Abstracts - Facultatea de Matematică

Faculty of Mathematics 

University “Al. I. Cuza” of Ias¸i 

Institute “Octav Mayer” of the 

Romanian Academy, Ias¸i, Romania 

International Conference on Applied Analysis 

and Differential Equations 

Ias¸i, Romania 

September 4–9, 2006 

ABSTRACTS

Invited Speakers 

Variational Methods in Metric Regularity and Implicit 

Multifunction Theorems 

D. AZÉ 

UMR CNRS MIP, Université Paul Sabatier 

118 Route de Narbonne, 31062 Toulouse cedex 4, France 

aze@mip.ups-tlse.fr 

Relying on the notion of strong slope introduced by De Giorgi, Marino and Tosques 

in the eighties, we develop a general framework leading to several characterizations of 

metric regularity of multifunctions in the complete metric space setting. We focus our 

attention on parametric metric regularity, leading to implicit multifunction theorems. 

We give several results on existence and regularity of the implicit multifunctions, and, 

surveying some recent results on this topic, we show how they can be derived from 

our general method. We also give an exact formula for the derivative of the implicit 

multifunction. At last, some applications are given to differential inclusions and to 

stability in nonsmooth analysis. 

Stochastic Taylor expansion and stochastic viscosity solutions 

for nonlinear SPDEs 

Rainer BUCKDAHN 

Université de Bretagne Occidentale, UFR Sciences et Techniques 

Laboratoire de Mathématiques, CNRS - UMR 6204 

buckdahn@univ-brest.fr 

In an earlier work R.Buckdahn and J.Ma (2001) introduced a notion of stochastic 

viscosity solution, inspired by earlier results of P.L.Lions and P.E.Souganidis (1998). By 

using a Doss-Sussmann-type transformation and the so-called backward doubly stochastic 

differential equations (BDSDEs) introduced by E.Pardoux and S.Peng (1994), they 

established the existence and uniqueness of stochastic viscosity solution to the stochastic 

partial differential equation (SPDE) 

u(t, x) = u(0, x) + � t 

g(t, x, u(t, x))dBt, 

0 (Au + f(x, u, σ∗∇u) (t, x)dt + � t 

0 

where B is a Brownian motion and A is the generator of a diffusion process. Our 

contribution extends the study of stochastic viscosity solutions to the class of SPDEs 

1

for which the dependence of the diffusion coefficient g on the solution u is replaced by 

that on its the spatial derivative ∇u . 

Our main tool is a new type of stochastic time-space Taylor expansion for Itô random 

fields which holds outside some universal null set N for every random expansion point. It 

generalizes the work on stochastic Taylor development by R. Buckdahn and J. Ma (2002) 

The second principal tool is the recently developed theory on BDSDEs (E. Pardoux, 

S. Peng, 1994). It is mainly used to prove existence of the stochastic viscosity solution. 

The talk is based on a common with with I. Bulla (Université de Brest, France) and 

J. Ma (Purdue University, U.S.A). 

Radius Theorems and Conditioning 

A. L. DONTCHEV 

Mathematical Reviews, Ann Arbor, Michigan 48107-8604, USA 

ald@ams.org 

As known from linear algebra, the distance from a nonsingular matrix A to the set of 

singular matrices is equal to the reciprocal of the norm of the inverse of A. The standard 

condition number of a matrix is then just the normalized reciprocal to the distance to 

singularity. This result is sometimes called the Eckart–Young theorem but the actual 

paper by Eckart and Young from 1936 is only indirectly related to the issue. In this talk 

we will take a quick tour through basic ideas that lead to expressions for the distances 

to irregularity of mappings. Then we outline generalizations in various directions and 

pose some open problems. 

On weak solutions of BSDEs 

Hans-Jürgen ENGELBERT 

Institute for Stochastics, Friedrich Schiller-University 

Jena, GERMANY 

engelbert@minet.uni-jena.de 

This talk is based on joint work with R. Buckdahn (Brest, France) and A. Ră¸scanu 

(Ia¸si, Romania) (cf. [1]–[4]). The main objective consists in introducing and discussing 

the concept of weak solutions of certain backward stochastic differential equations (BS- 

DEs): 

� � � 

T 

� 

Yt = E H(X) + f(s, X, Y )ds� 

� 

t 

Ft 

� 

, t ∈ [0, T ]. (1) 

condition H depends in functional form on a driving càdlàg process X, and the coefficient 

f (called the generator or driver of the BSDE) depends on time t and, in functional 

form, on X and the solution process Y . The functional f(t, x, y), (t, x, y) ∈ [0, T ] × 

D � [0, T ]; Rd+p� , is assumed to be bounded and continuous in (x, y) on the Skorohod 

2

space D � [0, T ]; R d+p� in the Meyer–Zheng topology. Using weak convergence in the 

Meyer–Zheng topology, we shall give a general result on the existence of a weak solution 

Y , with driving process X admitting a given distribution, defined on some filtered 

probability space (Ω, F, P ; F). By examples, we can show that there are, indeed, weak 

solutions which are not strong, i.e., are not solutions in the usual sense adapted to the 

filtration F X generated by X. We will also discuss pathwise uniqueness and uniqueness 

in law of the solution and conclude, similar to the Yamada–Watanabe theorem, that 

pathwise uniqueness and weak existence ensure the existence of a (uniquely determined) 

strong solution. Applying these concepts, finding a unique strong solution is divided into 

two subtasks: To prove pathwise uniqueness and to prove weak existence for BSDE (1). 

It turns out that pathwise uniqueness holds whenever every weak solution of BSDE (1) 

has a.s. continuous paths, and this condition is even necessary if the driving process X 

is a continuous local martingale satisfying the previsible representation property. 

References: 

[1] Buckdahn, R.; Engelbert, H.-J.; Ră¸scanu, A., On weak solutions of backward 

stochastic differential equations, Theory Probab. and its Appl. Vol. 49, No.1, 70–108 

(2004). 

[2] Buckdahn, R.; Engelbert, H.-J., A backward stochastic differential equations 

without strong solution, Theory Probab. and its Appl. Vol. 50, No.2 (2005). 

