT P t W 0 = 0, (t ∈ (0, T )).Also, we proved that W 0 ∈ N (T ) and W 0 ≠ 0. This is in contradiction withthe the assumption of the theorem and so the observability inequality (29)is true.□In the case ε = 1 (i.e. P 0 ∈ L(H)), using a compactness and uniquenessargument, we can prove that N (T ) = 0. More precisely, we obtained thefollowing result.Theorem 5. With the notations from the beginning of this section, we assumethat (12)-(13), with the observation (10), is exactly observable in timeτ > 0. If C 0 φ ≠ 0 for every eigenvector φ of A 2 a+P 0 then (15)-(17) is exactlyobservable in any time T > τ, i.e., there exists a constant k T > 0 such thatany solution v of (15)-(16) satisfies∫ T0‖C 0 ˙v(t)‖ 2 Ydt ≥ k 2 T([ ] )(‖v0 ‖ 2 H 1+ ‖v 1 ‖ 2 v0H), ∈ Hv 1 × H .1Proof. To prove this theorem is enough to[show]that N (T ) = {0} andw0to apply Theorem 3. In this purpose, let W 0 = be an element of N (T ).w 1

214 Nicolae Cîndea, Marius TucsnakLet G = (βI − Ãa) −1 ∈ L(H 1 × H, H 2 × H 1 ) be a compact operator for afixed β ∈ ρ(Ãa) ∩ ρ(A P ). The proof of Lemma 1 remains the same for thisG, henceC T ‖GW 0 ‖ 2 H 2 ×H 1≥ 1 2 k2 T ‖W 0 ‖ 2 H 1 ×H. (30)In a first step, we prove that N (T ) is a finite dimensional space, i.e.,the([unit ball])of the space (N (T ), ‖ · ‖ H1 ×H) is compact. In this purpose,w0nletbe a bounded sequence in (N (T ), ‖ · ‖wH1 ×H). Applying the1n n([ ])w0ninequality (30), we obtain thatis bounded in Hw 2 × H 1 . We recall1n nthat [ H 2 × H 1 ⊂ H 1 × H with compact embedding. Therefore, there existsf∈ Hg]2 × H 1 such that[w0n] [w f−→ in Hw 1n g]2 × H 1 and (31)[ ] [w0n f−→ in Hw 1n g]1 × H. (32)From CT t ∈ L(H 1 × H, Y ) and (31) we obtain that[ ] [ ]w0n w fCT t −→ CTw t in H1n g 1 × H.[ fTherefore, ∈ N (T ) and, so, N (T ) is a finite dimensional space.g]In a second step, we show that N (T ) is an invariant subspace of A P .Indeed, it is clear that for δ ∈ (0, T ), if W 0 ∈ N (T ) then T P t W 0 ∈ N (T −δ) foreach 0 < t < δ. Since A P commutes with (βI −A P ) −1 ∈ L(H 1 ×H, H 2 ×H 1 ),we have(βI − A P ) −1 TP t − IW 0 → A P (βI − A P ) −1 W 0 ,when t → 0. Therefore,t(T Pt −ItW 0 is a Cauchy family for the norm W ↦→)t‖(βI −A P ) −1 W ‖ in N (T −δ). From the Remark 2.11.3 in [13] it follows thatthe norms ‖((βI − A P ) −1 W )‖ and ‖(βI − Ãa) −1 W ‖ are equivalent. Hence,we obtain that N (T ) ⊂ D(A P ). Moreover,CT P t W 0 = 0, (t ∈ [0, T ]).

Internal exact observability of a perturbed Euler-Bernoulli equation 215After a differentiation with respect to t, the relation above becomesCT P t A P W 0 = 0, (t ∈ [0, T ]),and, therefore, N (T ) is A P -stable.Finally, we prove that N (T ) = {0}. Assume that N (T ) ≠ {0}. SinceN (T ) is finite dimensional[ ]and A P -stable, then it contains an eigenvector ofw0A P . Let W 0 = ∈ N (T ) be a eigenvector of Aw P . Then exists λ ≠ 0 such1thatA P W 0 = λW 0 ,and, so, w 1 is an eigenvector of A 2 a + P 0 . From the definition of N (T ) weobtain that C 0 w 1 = 0 which contradicts the assumption of Theorem 4 thatC 0 φ ≠ 0 for every eigenvector φ of A 2 a + P 0 .□Remark 1. In [2] the authors proved a result similar with Theorem 4, usinga spectral method completely different to the one presented in this section.4 A unique continuation result for bi-LaplacianThe aim of this section is to prove the following theorem :Theorem 6. Let a ∈ (0, ∞), b ∈ (L ∞ (Ω)) n , c ∈ L ∞ (Ω), µ ∈ R and letu ∈ H 4 (Ω) be a function such thatandThen u = 0 in Ω.∆ 2 u − a∆u + b · ∇u + cu = µ 2 u in Ω (33)u = ∆u = 0 on ∂Ω (34)u = 0 in O. (35)The key of the proof of Theorem 6 is a global Carleman estimate for bi-Laplacian (Proposition 1), which we obtained applying two times a particularcase of the global Carleman estimate proved by Imanuvilov and Puel in [5].Let Ω be an nonempty open set with a C 2 boundary or a rectangle. Lety ∈ H 2 (Ω) ∩ H0 1 (Ω) be the solution of the problem∆y = f, in Ω (36)y = 0, on ∂Ω, (37)

216 Nicolae Cîndea, Marius Tucsnakwhere f ∈ L 2 (Ω). We use the following classic lemma stated in [5], andproved in Fursikov-Imanuvilov [4].Lemma 2. Let O be an open and nonempty subset of Ω. Then there existsa function ψ ∈ C 2 (Ω) such thatψ(x) = 0, x ∈ ∂Ω (38)ψ(x) > 0, x ∈ Ω (39)|∇ψ(x)| > 0, x ∈ Ω \ O. (40)We consider the following weight functionϕ(x) = e λψ(x) , (41)where λ ≥ 1 will be chosen later.Using the definition of the function ϕ, and the properties of function ψgiven by Lemma 2, we have1ϕ(x) = 1e λψ(x) = e−λψ(x) ≤ 1 ≤ e 2λψ(x) = ϕ 2 (x), (42)for all λ ≥ 1.Theorem 7 is a particular case of the Carleman estimate proved in [5] forgeneral elliptic operators.Theorem 7. Assume that (38)-(41) are verified and let y ∈ H 2 (Ω) ∩ H0 1(Ω)be the solution of (36)-(37). Then there exists a constant C > 0 independentof s and λ, and parameters ̂λ > 1 and ŝ > 1 such that for all λ ≥ ̂λ and forall s > ŝ we have∫∫|∇y| 2 e 2sϕ dx + s 2 λ 2 |y| 2 ϕ 2 e 2sϕ dx ≤( 1Csλ 2 ∫ΩΩ|f| 2ϕ e2sϕ dx +Let u ∈ H 4 (Ω) be the solution of the problemwhere g ∈ L 2 (Ω).∫OΩ(|∇y| 2 + s 2 λ 2 ϕ 2 |y| 2) )e 2sϕ dx . (43)∆ 2 u − a∆u = g, in Ω (44)u = ∆u = 0, on ∂Ω, (45)