[3] Buckdahn, R.; Engelbert, H.-J., On the notion of weak solutions of backward 

stochastic differential equations, pp. 21, to appear in: Proceedings of the Fourth Colloquium 

on Backward Stochastic Differential Equations and Their Applications, Shanghai, 

P.R. China, May 29 – June 1, 2005. 

[4] Buckdahn, R.; Engelbert, H.-J., On the continuity of weak solutions of backward 

stochastic differential equations, manuscript, pp. 13, submitted. 

Regular and singular optimal controls 

H. O. FATTORINI 

University of California, Department of Mathematics 

Los Angeles, California 90095-1555 

hof@math.ucla.edu 

Let E be a Banach space, S(t) a strongly continuous semigroup in E. We deal 

mostly with the time optimal control problem for 

y ′ (t) = Ay(t) + u(t), y(0) = ζ (1) 

with point target condition y(T ) = y and control constraint �u(t)� = 1 a. e. A 

multiplier space Z is a space Z ⊃ E ∗ such that S(t) ∗ Z ⊆ E ∗ (t > 0). Pontryagin’s 

maximum principle for a control u(t) in 0 ≤ t ≤ T is 

〈S(T − t) ∗ z, u(t)) = max 

�u�≤1 〈S(T − t)∗ z, u〉 a.e. in 0 = t = T (2) 

3

with z in some multiplier space. Time optimal controls for (1) are divided into five 

classes determined by their adherence to the maximum principle (or lack thereof). A 

control u(t) satisfying (2) with z ∈ E ∗ is strongly regular. If z belongs to the space of 

multipliers Z(T ) ⊃ E ∗ characterized by 

� T 

�S(t) ∗ z�dt < 8 

0 

the control is regular. If z belongs to an arbitrary multiplier space, the control is weakly 

singular. If u(t) does not satisfy (2) for z in any multiplier space then u(t) is singular. 

Finally, a control not satisfying (2) in any subinterval [a, b] ⊆ [0, T ] is hypersingular. 

Enough examples are known to show that each class is nonempty and there are also 

results relating time optimality and membership in various classes. For instance: (a) 

if u(t) is time optimal and y ∈ D(A) then u(t) is regular, (b) regularity implies time 

optimality, (c) every time optimal control is weakly singular if E is a Hilbert space and 

S(t) is analytic, (d) weak singularity does not imply time optimality, although there are 

time optimal controls that are only weakly singular (not regular). This talk will be on 

what is known (and also what is not known) about each class of controls and on their 

relations. For instance, it is known that the condition y ∈ D(A) on the target does not 

guarantee strong regularity, but the only available counterexample lives in a nonsmooth 

space E (the sequence space ℓ 1 ) while it may be true that y ∈ D(A) implies strong 

regularity for a self adjoint infinitesimal generator A in a Hilbert space. 

Analyticity of stable invariant manifold and stabilization 

problem for semilinear parabolic equation 

A. V. FURSIKOV 

Department of Mechanics and Mathematics 

Moscow State University, 119992 Moscow, RUSSIA 

fursikov@mtu-net.ru 

In a bounded domain Ω ⊂ R d , d ≤ 3 we consider semilinear parabolic equation 

∂ty(t, x) − ∆y + f(y) = h(x), y|∂Ω = 0, y|t=0 = y0(x) (1) 

with analytic function f(y) = � ∞ 

k=1 fky k where |fk| ≤ γ0ρ k 0 and h(x) ∈ L2(Ω). We 

consider (1) in phase space V = H 2 (Ω) ∩ H 1 0(Ω). Let �y(x) ∈ V be an unstable steadystate 

solution of (1). Then orthogonal decomposition V = V+ ⊕ V− holds where finitedimensional 

subspace V+ is generated by eigenvectors of the operator Az = −∆z(x) + 

f ′ (�y(x))z(x), z∂Ω = 0 with non negative eigenvalues and V− is generated by eigenvectors 

with negative eigenvalues. 

Denote resolving operator of (1) by St(y0) : St(y0) ≡ y(t). Recall that stable 

invariant manifold M−(�y) of (1) is the manifold defined in an neigborhood of �y such that 

for y0 ∈ M−(�y) inclusion St(y0) ∈ M−(�y) is true for all t > 0, and �St(y0) − �y�V ≤ ce −αt 

4

as t → ∞ with some positive c, α. It is known that M−(�y) = �y + {y− + F (y−), y− ∈ 

O(V−)} where O(V−) is a neigborhood of origin in V− and F : O(V−) → V+ is a nonlinear 

operator. 

Theorem 1 The operator F (y−) is analytic: F (y−) = �∞ r=2 Fr(y−, . . . , y−), where 

Fr(y1, . . . , yr) : V− × · · · × V− → V+ are r-linear operators, and �Fr� ≤ γρr with 

γ > 0, ρ > 0 depending on γ0, ρ0. 

Theorem 1 has been applied for numerical stabilization of the semilinear equation 

from (1) with help of approach proposed in [1]. This stabilization is local since it was 

made under addition assumption that ��y − y0� is small enough. In the case of onedimensional 

Ω Theorem 1 has been obtained in [2]. 

The following result can be applied in the case of unlocal stabilization problem for 

equation from (1). 

Theorem 2 For each R > 0 there exist K > 0 and subspace VK ⊂ V− of codimension 

K such that restriction of F (y−) on VK can be extended analytically on the ball {y− ∈ 

VK : �y−�V− ≤ R} 

References: 

[1] A.V. Fursikov, Stabilizability of quasilinear parabolic equation by feedback boundary 

control, Matematicheskii Sbornik 192:4 (2001), 115-160 (in Russian); Sbornik: 

Mathematics, 192:4 (2001),593-639. 

[2] A.V. Fursikov, Analyticity of stable invariant manifold of 1D-semilinear parabolic 

equation. Proceedings of Joint Summer Research Conference on Control Methods in 

PDE-Dynamical Systems. AMS (to appear) 

The Complex Dynamics of Age-Structured Populations 

Mimmo IANNELLI 

Mathematics Department, University of Trento 

iannelli@science.unitn.it 

Modeling of age-structured populations is governed by the Gurtin–MacCamy system 

that takes into account age-structure and nonlinear effects on the vital rates. The 

framework of this problem allows to treat ecological mechanisms for the intra-specific 

interaction such as juvenile-adult competition, Allee effect, cannibalism. It is known 

that the models that take into account age-structure are related to delay equations 

(distributed delay, but also concentrated delay) and that stability of steady states is 

governed by transcendental characteristic equations that are not easy to analyze analytically. 

Thus numerical methods for the analysis of such characteristic equations are 

a powerful tool for exploring the behavior of the models, tracing asymptotical stability, 

Hopf bifurcations and possible chaos. Recent numerical approaches for characteristic 

roots of delay differential are based on the discretization of either the associated solution 

operator semigroup or its infinitesimal generator whose spectra are related to the 

characteristic roots. The idea is to turn the characteristic roots approximation problem 

into a corresponding eigenvalue problem for a suitable matrix. This approach has 

5

een applied to the specific case of the Gurtin–MacCamy model in order to explore 

its behaviour versus some significant parameters. In this talk we present the complex 

dynamics of the Gurtin–MacCamy model, the numerical method set up to locate the 

characteristic roots, the results we can draw on the behaviour of specific models. 

Variational analysis of evolution and partial differential 

inclusions 

Boris S. MORDUKHOVICH 

Department of Mathematics, Wayne State University 

Detroit, Michigan 48202, USA 

boris@math.wayne.edu 

This talk is devoted to optimal control problems governed by evolution/differential 

inclusions in infinite-dimensional spaces and also by semilinear partial differential inclusions. 

We pursue a twofold goal: to develop the method of discrete approximations 

for such problems and to derive necessary optimality conditions of the Euler-Lagrange 

type under natural assumptions. First we consider a generalized Bolza problems for 

infinite-dimensional differential inclusions with endpoint constraints. One of the principal 

differences between finite-dimensional and infinite-dimensional dynamic systems is 

the lack of compactness in infinite dimensions. Constructing well-posed discrete approximations 

and using advanced tools of variational analysis and generalized differentiation, 

we derive necessary conditions for discrete-time problems and then, by passing to the 

limit, for continuous-time evolution inclusions. A similar procedure is developed for optimal 

control problems of the Mayer type governed by constrained semilinear inclusions 

with unbounded operators generating compact semigroups. This particularly covers 

parabolic partial differential inclusions whose solutions are understood in the conventional 

mild sense. 

Slant Differentiability and Semismooth Methods for Operator 

Equations 

Zuhair NASHED 

Department of Mathematics, University of Central Florida 

Orlando, FL 32816, USA 

znashed@mail.ucf.edu 

Let X and Y be Banach spaces, and F : D ⊂ X → Y be a continuous mapping on an 

open domain D. The following concepts of slant differentiability and slanting function 

were introduced in [1]. A function F : D ⊂ X → Y is said to be slantly differentiable 

at x ∈ D if there exist a mapping f ◦ : D → L(X, Y ) such that the family f ◦ (x + h) 

6

of bounded linear operators is uniformly bounded in the operator norm for h sufficiently 

small and 

F (x + h) − F (x) − f 

lim 

h→0 

0 (x + h)h 

= 0. 

� h � 

The function f ◦ is called a slanting function for F at x. A function F : D ⊂ 

X → Y is said to be slantly differentiable in an open domain D0 ⊂ D if there exists 

a mapping f 0 : D → L(X, Y ) such that f ◦ is a slanting function for F at every point 

x ∈ D0. In this case, f ◦ is called a slanting function for F in D0. 

In this talk I will discuss operator-theoretic aspects of slant differentiability and related 

variants. Application to Newton-like methods, optimal control theory, and nonlinear 

ill-posed problems will be indicated. A unifying framework for semismooth analysis 

will be sketched and compared with the setting in [2] for smooth analysis. 

References: 

[1] X. Chen, Z. Nashed, and L.Qi, Smoothing methods and semismooth methods 

for nondifferentiable operator equations, SIAM J. Numer. Anal. 38 (2000), no. 4, 

1200-1216. 

[2] M. Z. Nashed, Differentiability and related properties of nonlinear operators: 

Some aspects of the role of differentials in nonlinear functional analysis, in L.B. Rall, 

ed., ”Nonlinear Functional Analysis and Applications”, Academic Press, New York, 

1971, pp. 103-309. 

Homogenization of periodic linear degenerate PDEs 

Étienne PARDOUX 

Université de Provence, Marseille 

pardoux@latp.univ-mrs.fr 

Our goal is to study, by a probabilistic method, the limit as ε → 0 of the solution 

uε (t, x) of an elliptic PDE in the regular bounded domain D ⊂ Rd ⎧ 

⎨ Lεu 

⎩ 

ε � 

(x) + f x, x 

� 

u 

ε 

ε (x) = 0, x ∈ D, 

u ε (x) = g(x), x ∈ ∂D, 

where f is bounded from above, and g is continuous, as well as the limit of uε (t, x), the 

solution of a parabolic PDE of the form 

⎧ 

⎨ ∂u 

⎩ 

ε (t, x) 

= Lεu 

∂t 

ε � 

1 

(t, x) + 

ε e(x 

x 

) + f(x, 

ε ε ) 

� 

u ε (t, x) 

u ε (0, x) = g(x), x ∈ R d , 

where 

Lε = 1 

2 

d� 

i,j=1 

aij( x 

ε ) 

∂2 + 

∂xi∂xj 

7 

d� 

i=1 

� 

1 x x 

bi( ) + ci( 

ε ε ε ) 

� 

∂ 

. 

∂xi

Homogenization, in particular in the periodic case, is an old problem. We refer to [3] 

and [8] for many results in this field, to [7] for the first presentation of the probabilistic 

approach to this problem (which will be our point of view), and to [9] for the modern 

PDE approach to homogenization. 

The novelty of our result lies in the fact that we allow the matrix a to degenerate 

(and even possibly to vanish) in some open subset D of R d . There is by now quite a vast 

literature concerning the homogenization of second order elliptic and parabolic PDEs 

with a possibly degenerating matrix of second order coefficients a, see among others [1], 

[2], [4], [6], [13]. But, as far as we know, in all of these works, either the coefficient a is 

allowed to degenerate in certain directions only, or else it may vanish on sets of measure 

zero only. It seems that our paper presents the first result where the matrix a is allowed 

to vanish on an open set. 