Internal exact observability of a perturbed Euler-Bernoulli equation 217Proposition 1. Let ψ ∈ C 2 (Ω) be a function such that (38)-(40) are verifiedand let ϕ given by (41). Let u ∈ H 4 (Ω) be a solution of (44)-(45). Thenthere exist ŝ > 1, ̂λ > 1 and a constant C > 0 independent of s ≥ ŝ andλ ≥ ̂λ, such thatsλ 2 ∫Ω+sλ 2 ∫(|∇(∆u)| 2 + s 3 λ 4 |∇u| 2 + s 5 λ 6 |u| 2 ϕ 2) (∫e 2sϕ ≤ COΩ|g| 2ϕ e2sϕ(|∇(∆u)| 2 + s 2 λ 2 ϕ 2 |∆u| 2 + s 3 λ 4 |∇u| 2 + s 5 λ 6 ϕ 2 |u| 2 )e 2sϕ ). (46)Proof. We denote y = ∆u and g 1 = g + a∆u. Then (44) and the lastpart of (45) can be written as∆y = g 1 , in Ω (47)y = 0, on ∂Ω (48)We can apply Theorem 7. Therefore, there exist s 1 > 1, λ 1 > 1 and aconstant C 1 > 0 independent of s and λ such that for all s ≥ s 1 , λ ≥ λ 1 thefollowing estimate is satisfied∫∫sλ 2 |∇y| 2 e 2sϕ dx + s 3 λ 4 |y| 2 ϕ 2 e 2sϕ dx ≤ΩΩ(∫∫C 1 |g 1 | 2 ϕ −1 e 2sϕ (dx + sλ 2 |∇y| 2 + s 3 λ 4 ϕ 2 |y| 2) )e 2sϕ dx ≤ΩO∫C 1(2 (|g| 2 ϕ −1 + a 2 |∆u| 2 ϕ 2 )e 2sϕ dx+Ω∫(sλ 2 |∇y| 2 + s 3 λ 4 ϕ 2 |y| 2) )e 2sϕ dxOwhere bellow we used (42). Replacing y with ∆u in the previous estimate,we obtain∫∫sλ 2 |∇(∆u)| 2 e 2sϕ dx + (s 3 λ 4 − 2a 2 C 1 ) |∆u| 2 ϕ 2 e 2sϕ dx ≤ΩΩ∫∫C 1(2 |g| 2 ϕ −1 e 2sϕ (dx + sλ 2 |∇(∆u)| 2 + s 3 λ 4 ϕ 2 |∆u| 2) )e 2sϕ dx . (49)ΩNow consider the problemO∆u = y, in Ω (50)u = 0, on ∂Ω, (51)

218 Nicolae Cîndea, Marius Tucsnakand apply Theorem 7. Then there exists a constant C 2 > 0, s 2 > 1, λ 2 > 1such that for s ≥ s 2 and λ ≥ λ 2 we have∫∫sλ 2 |∇u| 2 e 2sϕ dx + s 3 λ 4 |u| 2 ϕ 2 e 2sϕ dx ≤ΩΩ(∫∫C 2 |∆u| 2 ϕ −1 e 2sϕ (dx + sλ 2 |∇u| 2 + s 3 λ 4 ϕ 2 |u| 2) )e 2sϕ dx ≤ΩO(∫∫C 2 |∆u| 2 ϕ 2 e 2sϕ (dx + sλ 2 |∇u| 2 + s 3 λ 4 ϕ 2 |u| 2) )e 2sϕ dx , (52)ΩOwhere for the last part of the inequality bellow we used (42).We denote ̂λ = max{λ 1 , λ 2 } and ŝ = max{s 1 , s 2 }. For every s ≥ ŝ,λ ≥ ̂λ, combining (49) and (52) we havesλ 2 ∫Ω|∇(∆u)| 2 e 2sϕ dx+ s3 λ 4 − 2a 2 C 1sλ 2 (|∇u|C 2∫Ω2 + s 2 λ 2 |u| 2 ϕ 2 )e 2sϕ dx(∫−(s 3 λ 4 − a 2 (C 1 ) sλ 2 |∇u| 2 + s 3 λ 4 ϕ 2 |u| 2) )e 2sϕ dx ≤O∫∫C 1(2 |g| 2 ϕ −1 e 2sϕ (dx + sλ 2 |∇(∆u)| 2 + s 3 λ 4 ϕ 2 |∆u| 2) )e 2sϕ dx .ΩOFixing λ in above inequality, is easy to see that there exists a constantC > 0 such that (46) is verified. Therefore, the proof of the proposition iscomplete.□Proof of Theorem 6. The proof is a direct consequence of Theorem 1. Letus denote g = (µ 2 − a)u − b · ∇u ∈ L 2 (Ω). Applying Theorem 1 to (33)-(34)and using (35), we obtainsλ 2 ∫Ω∫(|∇(∆u)| 2 + s 3 λ 4 |∇u| 2 + s 5 λ 6 |u| 2 ϕ 2 )e 2sϕ dx ≤ CΩ|g| 2ϕ e2sϕ dx.We can easily verify that()|g(x)| 2 ≤ 2 µ 4 + ‖a‖ 2 L ∞ (Ω)|u(x)| 2 + 2‖b‖ 2 (L ∞ (Ω)) n|∇u(x)|2 ,(x ∈ Ω).

Internal exact observability of a perturbed Euler-Bernoulli equation 219Combining the above inequality with (42), we obtain∫(sλ 2 |∇(∆u)| 2 + s 4 λ 6 |∇u| 2 + s 6 λ 8 |u| 2 ϕ 2) e 2sϕ dx ≤Ω( ( ) ∫ ∫)2C µ 4 + ‖a‖ 2 L ∞ (Ω)|u| 2 ϕ 2 e 2sϕ dx + ‖b‖ 2 (L ∞ (Ω)) n |∇u| 2 e 2sϕ dx .ΩΩTaking s → ∞ in previous inequality, we easily obtain that u = 0 in Ω.□5 Proof of main resultsThe idea of the proofs of Theorem 1 and Theorem 2 is to apply the abstractresults proven in Section 2. In order to apply Theorem 4 or Theorem 5,we use the unique continuation result for the bi-Laplacian obtained in theprevious section.In the remaining part of this section, A a : H 1 → H denotes the followingoperatorH 1 = H 2 (Ω) ∩ H0 1 (Ω), H = L 2 (Ω),and P 0 ∈ L(H 1 , H)2A a ϕ = (∆ 2 − a∆) 1 2 ϕ, (a > 0, ϕ ∈ H1 )P 0 ϕ = b · ∇ϕ + cϕ, (ϕ ∈ H 1 ),2where a, b, c are as in Theorem 1 and H α are as in Section 2.(1)-(3) can be written asThereforeẅ(t) + A 2 aw(t) + P 0 w(t) = 0, t > 0 (53)w(0) = w 0 , ẇ(0) = w 1 . (54)Let Y = L 2 (O) and let C 0 ∈ L(H, Y ) be the bounded linear operator givenbyC 0 w(t) = w(·, t)| O .We consider the following output functiony(t) = C 0 ẇ(t) (55)

220 Nicolae Cîndea, Marius TucsnakTo prove Theorem 1 or Theorem 2 is nothing else than to prove the exactobservability of (53)-(55), the only difference between the two theorems beingthe exact observability time.Proof of Theorem 1. If we translate the result of Proposition 6 in theoperator notation introduced above, we obtain that{ A20 ψ + P 0 ψ = µ 2 ψC 0 ψ = 0implies ψ = 0.In other words, for every eigenvector ψ of A 2 0 +P 0 we have C 0 ψ ≠ 0. Thereforewe can apply Theorem 4.If b = 0 we can consider P 0 ∈ L(H) and then, applying Theorem 5 inthe case a = 0, we obtain the exact observability of (53)-(55) in a timearbitrarily small for every c ∈ L ∞ (Ω). In the case a > 0, applying Theorem3 and Theorem 4 we obtain the exact observability of (53)-(55) with noinformation about the observability time.If b and c are analytic functions we obtain the same results as in the caseb = 0, using John’s global Holmgren theorem (see for instance Rauch [12,Theorem 1, p.42]) to deduce that N (T ) = {0} for Theorem 4.□Proof of Theorem 2. The only difference between this proof and theprevious one is that we apply here Proposition 5.1 in [3] which give us theobservability of the pair (Ãa, C) in any time τ > 0 and, applying Theorem4, we obtain that (53)-(55) is exactly observable in any time τ > 0. □References[1] C. Bardos, G. Lebeau, and J. Rauch, Sharp sufficient conditions for theobservation, control and stabilization of waves from the boundary, SIAMJ. Control. and Optim., 30 (1992), pp. 1024–1065.[2] N. Cîndea and M. Tucsnak, Fast and strongly localized observation fora perturbed plate equation, in Optimal control of coupled systems ofpartial differential equations, vol. 158 of Internat. Ser. Numer. Math.,Birkhäuser Verlag, Basel, 2009, pp. 73–83.[3] N. Cîndea and M. Tucsnak, Local exact controllability for Berger plateequation, Mathematics of Control, Signals, and Systems (MCSS), 21(2009), pp. 93–110.