References: 

[1] R. De Arcangelis, F. Serra Cassano, On the homogenization of degenerate elliptic 

equations in divergence form, J. Math. Pures Appl. 71, 119–138, 1992. 

[2] M. Bellieud, G. Bouchitté, Homogénéisation de problèmes elliptiques dégénérés, 

CRAS Sér. I Math. 327, 787–792, 1998. 

[3] A. Bensoussan, J.L. Lions, G. Papanicolaou, Asymptotic analysis for periodic 

structures, Studies in Mathematics and its Applications, 5, North-Holland Publishing 

Co., Amsterdam-New York, 1978. 

[4] M. Biroli, U. Mosco, N. Tchou, Homogenization for degenerate operators with 

periodical coefficients with respect to the Heisenberg group, CRAS Sér. I Math. 322, 

439–44, 1996. 

[5] A. Diedhiou, É. Pardoux, Homogenization of semilinear hypoelliptic PDEs, Preprint. 

[6] J. Engström, L.–E. Persson, A. Piatnitski, P. Wall, Homogenization of random 

degenerated nonlinear monotone operators, Preprint. 

[7] M.I. Freidlin, The Dirichlet problem for an equation with periodic coefficients 

depending on a small parameter. (Russian) Teor. Verojatnost. i Primenen. 9, 133–139, 

1964. 

[8] V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of differential operators 

and integral functionals, Springer Verlag, 1994. 

[9] F. Murat, L. Tartar, Calculus of variations and homogenization, in Topics in the 

mathematical modelling of composite materials, 139–173, Progr. Nonlinear Differential 

Equations Appl., 31, Birkhäuser Boston, Boston, MA, 1997. 

[10] É. Pardoux, Homogenization of Linear and Semilinear Second Order Parabolic 

PDEs with Periodic Coefficients: A Probabilistic Approach, Journal of Functional Analysis 

167, 498-520, 1999. 

[11] É. Pardoux, A. Yu. Veretennikov, On Poisson equation and diffusion approximation 

1, Ann. Probab. 29, 1061-1085, 2001. 

[12] É. Pardoux, A. Yu. Veretennikov, On Poisson equation and diffusion approximation 

3, Ann. Probab., to appear. 

[13] F. Paronetto, Homogenization of degenerate elliptic–parabolic equations, Asymptot. 

Anal. 37, 21–56, 2004. 

8

Viability with probabilistic knowledge of initial condition, 

application to differential games 

Marc QUINCAMPOIX 

Département de Mathématiques, Université de Bretagne Occidentale 

6 Avenue Victor Le Gorgeu F-29200 Brest, France 

marc.quincampoix@univ-brest.fr 

We consider a deterministic control system with state constraints. The main specificity 

of the question we address now concerns with the case where the state-space is 

only unperfectly known by the controller: Instead of knowing the initial condition , he 

know only the probability measure of the initial condition. The problem is then reduced 

to a new problem where the state variable appears to be be a measure. We provide a 

result characterizing the compatibility of the constraints and the control systems with 

probabilistic knowledge of the state space. 

Then we use our approach to characterize the value function of an differential game 

with probabilistic knowledge of initial condition in term of the unique solution of a suitably 

defined Hamilton Jacobi Isaacs equation (written on the set of measures). This 

creates some difficulties mainly because the set of measure on R N is not finite dimensional 

and it is not a normed space: we will introduce and use the so-called Wasserstein 

distance between probability measures.Nevertheless, we give a new result of existence 

of a value for a differential game with unperfect information. 

Elliptic and parabolic PDE for measures 

Michael G. ROECKNER 

Fakultät für Mathematik, Universität Bielefeld 

Postfach 100131, D-33501 Bielefeld, Germany 

roeckner@math.uni-bielefeld.de 

Invariant measures and transition probabilities of continuous stochastic processes 

satisfy second order PDE of elliptic and parabolic type respectively, however, with coefficients 

of possibly low regularity. This motivates the study of such equations for 

measures from a purely analytic point of view. In the first part of the talk we shall 

start with reviewing existence, uniqueness and local regularity results in the elliptic 

case. In particular, the measures solving the PDE (essentially) always have densities 

with respect to Lebesgue measure. Subsequently, we shall present some recent global 

regularity results for these densities as well as conditions implying that they decay polynomially 

or exponentially at infinity. In the second part of the talk we shall pass to the 

parabolic case. Here the solutions will be measures on space time. Results on existence 

and local regularity are quite similar to the elliptic case. Results on uniqueness and 

global regularity have been established only very recently and are quite different from 

those in the elliptic case. Finally, it should be mentioned that though in this talk only 

9

the finite dimensional case is discussed, the same circle of problems is being analyzed in 

infinite dimensions and some of the above results have also been proved there. 

A continuation method for for a class of periodic evolution 

variational inequalities 

Michel THÉRA 

XLIM (UMR-CNRS 6172), Université de Limoges, France 

michel.thera@unilim.fr 

In this presentation we will give new existence results for finite variational inequalities 

and we will derive the existence of solutions of periodic solutions for a class of evolution 

variational inequalities. 

Optimization of fundamental mechanical structures 

Dan TIBA 

Institute of Mathematics, Bucharest 

dan.tiba@imar.ro 

We study structures like beams, plates, arches, curved rods and shells. For shells 

and curved rods, we introduce new models of generalized Naghdi type and of asymptotic 

type. 

The optimization problems concern the thickness for beams and plates or the geometry 

for arches, curved rods and shells. They enter the class of control by coefficients 

problems, known for their high nonconvexity and their stiff character. 

Our approach allows the relaxation of the regularity hypotheses on the geometry and 

gives a partial answer for the “locking problem” in the case of arches and curved rods. 

We also present numerical examples with have a clear physical interpretation, which 

validates our methods and results. 