Internal exact observability of a perturbed Euler-Bernoulli equation 221[4] A. V. Fursikov and O. Y. Imanuvilov, Controllability of evolution equations,vol. 34 of Lecture Notes Series, Seoul National University ResearchInstitute of Mathematics Global Analysis Research Center, Seoul, 1996.[5] O. Y. Imanuvilov and J.-P. Puel, Global Carleman estimates for weaksolutions of elliptic nonhomogeneous Dirichlet problems, Int. Math. Res.Not., (2003), pp. 883–913.[6] S. Jaffard, Contrôle interne exact des vibrations d’une plaque rectangulaire,Portugal. Math., 47 (1990), pp. 423–429.[7] V. Komornik, On the exact internal controllability of a Petrowsky system,J. Math. Pures Appl. (9), 71 (1992), pp. 331–342.[8] G. Lebeau, Contrôle de l’équation de Schrödinger, J. Math. Pures Appl.(9), 71 (1992), pp. 267–291.[9] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmesdistribués. Tome 1, vol. 8 of Recherches en Mathématiques Appliquées,Masson, Paris, 1988.[10] L. Miller, Controllability cost of conservative systems: resolvent conditionand transmutation, Journal of Functional Analysis, 218 (2005),pp. 425–444.[11] A. Pazy, Semigroups of linear operators and applications to partial differentialequations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.[12] J. Rauch, Partial differential equations, vol. 128 of Graduate Texts inMathematics, Springer-Verlag, New York, 1991.[13] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups,Birkhäuser Advanced Texts / Basler Lehrbücher, BirkhäuserBasel, 2009.[14] Y. H. Wang, Global uniqueness and stability for an inverse plate problem,J. Optim. Theory Appl., 132 (2007), pp. 161–173.

Annals of the Academy of Romanian ScientistsSeries on Mathematics and its ApplicationsISSN 2066 - 6594 Volume 2, Number 2 / 2010SUFFICIENT OPTIMALITYCONDITIONS FOR THEMOREAU-YOSIDA-TYPEREGULARIZATION CONCEPTAPPLIED TO SEMILINEAR ELLIPTICOPTIMAL CONTROL PROBLEMS WITHPOINTWISE STATE CONSTRAINTS ∗Klaus Krumbiegel † Ira Neitzel ‡ Arnd Rösch §AbstractWe develop sufficient optimality conditions for a Moreau-Yosidaregularized optimal control problem governed by a semilinear ellipticPDE with pointwise constraints on the state and the control. We makeuse of the equivalence of a setting of Moreau-Yosida regularization to aspecial setting of the virtual control concept, for which standard secondorder sufficient conditions have been shown. Moreover, we present anumerical example, solving a Moreau-Yosida regularized model problemwith an SQP method.MSC: 49K20, 49M25, 49M29∗ Accepted for publication in revised form on September 1, 2010.† krumbieg@wias-berlin.de Weierstrass Institute for Applied Mathematics andStochastics, Nonlinear Optimization and Inverse Problems, Mohrenstrasse 39, D-10117Berlin;‡ neitzel@math.tu-berlin.de Technische Universität Berlin, Fakultät II - Mathematikund Naturwissenschaften, Str. des 17. Juni 136, D-10623 Berlin§ arnd.roesch@uni-due.de Universität Duisburg-Essen, Department of Mathematics,Forsthausweg 2, D-47057 Duisburg222

Sufficient optimality conditions for Moreau-Yosida-type regularization 223keywords: Optimal control, semilinear elliptic equation, state constraints,regularization, Moreau-Yosida approximation, virtual control, sufficient optimalityconditions1 IntroductionIn this paper we consider the following class of semilinear optimal controlproblems with pointwise state and control constraintsminJ(y, u) := 1 2 ‖y − y d‖ 2 L 2 (Ω) + ν 2 ‖u‖2 L 2 (Ω)⎫Ay + d(x, y) = u∂ nA y = 0in Ωon Γ⎪⎬(P)u a ≤ u(x) ≤ u b a.e. in Ωy(x) ≥ y c (x) a.e. in ¯Ω.⎪⎭The precise assumptions on the given setting are stated in Assumption 1.Due to the nonlinearity of the state equation the above model problem is ofnonconvex type, which makes it necessary to consider sufficient optimalityconditions ensuring local optimality of stationary points. We point out theresults in [7, 8, 9] where second order sufficient conditions were established forsemilinear elliptic control problems with pointwise state constraints. However,it is well known that Lagrange multipliers with respect to pointwisestate constraints are in general only regular Borel measures, cf. [1, 4, 5].The presence of these measures in the optimality system complicates thenumerical treatment of such problems significantly, since a pointwise evaluationof the complementary slackness conditions is not possible. For thatreason, several regularization concepts to overcome this lack of regularityhave been developed in the recent past. We mention for example the penalizationmethod by Ito and Kunisch, [16], Lavrentiev regularization by Meyer,Rösch, and Tröltzsch, [20], as well as interior point methods, cf. [28] and thereferences therein. Special methods have been developed for boundary controlproblems, such as an extension of Lavrentiev regularization by a sourceterm representation of the control, see [31] and [24], and the virtual controlapproach [17]. This approach has been extended to distributed controlsin [10] and turned out to be suitable for problems were control and stateconstraints are active simultaneously. Efficient optimization algorithms are

224 Klaus Krumbiegel, Ira Neitzel, Arnd Röschavailable for all these regularized problems, see section 6 for detailed information.Concerning second order sufficient conditions for Lavrentiev regularizedproblems, we point out the results in [26]. For the Moreau-Yosida regularizationconcept, one can easily see that a classical second order analysis is notpossible due to the fact that the regularized objective function is not twicedifferentiable.However, interpreting a specific setting of the virtual control concept,i.e. φ = 0, as a Moreau-Yosida regularization, we are able to derive a sufficientoptimality condition for the Moreau-Yosida regularization making useof classical second order sufficient conditions for the virtual control concept.This condition ensures local optimality of controls satisfying the first orderoptimality conditions of Moreau-Yosida regularized problems. These resultsare not strictly limited to problem (P). In section 5, we therefore give examplesof problem classes to which the theory can be extended, includingboundary control problems as well as problems governed by parabolic PDEs.2 Assumptions and properties of the state equationWe begin by briefly laying out the setting of the optimal control problem andstating some properties of the problem and the underlying PDE. Throughoutthe paper, we will use the following notation: By ‖ · ‖ we denote the usualnorm in L 2 (Ω), and (·, ·) is the associated inner product. The L ∞ (Ω)-normis specified by ‖ · ‖ ∞ .Assumption 1.• The function y d ∈ L 2 (Ω) and y c ∈ L ∞ (Ω) are given functions andu a < u b , ν > 0 are real numbers.• Ω denotes a bounded domain in R N , N = {2, 3}, which is convex orhas a C 1,1 -boundary ∂Ω.• A denotes a second order elliptic operator of the formAy(x) = −N∑∂ xj (a ij (x)∂ xi y(x)),i,j=1

Sufficient optimality conditions for Moreau-Yosida-type regularization 225where the coefficients a ij belong to C 0,1 ( ¯Ω) with the ellipticity conditionN∑a ij (x)ξ i ξ j ≥ θ|ξ| 2 ∀(x, ξ) ∈ Ω × R N , θ > 0.i,j=1Moreover, ∂ nA denotes the conormal-derivative associated with A.• The function d = d(x, y): Ω ×R is measurable with respect to x ∈ Ω forall fixed y ∈ R, and twice continuously differentiable with respect to y,for almost all x ∈ Ω. Moreover, d yy is assumed to be a locally boundedand locally Lipschitz continuous function with respect to y, i.e. thefollowing Carathéodory type conditions hold true: there exists K > 0such that‖d(·, 0)‖ ∞ + ‖d y (·, 0)‖ ∞ + ‖d yy (·, 0)‖ ∞ ≤ Kand for any M > 0 there exists L M > 0 such that‖d yy (·, y 1 ) − d yy (·, y 2 )‖ ∞ ≤ L M |y 1 − y 2 |for all y i ∈ R with |y i | ≤ M, i = 1, 2.Additionally, we assume that d y (x, y) is nonnegative for almost all x ∈Ω and y ∈ R and positive on a set E Ω ×R, where E Ω ⊂ Ω is of positivemeasure.Under the previous assumptions, we can deduce the following standardresult for the state equation in problem (P):Theorem 1. Under Assumption 1 the semilinear elliptic boundary valueproblemAy + d(x, y) = u in Ω(1)∂ nA y = 0 on Γadmits for every right hand side u ∈ L 2 (Ω) a unique solution y ∈ H 1 (Ω) ∩C( ¯Ω).The proof can be found e.g. in [6]. Based on this theorem, we introducethe control-to-state operatorG: L 2 (Ω) → H 1 (Ω) ∩ C( ¯Ω), u ↦→ y, (2)that assigns to each u ∈ L 2 (Ω) the weak solution y ∈ H 1 (Ω) ∩ C( ¯Ω) of (1).For future reference, we will provide results concerning differentiability of thecontrol-to-state operator, that can be found in, e.g., [30].

226 Klaus Krumbiegel, Ira Neitzel, Arnd RöschTheorem 2. Let Assumption 1 be fulfilled. Then the mapping G: L 2 (Ω) →H 1 (Ω) ∩ C( ¯Ω), defined by G(u) = y is of class C 2 . Moreover, for all u, h ∈L 2 (Ω), y h = G ′ (u)h is defined as the solution ofAy h + d y (x, y)y h = hin Ω∂ nA y h = 0 on Γ.(3)Furthermore, for every h 1 , h 2 ∈ L 2 (Ω), y h1 ,h 2= G ′′ (u)[h 1 , h 2 ] is the solutionofAy h1 ,h 2+ d y (x, y)y h1 ,h 2= −d yy (x, y)y h1 y h2 in Ω(4)∂ nA y h1 ,h 2= 0 on Γ,where y hi = G ′ (u)h i , i = 1, 2.Due to the convexity of the cost functional with respect to the control uand the associated state y = G(u), the existence of at least one solution ofproblem (P) can be obtained by standard arguments, assuming that the setof feasible controls is nonempty. For future references, we define the set ofadmissible controls handling the box constraintsU ad = {u ∈ L 2 (Ω): u a ≤ u ≤ u b a.e. in Ω}. (5)Relying on the standard assumption of a Mangasarian-Fromovitz constraintqualification, sometimes called linearized Slater condition, of the existence ofa control u 0 ∈ U ad and a constant τ > 0 such thatG(ū) + G ′ (ū)(u 0 − ū) ≥ y c + τ (6)for the pure state constraints, we obtain the following first order necessaryoptimality conditions for a locally optimal control ū:Theorem 3. Assume that condition (6) is satisfied, and let ū be a solutionof problem (P) and let ȳ = Gū be the associated state. Then, a regular Borelmeasure ¯µ := ¯µ Ω + ¯µ Γ ∈ M( ¯Ω) and an adjoint state ¯p ∈ W 1,s (Ω), s

Sufficient optimality conditions for Moreau-Yosida-type regularization 227∫(¯p + νū , u − ū) ≥ 0, ∀u ∈ U ad (8)(y c − ȳ)d¯µ = 0, ȳ(x) ≥ y c (x) for all x ∈ ¯Ω¯Ω∫ϕd¯µ ≥ 0 ∀ϕ ∈ C( ¯Ω) + ,(9)¯Ωwhere C( ¯Ω) + is defined by C( ¯Ω) + := {y ∈ C( ¯Ω) | y(x) ≥ 0 ∀ x ∈ ¯Ω}.Here and in the following, A ∗ denotes the formally adjoint operator tothe differential operator A. This result can be obtained adapting the theoryof Casas, cf. [6].With the help of the classical reduced Lagrange functional∫L(u, µ) = J(G(u), u) + (y c − G(u)) dµ,the second order sufficient condition¯Ω∂ 2 L∂u 2 (ū, ¯µ)h2 ≥ α‖h‖ 2 , α > 0, ∀h ∈ L 2 (Ω) (10)guarantees ū to be a local minimum of (P) since the quadratic growth conditionJ(G(u), u) ≥ J(G(ū), ū) + β‖u − ū‖ 2is satisfied for a constant β > 0 for all u ∈ U ad in a sufficiently small L 2 -neighborhood of ū.Remark 1. Condition (10) is a strong second order sufficient condition. Aweaker formulation is possible along the lines of, e.g., [7], but also rathertechnical. Moreover, while weaker conditions are important for theoreticalinvestigations, they are more difficult to verify in numerical computations.A nice presentation of general results can be found in the book of Bonnansand Shapiro, [3, Section 2.3], that also explicitely takes into account possiblynon-unique Lagrange multipliers.3 Regularization approachesThe main focus of this paper is on regularized versions of problem (P). In thissection we present the two regularization approaches we will examine in this

228 Klaus Krumbiegel, Ira Neitzel, Arnd Röschpaper, the Moreau-Yosida approximation on the one hand and the virtualcontrol concept on the other. We will elaborate that the simple version ofMoreau-Yosida regularization is equivalent to a special setting of the virtualcontrol concept.3.1 Moreau-Yosida regularizationThe penalization technique by Ito and Kunisch, [16], based on a Moreau-Yosida approximation of the Lagrange multipliers with respect to the stateconstraints, applied to problem (P), leads to the following family of regularizedproblemsmin J MY (y γ , u γ ) := J(y γ , u γ ) + γ 2Ay γ + d(x, y γ ) = u γ∂ nA y γ = 0∫Ω⎫((y c − y γ ) + ) 2 dx⎪⎬in Ωon Γu a ≤ u γ (x) ≤ u b a.e. in Ω,⎪⎭(P MY )where γ > 0 is a regularization parameter that is taken large. Note, thatthe mapping (·) + denotes the positive part of a measurable function, i.e.(f) + := max{0, f}.Introducing a reduced formulation of problem (P MY ) by the control-to-statemapping G in (2) for the state equation, the following existence theorem canbe proven since the set of admissible controls is nonempty. We refer to, e.g.,[30] for details.Theorem 4. Under Assumption 1, the regularized optimal control (P MY )admits at least one (globally) optimal control ū γ with associated optimal stateȳ γ = G(ū γ ).Due to the nonlinearity of the state equation, the optimal control problemis nonconvex and one has to take into account the existence of multiplelocally optimal controls. Forthcoming, let ū γ be a locally optimal controlof problem (P MY ) with associated state ȳ γ = G(ū γ ). Using the classicalLagrange formulation, straight forward computations yield the following firstorder necessary optimality conditions, cf. [16] for the linear-quadratic setting.