Theoretical approaches and numerical implementations of 

quantum control 

Gabriel TURINICI 

CEREMADE, Universite Paris Dauphine 

Place du Marechal De Lattre De Tassigny, 75775 PARIS CEDEX 16, FRANCE 

gabriel.turinici@dauphine.fr 

The control of quantum phenomena is being increasingly explored, both theoretically 

and in the laboratory and has reached a development allowing to consider practically 

10

elevant circumstances : selective dissociation, creation of particular molecular states, 

remote detection, high frequency laser generation, etc. 

These models are expressed in terms of equations that are bi-linear (the control multiplies 

the state). After an initial presentation of the applications and mathematical 

implications of the experimental settings, we will discuss in this talk two types of results: 

ensemble controllability and numerical algorithms. While in the first, geometrical 

Lie group controllability theory is used, in the second we will deal with numerical algorithms 

and their relationship with the designing of optimal control cost functionals. 

Relationships with both theoretical works on PDE control and experimental algorithms 

will also be presented. 

Local uniform linear openness of multifunctions and calculus of 

Bouligand–Severi tangent sets 

Corneliu URSESCU 

“Octav Mayer” Institute of Mathematics, Romanian Academy, Ia¸si Branch 

8 Carol I blvd., 700506 Ia¸si, România 

corneliuursescu@yahoo.com 

Let X and Y be linear spaces, and let F : X → Y be a multifunction. For every 

tangency concept τ and for every point (a, b) ∈ X × Y the equality 

graph(τF (a, b)) = τgraph(F )(a, b) 

defines a multifunction τF (a, b) : X → Y , and moreover, the equality 

τ F −1 (b)(a) = (τF (a, b)) −1 (0) 

is expected. Here, F −1 : Y → X stands for the inverse of the multifunction F . In 

the following we consider the specific tangency concept K which originates from the 

tangent half-lines considered by Bouligand (1931) and Severi (1931). The multifunction 

KF (a, b) was introduced by Aubin (1981). Because of the elementary inclusion 

the equality 

K F −1 (b)(a) ⊆ (KF (a, b)) −1 (0), 

K F −1 (b)(a) = (KF (a, b)) −1 (0) 

is strongly expected. This equality is established at all points (a, b) ∈ graph(F ) assuming 

local uniform linear openness of F . Local uniform linear openness of F is investigated 

through a metric result concerning local uniform ω−openness of F . It is also established 

the equivalence between local uniform ω−openness of F and local metric regularity of 

F . 

11

Short Communications 

Internal stabilization of nonnegative solutions of parabolic 

systems 

Sebastian ANIT¸ A 

Faculty of Mathematics, “Al. I. Cuza” University of Ia¸si 

“O. Mayer” Mathematics Institute of the Romanian Academy, Ia¸si, Romania 

sanita@uaic.ro 

We investigate the internal stabilization of nonnegative solutions of a linear parabolic 

equation. We find a necessary condition and a sufficient condition for the zero-stabilizability 

of the nonnegative solution with respect to the value of the principal eigenvalue of 

a certain elliptic operator. The value of this principal eigenvalue is strongly related to 

the convergence rate of the nonnegative solution. We estimate this principal eigenvalue 

as a function of the shape of the domain and the support of the control. The approach 

from the case of the parabolic equation is used also in the case of a system of reactiondiffusion 

type with two components. This models the dynamic of prey-predator type 

system. Several stabilization strategies are compared. 

Influence of variable permeability on vortex instability of a 

horizontal non-darcy free or mixed convection flow in a 

saturated porous medium 

A.A. BAKER, A.M. ELAIW and F.S. IBRAHIM 

Department of Mathematics, Faculty of Science 

Al-Azhar University, Assiut, EGYPT 

a a baker@yahoo.com 

Department of Mathematics, Faculty of Science 

Assiut University, Assiut, EGYPT 

A linear stability theory is used to analyze the vortex instability of free or mixed 

convection boundary layer flow in a saturated porous medium. The non-Darcian effects, 

which include the inertia force and surface mass flux are examined. The variation of 

permeability in the vicinity of the solid boundary is approximated by an exponential 

function. The variation rate itself depends slowly on the streamwise coordinate, such as 

to allow the problem to possess a set of solutions, invariant under a group of transformations. 

Velocity and temperature profiles as well as local Nusselt number for the base 

flow are presented for the uniform permeability UP and variable permeability VP cases. 

An implicit finite difference method is used to solve the base flow and the resulting 

12

eigenvalue problems are solved numerically. The critical Peclet and Rayleigh numbers 

and the associated wave numbers for both UP and VP cases are obtained. The results 

indicate that the inertial coefficients reduces the heat transfer rate and destabilizes 

the flow to the vortex mode of disturbance. Moreover, the variable permeability effect 

tends to increase the heat transfer rate and destabilize the flow to the vortex mode of 

disturbance. 

Endpoint Strichartz estimates in 3d for nonselfadjoint 

Schrödinger operators 

Marius BECEANU 

Department of Mathematics, University of Chicago 

5734 S. University Ave., Chicago, IL 60637 

mbeceanu@uchicago.edu 

Endpoint Strichartz estimates in R3 are obtained for nonselfadjoint Schrödinger operators. 

Consider operators in R3 of the form H = H0 + V , where 

H0 = 

� −∆ + µ 0 

0 ∆ − µ 

� 

, V = 

� −U −W 

W U 

We assume that U and W are real-valued and have a suitable amount of polynomial 

decay: |U| + |W | ≤ C〈x〉 −7/2− , where 〈x〉 = 1 + |x|. 

The operator H has σ(H) ⊂ R ∪ iR and σess(H) = (−∞, −µ] ∪ [µ, ∞). We make the 

spectral assumption that H has no eigenvalues in the set (−∞, −µ) ∪ (µ, ∞) and that 

the thresholds ±µ are also regular, meaning that I +(H0 −µ±i0) −1 V : L 2,−1− → L 2,−1− 

is invertible. Then σc(H) = σess(H). 

Let Pc = 1 − Ppp be the Riesz projection on the continuous spectrum of the operator 

H. Under these conditions, the following estimates hold for all admissible pairs (q, r) 

and (�q ′ , �r ′ ), where 2 ≤ q, r ≤ ∞, 1 ≤ �q ′ , �r ′ ≤ 2, 1 

q 

�e itH Pcf� L q 

t Lr x 

+ 3 

2r 

= 3 

4 

� � 

� 

≤ �f�2, � 

� e −isH � 

� 

PcF (s)ds� 

� 

2 

� � 

� 

� 

� e 

s

Boundary value problems with non-surjective φ-Laplacian and 

one-sided bounded nonlinearity 

Cristian BEREANU 

Département de Mathématique, Université Catholique de Louvain 

B-1348 Louvain-la-Neuve, Belgium 

bereanu@math.ucl.ac.be 

Using Leray-Schauder degree theory we obtain various existence results for nonlinear 

boundary value problems 

(φ(u ′ )) ′ = f(t, u, u ′ ), l(u, u ′ ) = 0 

where l(u, u ′ ) = 0 denotes the periodic, Neumann or Dirichlet boundary conditions on 

[0, T ], φ : R → ] − a, a[ is a homeomorphism, φ(0) = 0. 

This is a joint work with Prof. Dr. Jean Mawhin. 

Markov processes associated with L p -resolvents and 

applications to stochastic differential equations on Hilbert 

space 

Lucian BEZNEA 

“Simion Stoilow” Institute of Mathematics of the Romanian Academy 

P.O. Box 1-764, RO-014700, Bucharest, Romania 

lucian.beznea@imar.ro 

The talk is based on joint works with Nicu Boboc and Michael Röckner. 

We give general conditions on a generator of a C0-semigroup (resp. of a C0-resolvent) 

on L p (E, µ), p ≥ 1, where E is an arbitrary (Lusin) topological space and µ a σ-finite 

measure on its Borel σ-algebra, so that it generates a sufficiently regular Markov process 

on E. We present a general method how these conditions can be checked in many 

situations. Applications to solve stochastic differential equations on Hilbert space in 

the sense of a martingale problem are given. 

On the use of Korn’s type inequalities in the existence theory 

for Cosserat elastic surfaces with voids 

Mircea BÎRSAN 

University “A.I. Cuza” Iasi, Faculty of Mathematics 

Bd. Carol I, no. 11, 700506, Iasi, Romania 

bmircea@uaic.ro 

14

The theory of Cosserat surfaces is a direct approach to the mechanics of thin elastic 

shells in which the shell-like body is modeled as a two-dimensional continuum endowed 

with a deformable vector assigned to every point of the surface. A detailed analysis 

of the theory of Cosserat shells is included in the classical monograph by Naghdi [1] 

and in the more recent book of Rubin [2]. The differential equations governing the 

deformation of porous Cosserat shells have been presented by Birsan [3]. The porosity 

of the material is described using the Nunziato-Cowin theory for elastic media with 

voids. In the present paper, we investigate the existence of solutions to the boundary 

value problems associated to the deformation of Cosserat shells with voids. We establish 

first the inequalities of Korn’s type which are valid for Cosserat surfaces and employ 

them to show the existence of solution to the variational equations in elastostatics. 

The proof of the Korn’s type inequality rely in a crucial manner on a lemma of J.L. 

Lions [4]. We employ a method similar to that used in the classical linear shell theory by 

Ciarlet [5]. For the dynamic problems, we show the existence, uniqueness and continuous 

dependence of solution to the boundary-initial-value problem using the results of the 

semigroup of linear operators theory. 

References: 

1. Naghdi, P. M.: The Theory of Shells and Plates. In: Handbuch der Physik, Vol. 

VI a/2, pp. 425-640, Springer-Verlag, Berlin Heidelberg New York (1972). 

2. Rubin, M. B.: Cosserat Theories: Shells, Rods, and Points. Kluwer Academic 

Publishers, Dordrecht (2000). 

3. Birsan, M.: On the theory of elastic shells made from a material with voids. 

International Journal of Solids and Structures 43, 3106-3123 (2006). 

4. Duvaut, G., Lions, J. L.: Inequalities in Mechanics and Physics. Springer-Verlag, 

Berlin (1976). 

5. Ciarlet, P. G.: Mathematical Elasticity, Vol. III: Theory of Shells. North-Holland, 

Amsterdam (2000). 

On the dynamical reconstruction of control in systems of 

ordinary differential equations 

M. BLIZORUKOVA, V. MAKSIMOV and N. FEDINA 

Institute of Mathematics and Mechanics 

Ural Branch of Russian Academy of Sciences 

S. Kovalevskaya Str. 16, Ekaterinburg, 620219, RUSSIA 

msb@imm.uran.ru 

Problems of dynamical identification of unknown control acting upon a nonlinear 

system described by ordinary differential equations are discussed. Under the measurements 

(with errors) of phase state of the system, a regularizing algorithm that allows to 

identify the control is indicated. The algorithm is stable with respect to informational 

noises and computational errors. The procedure suggested is based on the combination 

of the methods of the theory of guaranteed control and of the theory of ill-posed 

problems. 

15

Time periodic viscosity solutions of Hamilton–Jacobi equations 

Mihai BOSTAN 

Université de Franche-Comté, France 

mihai.bostan@math.univ-fcomte.fr 

We study the existence of time periodic viscosity solution of first order Hamilton– 

Jacobi equations. Existence results are presented under usual hypotheses. The main 

idea is to reduce the analysis of time periodic problems to the study of stationary 

problems obtained by averaging the source term over a period. We investigate also the 

long time behavior and high frequency asymptotic behavior (leading to homogenization 

problems). Most of these results can be generalized in the framework of almost-periodic 

viscosity solutions. 

Brézis–Haraux - type approximation of the range of a 

monotone operator composed with a linear mapping 

Radu Ioan BOT¸ , Sorin Mihai GRAD, Gert WANKA 

Chemnitz University of Technology, Germany 

bot@mathematik.tu-chemnitz.de 

Given two monotone operators, the sum of their ranges is usually larger than the 

range of their sum, but there are some situations where these sets are almost equal, i.e. 

their interiors and closures coincide. The problem of finding conditions under which the 

sum of the ranges of two monotone operators is almost equal to the range of their sum 

is known as the Brézis–Haraux approximation problem ([1]). We give a Brézis–Haraux 

- type approximation of the range of the monotone operator TA = A ∗ ◦ T ◦ A where 

A is a linear continuous mapping between two non-reflexive Banach spaces and T is a 

maximal monotone operator. Then we specialize the result for a Brézis–Haraux - type 

approximation of the range of the subdifferential of the precomposition to A of a proper 

convex lower semicontinuous function defined on a non-reflexive Banach space, which is 

proven to hold under a weak sufficient condition. This extends and corrects some older 

results due to Riahi ([2]) that consist in the approximation of the range of the sum of 

the subdifferentials of two proper convex lower semicontinuous functions. Finally we 

give two applications, one in optimization and the other to a complementarity problem. 