Sufficient optimality conditions for Moreau-Yosida-type regularization 229Proposition 1. Let (ȳ γ , ū γ ) be a locally optimal solution of problem (P MY ).Then, there exists a unique adjoint state ¯p γ ∈ H 1 (Ω) ∩ C( ¯Ω) such that thefollowing optimality system is satisfiedAȳ γ + d(x, y γ ) = ū γ∂ nA ȳ γ = 0A ∗ ¯p γ + d y (x, ȳ γ )¯p γ = ȳ γ − y d − ¯λ γ∂ nA ∗ ¯p γ = 0(11)(¯p γ + νū γ , u − ū γ ) ≥ 0 ∀u ∈ U ad (12)¯λ γ = γ(y c − ȳ γ ) + ∈ L 2 (Ω) (13)Convergence analysis as γ tends to infinity is discussed in [22]. Convergenceresults of the Moreau-Yosida approximation applied to control andstate constrained optimal control problems governed by semilinear parabolicPDEs are derived in [23].3.2 Virtual control conceptIn this section, we will apply the so called virtual control concept, first introducedin [17]. Instead of problem (P), we will investigate a family ofregularized optimal control problems with mixed control-state constraints:min J V C (y ε , u ε , v ε ) := J(y ε , u ε ) + ψ(ε) ⎫2 ‖v ε‖ 2 L 2 (Ω)Ay ε + d(x, y) = u ε + φ(ε)v ε in Ω ⎪⎬∂ nA y ε = 0 on Γ(P V C )u a ≤ u ε (x) ≤ u b a.e. in Ωy ε (x) ≥ y c (x) − ξ(ε)v ε a.e. in Ω,with a regularization parameter ε > 0 and positive and real valued parameterfunctions ψ(ε), φ(ε) and ξ(ε). The remaining given quantities are defined asfor problem (P), see Assumption 1.Denoting a local optimal control of (P) by ū, we point out that the pair(ū, 0) is feasible for all problems (P V C ). Then, using a continuous control-tostatemapping, the existence of at least one pair of optimal controls (ū ε , ¯v ε )can be proven by standard arguments.The existence of regular Lagrange multipliers with respect to mixed controlstateconstraints is known from e.g. [25] and [27], assuming that a constraint⎪⎭

230 Klaus Krumbiegel, Ira Neitzel, Arnd Röschqualification is satisfied. For (P V C ), constraint qualifications are not necessarysince the problem can be transformed into a purely control constrainedproblem with u a ≤ u ε ≤ u b and w := ξ(ε)v ε + y ε ≥ y c , cf. [21] for a Lavrentievregularized problem without constraints on the control u. Based onthis, the following first order necessary optimality conditions for (P V C ) areobtained in a straight forward manner.Proposition 2. Let (ū ε , ¯v ε ) be an optimal solution of (P V C ) and let ȳ ε be theassociated state. Then, there exist a unique adjoint state ¯p ε ∈ H 1 (Ω) ∩ C( ¯Ω)and a unique Lagrange multiplier ¯µ ε ∈ L 2 (Ω) so that the following optimalitysystem is satisfiedAȳ ε + d(x, ȳ ε ) = ū ε + φ(ε)¯v ε∂ nA ȳ ε = 0A ∗ ¯p ε + d y (x, ȳ ε )¯p ε = ȳ ε − y d − ¯µ ε∂ nA ∗ ¯p ε = 0(14)(¯p ε + νū ε , u − ū ε ) ≥ 0, ∀u ∈ U ad (15)φ(ε)¯p ε + ψ(ε)¯v ε − ξ(ε)¯µ ε = 0, a.e. in Ω (16)(¯µ ε , y c − ȳ ε − ξ(ε)¯v ε ) = 0, ¯µ ε ≥ 0, ȳ ε ≥ y c − ξ(ε)¯v ε a.e. in Ω. (17)The convergence of a sequence of regularized optimal controls ū ε to anoptimal solution of the original problem (P) and the uniqueness of dual variableswas discussed in [18].3.3 Equivalence of the conceptsIn this section, we will point out the equivalence of the Moreau-Yosida approximationto a special case of the virtual control concept. More precisely,we will demonstrate that the two optimal control problems admit the sameoptimal controls ū ε = ū γ and we will then call the regularization conceptsand the respective optimal control problems equivalent.We observe the problems (P V C ) for the specific choice φ(ε) ≡ 0, i.e.:min J V C (y ε , u ε , v ε ) := J(y ε , u ε ) + ψ(ε) ⎫2 ‖v ε‖ 2 L 2 (Ω)Ay ε + d(x, y) = u ε in Ω⎪⎬∂ nA y ε = 0 on Γ(Q V C )u a ≤ u(x) ≤ u b a.e. in Ωy ε (x) ≥ y c (x) − ξ(ε)v ε a.e. in Ω,⎪⎭

Sufficient optimality conditions for Moreau-Yosida-type regularization 231As one can easily see, there is no longer a coupling of both control variablesby the state equation of the problem.First, we consider both types of problems (Q V C ) and (P MY ) withoutany notice on the optimality conditions. We start investigating the mixedcontrol-state constraints in (Q V C ) pointwise, where we split the domain Ωinto two disjoint subsets Ω = Ω 1 ∪ Ω 2 :Ω 1 := {x ∈ Ω : y c (x) − y ε (x) < 0 a.e. in Ω}Ω 2 := {x ∈ Ω : y c (x) − y ε (x) ≥ 0 a.e. in Ω}.Initially, we consider Ω 1 . The mixed constraints are given by y c (x)−y ε (x) ≤ξ(ε)v ε (x) a.e. in Ω. Due to the minimization of the L 2 -norm of the virtualcontrol v ε in the objective of (Q V C ), we deriveConsidering Ω 2 , the inequalityhas to be satisfied.deducev ε ≡ 0 a.e. in Ω 1 .ξ(ε)v ε (x) ≥ y c (x) − y ε (x) ≥ 0Choosing the virtual control as small as possible, wev ε = 1ξ(ε) (y c − y ε ) a.e. in Ω 2 .Concluding, the mixed control-state constraints can be replaced by the equationv ε = 1ξ(ε) (y c − y ε ) + .Thus, the optimal control problem (Q V C ) can be rewritten equivalently inthe formmin J(y ε , u ε ) + ψ(ε)2ξ(ε) 2 ‖(y c − y ε ) + ‖ 2 L 2 (Ω)Ay ε + d(x, y ε ) = u ε in Ω∂ nA y ε = 0 on Γu a ≤ u ε (x) ≤ u b a.e. on Ω.Consequently, we formulate the following result.

232 Klaus Krumbiegel, Ira Neitzel, Arnd RöschCorollary 1. For the specific parameter function φ(ε) ≡ 0, the problem(P V C ) is equivalent to the problem (P MY ) arising by the Moreau-Yosida regularization,if the regularization parameter γ > 0 is defined by γ := ψ(ε)ξ(ε) 2 .For the sake of completeness, we will additionally elaborate on the equivalenceby the different first order necessary optimality conditions. Due toProposition 2 and φ(ε) ≡ 0, an optimal control (ū ε , ¯v ε ) of (Q V C ) satisfiesAȳ ε + d(x, ȳ ε ) = ū ε∂ nA ȳ ε = 0A ∗ ¯p ε + d y (x, ȳ ε )¯p ε = ȳ ε − y d − ¯µ ε∂ nA ∗ ¯p ε = 0(18)(¯p ε + νū ε , u − ū ε ) ≥ 0, ∀u ∈ U ad (19)ψ(ε)¯v ε − ξ(ε)¯µ ε = 0, a.e. in Ω (20)(¯µ ε , y c − ȳ ε − ξ(ε)¯v ε ) = 0, ¯µ ε ≥ 0, ȳ ε ≥ y c − ξ(ε)¯v ε a.e. in Ω (21)Since the multiplier ¯µ ε is a regular function, it is well known that the complementaryslackness conditions in (21) are equivalent to¯µ ε − max{0, ¯µ ε + c(y c − ȳ ε − ξ(ε)¯v ε )} = 0for every c > 0. Using the specific choice c = ψ(ε)ξ(ε) 2 , we obtain¯µ ε = max{0, ψ(ε)ξ(ε) 2 (y c − ȳ ε )} = ψ(ε)ξ(ε) 2 (y c − ȳ ε ) + .instead of (20) and (21). Due to (20), the virtual control satisfies¯v ε = ξ(ε)ψ(ε) ¯µ ε = 1ξ(ε) (y c − ȳ ε ) + . (22)By means of Proposition 1, it is easily seen that the optimality systems of(P MY ) and (Q V C ) are equivalent and we conclude with the following result.Corollary 2. Let (ȳ ε , ū ε , ¯v ε ) be a stationary point of (P V C ). If we setφ(ε) ≡ 0, then the virtual control can be represented by ¯v ε = 1/ξ(ε)(y c −ȳ ε ) + .Moreover, (ȳ ε , ū ε ) is also a stationary point of (P MY ) for the specific choiceγ = ψ(ε) . Conversely, a stationary point of (P MY ) is also a stationary pointξ(ε) 2of (P V C ) if the conditions above are satisfied.

Sufficient optimality conditions for Moreau-Yosida-type regularization 2334 Sufficient optimality conditions for the Moreau-Yosida approximationNow we will formulate a sufficient optimality condition for the Moreau-Yosidaapproximation based on a second order sufficient optimality condition forthe respective equivalent virtual control concept (Q V C ). We first define theLagrangian of problem (Q V C ) byL V C (u, v, µ) = 1 2 ‖G(u) − y d‖ 2 + ν 2 ‖u‖2 + ψ(ε)2 ‖v‖2∫+ (y c − G(u) − ξ(ε)v)µ dxΩ(23)using the control-to-state operator G, given in (2). Straight forward computationsshow that the second derivative of the Lagrangian is given by∂ 2 L V C (u, v, µ)∂(u, v) 2 [h 1 , h 2 ] =(G ′ (u)h u,1 , G ′ (u)h u,2 )++ (G(u) − y d , G ′′ (u)[h u,1 , h u,2 ]) + ν(h u,1 , h u,2 )++ ψ(ε)(h v,1 , h v,2 ) − (G ′′ (u)[h u,1 , h u,2 ], µ)(24)for h i = (h u,i , h v,i ) ∈ L 2 (Ω) 2 , i = 1, 2. In the sequel, let (ū ε , ¯v ε ) be a localsolution of (Q V C ) with associated Lagrange multiplier ¯µ ε , i.e. (18)-(21) aresatisfied. We proceed with establishing the second order sufficient optimalitycondition.Assumption 2. There exists a constant α ≥ 0 such thatis valid for all h u ∈ L 2 (Ω).∂ 2 L V C (ū ε , ¯v ε , ¯µ ε )∂(u, v) 2 [h u , h v ] 2 ≥ α‖h u ‖ 2 + ψ(ε)‖h v ‖ 2 (25)Remark 2. Condition (25) can be deduced from the strong second ordersufficient condition (10) for the unregularized problem (P), cf. [18].Remark 3. Note, that the coercivity condition of the second derivative of theLagrangian with respect to directions h v ∈ L 2 (Ω) is satisfied by construction,see (24). Coercivity with respect to directions h u can again be formulatedwith the help of strongly active sets, cf. [7]. However, the strong formulation(25) matches the strong formulation (10) for the unregularized problem.

Sufficient optimality conditions for Moreau-Yosida-type regularization 235Using the SSC of Assumption 2, we obtainL V C (u, v, ¯µ ε ) ≥ J V C (G(ū ε ), ū ε , ¯v ε ) + α‖u − ū ε ‖ 2 + ψ(ε)‖v − ¯v ε ‖ 2( )+ 1 ∂ 2 L V C (ũ,ṽ,¯µ ε)2− ∂2 L V C (ū ε,¯v ε,¯µ ε)(u − ū∂(u,v) 2 ∂(u,v) 2 ε , v − ¯v ε ) 2 .One can easily see that the second derivative (24) is independent of thevirtual control v since the control-to-state operator is only applied to thecontrol variable u and linear mixed control-state constraints are considered.Moreover, one can prove under Assumption 1 that the second derivative ofthe Lagrangian (24) is locally Lipschitz continuous with respect to u, i.e.there exists a positive constant C L such that the estimate( ∂ 2 L V C (u 1 , v, µ)∣ ∂(u, v) 2 − ∂2 L V C ) ∣(u 2 , v, µ) ∣∣∣∂(u, v) 2 h 2 ≤ C L ‖u 1 − u 2 ‖‖h‖ 2holds true for ‖u 1 − u 2 ‖ ≤ δ and δ > 0 sufficiently small, see for instance[30, Lemma 4.24]. By means of the Lipschitz property concerning u and theindependency of v, see (24), we concludeL V C (u, v, ¯µ ε ) ≥ J V C (G(ū ε ), ū ε , ¯v ε ) + α‖u − ū ε ‖ 2 + ψ(ε)‖v − ¯v ε ‖ 2−c‖u − ū ε ‖(‖u − ū ε ‖ 2 + ‖v − ¯v ε ‖ 2 )≥ J V C (G(ū ε ), ū ε , ¯v ε ) + (α − cδ)‖u − ū ε ‖ 2 ++(ψ(ε) − cδ)‖v − ¯v ε ‖ 2 ,provided that ‖u−ū ε ‖ ≤ δ. For sufficiently small δ > 0, we find a positiveconstant β > 0 such that the assertion is fulfilled.Forthcoming, we will rewrite the second order sufficient optimality conditionof problem (Q V C ) in terms of the equivalent Moreau-Yosida regularization(P MY ) using relations between the respective variables derived in theprevious section.Due to Corollary 2, the control ū ε satisfies the first order optimalityconditions (11)-(13) of (P MY ) with γ = ψ(ε)ξ(ε) 2 . Thus, we setū λ = ū ε ,¯λγ = ¯µ ε = ψ(ε)ξ(ε) 2 (y c − ȳ ε ) +

236 Klaus Krumbiegel, Ira Neitzel, Arnd Röschand the SSC (25) of Assumption 2 yields the following‖G ′ (ū γ )h u ‖ 2 + ν‖h u ‖ 2 +(G(ū γ ) − y d , G ′′ (ū γ )h 2 u)−(¯λ γ , G ′′ (ū γ )h 2 u) ≥ α‖h u ‖ 2 (27)for all h u ∈ L 2 (Ω), written in terms of the Moreau-Yosida regularization.Summarizing, one ends up withJ ′′ (G(ū γ ), ū γ )h 2 u − (¯λ γ , G ′′ (ū γ )h 2 u) ≥ α‖h u ‖ 2 .Concluding, we can state the following result.Theorem 5. Let ū γ ∈ U ad , with associated state ȳ γ = G(ū γ ), be a controlsatisfying the first order necessary optimality conditions (11)-(13). Additionally,there exists a constant α > 0 such that for all h u ∈ L 2 (Ω) the followingcondition is fulfilled:J ′′ (G(ū γ ), ū γ )h 2 u − γ((y c − G(ū γ )) + , G ′′ (ū γ )h 2 u) ≥ α‖h u ‖ 2 , (28)i.e. there exists a constant α > 0 such that∫(yh 2 u− ¯p γ d yy (x, ȳ γ )yh 2 u+ νh 2 u) dx ≥ α‖h u ‖ 2 (29)Ωis satisfied for all (h u , y hu ) ∈ L 2 (Ω) × H 1 (Ω) with y hu = G ′ (ū γ )h u , and ¯p γdefined in (11).Then, there exist constants β > 0 and δ > 0 so that the quadratic growthconditionJ MY (G(u γ ), u γ ) ≥ J MY (G(ū γ ), ū γ ) + β‖u γ − ū γ ‖ 2 (30)holds for all u γ ∈ U ad with ‖u γ − ū γ ‖ ≤ δ. In particular, (G(ū γ ), ū γ ) is alocally optimal solution of (P MY ).Proof. Due to Corollary 2 and (22), the pair (ū γ , ¯v γ := 1ξ(ε) (y c − ȳ γ ) + ) satisfiesthe first order optimality conditions (18)-(21) of problem (Q V C ), wherethe parameter functions ψ(ε) and ξ(ε) are chosen in a way such that γ = ψ(ε) .ξ(ε) 2The associated Lagrange multiplier in the optimality conditions is denoted