References: 

[1] Brézis, H., Haraux, A. Image d’une somme d’opérateurs monotones et applications, 

Israel Journal of Mathematics No. 23 (1976), 165-186. 

[2] Riahi, H. On the range of the sum of monotone operators in general Banach 

spaces, Proceedings of the American Mathematical Society No. 124 (1996), 3333–3338. 

16

On a wrong solution to a trivial optimal control problem in 

mathematical economics 

S¸tefan MIRICĂ and Touffik BOUREMANI 

Faculty of Mathematics, University of Bucharest 

Academiei 14, 010014 Bucharest, ROMANIA 

bouremani@yahoo.com, mirica@fmi.unibuc.ro 

This work is intended, in the first place, as a warning to the increasing number of 

authors that try to solve concrete optimal control problems without enough knowledge 

and even basic mathematical abilities; secondly, our aim is to show that the Dynamic 

Programming approach in [2] is much more efficient than the PMP approach, in the 

study of this type of problems. 

A tipical example is the one in the recent paper, [1], in which the authors believe 

that they solved the problem of minimizing the cost functional 

subject to: 

� T 

J(P (.), I(.)) := e −ρt [h(I(t)) + K(P (t))]dt 

0 

I ′ (t) = P (t) − D(t, I(t)), P (t) ≥ D(t, I(t)) > 0, 

a.e. ([0, T ], I(0) = I0, I(T ) = IT that arise from some (not very convincing) “model in 

mathematical economics”. 

Applying a non-existent (in [3]) variant of Pontryagin’s Maximum Principle (PMP) 

as a “necessary optimality condition”, the authors of [1] conclude in their Th.1 that 

in the particular case in which: K(p) := 1 

2 kp2 , h(I) := 1 

2 hI2 , D(t, I) := d1(t) + 

d2I, k, h, d2, d1(t) > 0, the “optimal trajectory” of the problem above is of the form: 

I ∗ (t) := a1e m1t + a2e m2t + Q(t). 

On the other hand, using the Dynamic Programming approach in [2] (adapted to 

non-autonomous problems) we prove that the only optimal trajectories are the constant 

functions, � I(t) ≡ I0, hence the solution in [1] is wrong and the problem above is rather 

“trivial”. Moreover, we also identify the main scientific errors made by the authors 

of [1], among which, the first one is the fact that they apply the “classical form” of 

PMP to a problem defined by a differential inclusion since in this case the set of control 

parameters is variable (depending on the time and the state). 

17

Derived cones to reachable sets of discrete hyperbolic 

inclusions 

Aurelian CERNEA 

Faculty of Mathematics and Informatics, University of Bucharest 

Academiei 14, 010014 Bucharest, Romania 

acernea68@yahoo.com 

We consider a multiparameter discrete inclusion that describes Roesser’s model and 

we prove that the reachable set of a certain variational multiparameter inclusion is a 

derived cone in the sense of Hestenes to the reachable set of the discrete inclusion. This 

result allows to obtain sufficient conditions for local controllability along a reference 

trajectory and a new proof of the minimum principle for an optimization problem given 

by a hyperbolic discrete inclusion with end point constraints. 

Numerical solution of variational problems using Walsh 

wavelet packets 

Heena D. CHALISHAJAR and Pragna KANTAWALA 

Department of Applied Mathematics, Babaria Institute of Technology (BIT) 

Varnama-391240, Vadodara, Gujarat, India 

heena14672@yahoo.co.in 

Department of Applied Mathematics, Faculty of Technology and Engineering 

M.S. University of Baroda, Vadodara- 390001, Gujarat, India 

pskmsu@yahoo.com 

In 2004, Glabisz [2] solved linear boundary value problems using the operational 

matrices obtained from Walsh wavelet packets i.e. using Walsh Bases and Haar bases. 

In this paper the authors are solving the variational problems using the operational 

matrices for Walsh wavelet packets i.e. using Walsh bases and Haar bases. An illustrative 

example has been included. Also a comparative study has been made for 

the solutions obtained using the Haar bases, Walsh bases, the Exact solution and the 

solution obtained by using Walsh functions. 

References: 

[1] Chalishajar H.D and Kantawala P.S – A Direct Method for solving Variational 

Problems using Walsh Wavelet packets and Error Estimates, Modern Mathematical 

Models, Methods and Algorithms for Real World Systems, Anamaya Publishers, India 

pp. 235-245 (2006) 

[2] Glabisz Wojciech – The use of Walsh wavelet packets in linear boundary value 

problems, Computers and Structures, Vol.82, 131-141 (2004). 

18

Controllability of Damped Second Order Initial Value Problem 

for a class of Differential Inclusions with Nonlocal Conditions 

on Noncompact Intervals 

Dimplekumar N. CHALISHAJAR 

Department of Applied Mathematics, Sardar Vallabhbhai Patel Institute of 

Technology (SVIT), Gujarat University 

Vasad-388 306, DIST: Anand, Gujarat State, INDIA 

dipu17370@yahoo.com 

In this article, we investigate sufficient conditions for controllability of second-order 

semi-linear initial value problem with nonlocal conditions for the class of differential 

inclusions in Banach spaces using the theory of strongly continuous cosine families. We 

shall rely on a fixed point theorem due to Ma for multi-valued maps. An example is 

provided to illustrate the result. This work is motivated by the paper of Benchohra and 

Ntouyas [1] and Benchohra, Gatsori and Ntouyas [2]. 

We have considered the following second order inclusion system with non local conditions 

� 

y ′′ (t) − Ay(t) ∈ Gy ′ (t) + Bu(t) + F (t, yt, y ′ (t)), t ∈ J 

y(0) + g(y) = φ, y ′ (0) = y0. 