Sufficient optimality conditions for Moreau-Yosida-type regularization 237by ¯µ γ . Due to the former argumentation, one can easily see, that (28) impliesthe coercivity condition (25) in the point (ū γ , ¯v γ , ¯µ γ ), i.e.∂ 2 L V C (ū γ , ¯v γ , ¯µ γ )∂(u, v) 2 [h u , h v ] ≥ α‖h u ‖ 2 + ψ(ε)‖h v ‖ 2for all h u ∈ L 2 (Ω). Thus, Assumption 2 is satisfied and we proceed byapplying Proposition 26. Hence, there exist constants β > 0 and δ > 0 suchthatJ V C (G(u), u, v) ≥ J V C (G(ū γ ), ū γ , ¯v γ ) + β(‖u − ū γ ‖ 2 + ‖v − ¯v γ ‖ 2 )for all feasible (u, v) of problem (Q V C ) with ‖u − ū γ ‖ ≤ δ. Now, we consideran arbitrary control u ∈ U ad with ‖u − ū γ ‖ ≤ δ. Furthermore, the pair ofcontrols (u, v := 1ξ(ε) (y c − G(u)) + ) is feasible for problem (Q V C ) sinceξ(ε)v = (y c − G(u)) + ≥ y c − G(u).By means of the equivalence of the problems (P MY ) and (Q V C ) and γ = ψ(ε)ξ(ε) 2 ,we deduceJ MY (G(ū γ ), ū γ )=J V C (G(ū γ ), ū γ ¯v γ ) and J MY (G(u), u)=J V C (G(u), u, v).Concluding, we obtain the assertionJ MY (G(u), u) ≥ J MY (G(ū γ ), ū γ ) + β‖u − ū γ ‖ 2for all u ∈ U ad with ‖u − ū γ ‖ ≤ δ.The quadratic growth condition (30) for Problem (P MY ) from the lasttheorem has essentially been proven under condition (28) for the regularizedproblem formulation. In [18], we have deduced second-order sufficient conditionsfor the regularized problem (P V C ) on assumptions on the unregularizedproblem (P) only. By the previously shown equivalence of the two regularizationconcepts the same is true for (P MY ). As an analogue to [18, Theorem4.5] we obtain that (28) also follows from (10):Corollary 3. Let ū fulfill the first order necessary optimality conditions ofTheorem 3 with unique dual variables ¯µ and ¯p, as well as the second ordersufficient condition (10). Then there exists a constant α > 0 such that∫(yh 2 u− ¯p γ d yy (x, ȳ γ )yh 2 u+ νh 2 u) dx ≥ α‖h u ‖ 2Ωis fulfilled for all h u ∈ L 2 (Ω) provided that γ is sufficiently large.

238 Klaus Krumbiegel, Ira Neitzel, Arnd Rösch5 GeneralizationsIn this section we want to point out that the theory presented in this papercan be generalized to large classes of semilinear optimal control problems.Let us start with an elliptic boundary control problem. The virtual controlformulation with φ(ε) = 0 is given bymin J(y ε , u ε , v ε ) := α 12 ‖y ε − y d,Ω ‖ 2 L 2 (Ω) + α 22 ‖y ⎫ε − y d,Γ ‖ 2 L 2 (Γ )+ ν 2 ‖u ε‖ 2 L 2 (Γ ) + ψ(ε)2 ‖v ε‖ 2 L 2 (Ω)⎪⎬Ay ε + d(x, y ε ) = 0 in Ω∂ nA y ε + b(x, y ε ) = u εon Γu a ≤ u ε (x) ≤ u b a.e. in Γy ε (x) ≥ y c (x) − ξ(ε)v ε a.e. in Ω,⎪⎭(Q V C1 )and the corresponding equivalent Moreau-Yosida regularization is presentedbymin J(y γ , u γ , v γ ) := α 12 ‖y γ − y d,Ω ‖ 2 L 2 (Ω) + α 22 ‖y ⎫γ − y d,Γ ‖ 2 L 2 (Γ )+ ν 2 ‖u γ‖ 2 L 2 (Γ ) + γ 2 ‖(y c − y γ ) + ‖ 2 L 2 (Ω)Ay γ + d(x, y γ ) = 0∂ nA y γ + b(x, y γ ) = u γin Ωon Γu a ≤ u γ (x) ≤ u b a.e. in Γ.⎪⎬⎪⎭(P MY1 )The theory presented in section 3 can be adapted by only changing thecorresponding sets. The results of section 4 depend on the dimension of thedomain. For dimension N = 3 we get a two norm discrepancy in the secondorder sufficient optimality condition of proposition 3 in the virtual controlapproach, but only for the original control u. Of course, the correspondingsufficient optimality condition for the Moreau-Yosida regularization in Theorem5 contains a two norm setting, too. Let us mention that in this casesufficient optimality conditions for the unregularized problems are challengingdue to regularity problems. Therefore, Corollary 3 is then not verified byour theory.

Sufficient optimality conditions for Moreau-Yosida-type regularization 239It is also possible to generalize the theory to the regularized version ofparabolic optimal control problems likemin J(y, u) := α 12 ‖y − y d‖ 2 L 2 (Q) + α 22 ‖y(T ) − y ⎫T ‖ 2 L 2 (Ω)+ α 32 ‖y − y Σ‖ 2 L 2 (Σ) + ν 2 ‖u‖2 L 2 (Q)y t + Ay + d(t, x, y) = u in Q = (0, T ) × Ω⎪⎬(P∂ nA y + b(t, x, y) = 0 on Σ = (0, T ) × Γ2 )u a ≤ u(t, x) ≤ u b a.e. in Qy(t, x) ≥ y c (t, x) a.e. in Q,y(0) = y 0 .Due to the weaker differentiability properties of parabolic control-to-stateoperators, a two norm discrepancy will have to be taken into account inproposition 3 and theorem 5 for spatial dimensions greater than one. Similarlyto the elliptic problem, Corollary 3 is then not verified.Moreover, it is possible to discuss more general objectives and nonlinearitiesin the partial differential equations with respect to the control u.However, then the discussion of the differentiability of the control-to-statemapping becomes more involved. In addition, one needs several technicalassumptions on the nonlinearities to get the desired results. Such assumptionsare essentially that ones that were needed for the derivation of sufficientsecond order conditions, see [26]. These discussions go beyond the scope ofthe paper.6 Numerical exampleIn this section, we present a numerical example and motivate how the theoreticalresults shown in this article are used in numerical computations. Weaim at solving the optimal control problemmin J(y, u) := 1 2 ‖y − y d‖ 2 L 2 (Ω) + ν ⎫2 ‖u‖2 L 2 (Ω)∆y + y + y 3 = u + f in Ω ⎪⎬∂ n y = 0 on Γ(PT)u a ≤ u(x) ≤ u b a.e. in Ωy(x) ≥ y c (x) a.e. in ¯Ω,⎪⎭⎪⎭

240 Klaus Krumbiegel, Ira Neitzel, Arnd Röschwith Ω = [0, 1] 2 and Tikhonov regularization parameter ν = 1 · 10 −3 , wherethe remaining data is chosen such that{ū(x) = Π [ua,ub ] − p(x) }νwith u a = 150 and u b = 850 is an optimal control with associated optimalstate ȳ, adjoint state p, and Lagrange multiplier µ, given byȳ(x) = −16x 4 1 + 32x 3 1 − 16x 2 1 + 1,p(x) = 2x 3 1 − 3x 2 1,µ(x) = max{0, ȳ(x 1 = 0.2) − ȳ(x)}.It can be verified that this is obtained withy c (x) = min{ȳ(x 1 = 0.2), ȳ(x)},f = −∆ȳ + ȳ + ȳ 3 − ū,y d = ∆p − p − 3ȳ 2 p + ȳ − µ.The second order sufficient conditions are also satisfied, which is easily provenby computing −p(x)d yy (x, ȳ(x)) ≥ 0 on [0, 1] 2 , that guarantees (29). Notice,that the active sets associated to the pure state constraints and active set correspondingto the control constraints are not disjoint, so that regularizationby the virtual control approach is reasonable.We solve this problem with the help of the Moreau-Yosida regularizationapproach, i.e. the virtual control approach with φ = 0, and denote the regularizedproblem by (PT MY ). We apply an SQP method, cf. for instance[15] and [29]. We point out that a key argument in the proof of convergenceof SQP methods are second order sufficient conditions, which are now guaranteedfor the Moreau-Yosida regularized problem, and it is reasonable toinvestigate the convergence behavior of the solution algorithm.For completeness, let us mention that a primal-dual active set strategyis used for solving the linear quadratic subproblems, see e.g. [2, 11, 12,19] and the references therein. Moreover, all functions are discretized bypiecewise linear ansatz functions, defined on a uniform finite element mesh.The number of intervals in one dimension, denoted by N, is related to themesh size by h = √ 2N. In the following all computations are performed withN = 192. The Figures 1-4 show the numerical solution of the Moreau-Yosida

Sufficient optimality conditions for Moreau-Yosida-type regularization 241Figure 1: Control u γFigure 2: State y γapproximation of problem (PT) for the fixed penalization parameter γ =1 · 10 5 . In Figure 4 one can see irregularities of the multiplier approximationon the boundary and in the parts of the domain, where the active sets of theoriginal problem (PT) associated to the different constraints are not disjoint.We obtain the following error of the numerical solution of problem (PT MY ):‖u γ −ū‖ ≈ 3.1426e−02, ‖y γ −ȳ‖≈2.7497e−05, ‖p γ −p‖ ≈ 1.5147e−04. (31)The convergence behavior of the SQP method is presented in Table 1. Wedisplay the value of the cost functional J MY for each step of SQP algorithmas well as the relative difference between two iterates, which is defined byδ γ = 1 3(‖u (n)γ− u (n+1)γ ‖‖u (n+1)γ ‖+ ‖y(n) γ− y γ(n+1) ‖‖y γ(n+1) ‖+ ‖p(n) γ− p (n+1)γ‖p γ(n+1) ‖)‖.This quantity is used for a termination condition of the SQP method. In allnumerical tests the algorithm stops if δ < 1 · 10 −6 . In addition the numberof iterations of the primal-dual active set strategy is shown.We also test the regularization algorithm for increasing regularization parameters,noting that convergence of ū γ towards ū is discussed in, e.g., [22]. Wemention Hintermüller and Kunisch in [14, 13], where path-following methodsassociated to the Moreau-Yosida regularization parameter are developed. Inthis numerical test, we use only a simple nested approach: the numerical solutionof the problem is taken as the starting point for the SQP-method withrespect to the next regularization parameter. The convergence behavior forincreasing regularization parameters γ is displayed in Table 2. As expected,

242 Klaus Krumbiegel, Ira Neitzel, Arnd Röschit SQP J MY δ γ #it AS1 1.497214e + 02 1.414707e + 00 132 1.766473e + 02 3.585474e − 01 323 1.767212e + 02 4.605969e − 02 124 1.767218e + 02 7.115167e − 04 65 1.767218e + 02 2.087841e − 07 1Table 1: Convergence of SQP-method for (PT MY )Figure 3: Adjoint state p γFigure 4: Approximation of Lagrangemultiplier λ γthe errors ‖ū γ − ū‖ and ‖ȳ γ − ȳ‖ are decreasing for increasing parameters γ.Moreover, an influence of the discretization error is visible in the differenceof the controls.7 ConclusionsIn this article, we have investigated the well-known Moreau-Yosida regularizationconcept for state-constrained optimal control problems governed bysemilinear elliptic equations with respect to sufficient optimality conditions.These are important in numerous ways such as convergence of numericalalgorithms, stability with respect to perturbations, and also discretizationerror estimates, and hence play an essential role in the analysis of numericalmethods for nonlinear optimal control problems. For the Moreau-Yosida

Sufficient optimality conditions for Moreau-Yosida-type regularization 243γ ‖ū γ − ū‖ ‖ȳ γ − ȳ‖ #it SQP #it AS20 5.114051e − 01 1.495545e − 02 8 3440 3.161757e − 01 7.893723e − 03 3 580 1.853852e − 01 4.056569e − 03 2 3160 1.059317e − 01 2.053420e − 03 2 3320 6.072430e − 02 1.030834e − 03 2 3640 3.559594e − 02 5.152590e − 04 2 21280 2.176439e − 02 2.568771e − 04 2 32560 1.442802e − 02 1.277243e − 04 2 25120 1.093492e − 02 6.323253e − 05 2 210240 9.458933e − 03 3.104869e − 05 2 220480 8.883362e − 03 1.500880e − 05 2 240960 8.656709e − 03 7.063147e − 06 2 3Table 2: Convergence of (PT MY )regularization, a standard second order analysis is not possible, since the regularizedobjective function is not twice differentiable. However, by the equivalenceof the Moreau-Yosida regularization to a specific setting of the virtualcontrol concept, we were able to bypass these restrictions. As a byproduct ofour analysis, we obtained that a sufficient condition for the Moreau-Yosidaregularization can be deduced from an SSC for the unregularized problem.References[1] J.-J. Alibert and J.-P. Raymond. Boundary control of semilinear ellipticequations with discontinuous leading coefficients and unboundedcontrols. Numer. Funct. Anal. and Optimization, 3 & 4:235–250, 1997.[2] M. Bergounioux, K. Ito, and K. Kunisch. Primal-dual strategy for constrainedoptimal control problems. SIAM J. Control and Optimization,37:1176–1194, 1999.[3] J.F. Bonnans and A. Shapiro. Perturbation Analysis of OptimizationProblems. Springer, New York, 2000.

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ANNALS OF THE ACADEMYOF ROMANIAN SCIENTISTSSeries on MATHEMATICS AND ITS APPLICATIONSCONTENTSMihail MEGAN, Codruța STOICAConcepts of dichotomy for skew‐evolution semiflows in Banach spaces .......................................125Vasile DRĂGAN, Toader MOROZANRobust stability and robust stabilization of discrete‐time linear stochastic systems .....................141Peter PHILIPAnalysis, optimal control, and simulation of conductive‐radiative heat transfer .........................171Nicolae CÎNDEA, Marius TUCSNAKInternal exact observability of a perturbed Euler‐Bernoulli equation of arbitrary order ................205Klaus KRUMBIEGEL, Ira NEITZEL, Arnd RÖSCHSufficient optimality conditions for the Moreau‐Yosida‐typeregularization concept applied to semilinear elliptic optimalcontrol problems with pointwise state constraints.........................................................................222ISSN 2066 – 6594