Here the state y(t) takes values in Banach space E and the control u ∈ L 2 (J, U), the 

space of admissible controls, where J = (0, ∞). 

Our aim is to study the exact controllability of the above abstract system which will 

have applications to many interesting systems including PDE systems. We reduce the 

controllability problem (1) to the search for fixed points of a suitable multi-valued map 

on a subspace of the Frechet space C(J, E). 

References: 

[1] Benchohra, M. and Ntouyas, S. K., Controllability for an infinite time horizone 

of second order differential inclusions in Banach spaces with Nonlocal conditions, J. 

Optim. Theory Appl. 109 (2001), 85–98. 

[2] Benchohra, M., Gatsori, E. P. and Ntouyas, S. K., Nonlocal quasilinear damped 

differential inclusions, Electronic journal of Differential Equations, Vol.2002 (2002), 

No.7, 1–14. 

A (p − q) coupled system in elliptic nonlinear boundary value 

problems 

L. CONSIGLIERI 

Department of Mathematics and CMAF 

Sciences Faculty of University of Lisbon, 1749-016 Lisboa, Portugal 

lcconsiglieri@fc.ul.pt 

19 

(1)

In the present work we deal with a problem motivated by the solid and/or fluid 

thermomechanics, and we establish an existence result of a weak solution. For Ω an 

open bounded set of R n (n > 1) with a sufficiently smooth boundary ∂Ω constituted 

by two disjoint complementary open subsets Γ0 and Γ, we study the elliptic boundary 

value problem: find u, e : ¯ Ω → R and τ : ¯ Ω → R n such that 

−∇ · τ = f(e, u, ∇u) in Ω; 

τ ∈ ∂F(e, ∇u) in Ω; 

−τ · n ∈ ∂G(e, u) on Γ; 

−∇ · A(e, ∇e) = g(u, e, ∇e) + τ · ∇u in Ω; 

A(e, ∇e) · n + γ(e) = −(τ · n)u on Γ; 

u = e = 0 on Γ0 := ∂Ω \ ¯ Γ. 

Here n denotes the unit outward normal vector to Γ. The (p − q) structure is related to 

the functionals F and A. The proof is based on variational and compactness methods. 

Some remarks on functional equations and their applications 

C. CORDUNEANU 

Department of Mathematics, University of Texas, Arlington, USA 

concord@uta.edu 

We shall consider functional differential equations with causal operators,and deal 

with the following type of problems: 

1) Some existence theorem,with special concern on global existence and uniqueness; 

2) Neutral functional/functional differential equations with causal operators and 

their reduction to equations of normal form; 

3) Control problems for some classes of functional equations with causal operators; 

4) A duality principle for dynamical systems described by functional differential 

equations. 

The following papers of the author are taken as departing point for conducting the 

research: 

[1] Functional Equations with Causal operators. Taylor and Francis, London, 2002; 

[2] A duality principle for dynamical systems described by functional equations. 

Nonlinear Dynamics and Systems Theory, vol. 5 (2005). 

On nonlinear mixed Volterra–Fredholm integrodifferential 

equations in Banach spaces 

M.B. DHAKNE and S.D. KENDRE 

Department of Mathematics, Dr.Babasaheb Ambedkar Marathwada University, 

Aurangabad-431004, India (M.S.), mbdhakne@yahoo.com 

Department of Mathematics, University of Pune, Pune-411007, India (M.S.) 

20

Let X be a Banach space with norm � . �. Let B = C([0, α], X) be the Banach space 

of all continuous functions from [0, α] into X endowed with supremum norm � x �B = 

sup {� x(t) � : t ∈ [0, α]}. In the present paper, we investigate the global existence of 

mild solutions of nonlinear mixed Volterra-Fredholm integrodifferential equation of the 

type 

x ′ � 

(t) + Ax(t) = f t, x(t), 

� t 

� α 

k(t, s, x(s))ds, 

0 

x(0) = x0 ∈ X; t ∈ [0, α], 

0 

� 

h(t, s, x(s))ds , 

where −A is the infinitesimal generator of a strongly continuous semigroup of bounded 

linear operators T (t) in X, f : [0, α] × X × X × X → X, k, h: [0, α] × [0, α] × X → X 

are continuous functions and x0 is a given element of X. The main tool employed in 

our analysis is based on an application of the Leray-Schauder alternative and rely on a 

priori bounds of solutions. 

References: 

[1] M. B. Dhakne; Global existence of solutions of nonlinear functional integral equations, 

Indian J. Pure. Appl. Math., 30(7) (1999), 735-744. 

[2] J. Dugundji and A. Granas; Fixed point theory, Vol.1 Monografie Matematyczne, 

PWN, Warrsw (1982). 

[3] J. W. Lee and D. O’Regan; Existence results for differential delay equations, I. 

J. Differential equations, 102 (1993), 342-359. 

Singularly perturbed evolution inclusions 

Tzanko DONCHEV 

Department of Mathematics, University of Architecture and Civil Engineering 

1 “Hr. Smirnenski” str., 1046 Sofia, Bulgaria 

tdd51us@yahoo.com 

In this paper we study a control evolution system described by two evolution inclusions: 

a “slow” and a “fast” one: 

˙x(t) + A1x ∈ F (x, y, u(t)), x(0) = x 0 , u(t) ∈ U 

ε ˙y(t) + A2y ∈ G(x, y, u(t)), y(0) = y 0 , t ∈ I = [0, 1]. 

We describe the limit of the solution set as the small parameter ε tends to 0 + . We 

present an example of concrete system, where our results are applicable. 

21

Robust ℓ-step receding horizon control of sampled-data 

nonlinear systems with bounded additive disturbances with 

application to a HIV/AIDS model 

A. M. ELAIW 

Al-Azhar University (Assiut), Faculty of Science 

Department of Mathematics, Assiut, EGYPT 

a m elaiw@yahoo.com 

In this paper, a robust receding horizon control for multirate sampled-data nonlinear 

systems with bounded additive disturbances is presented. Sufficient conditions 

are established which guarantee that the ℓ-step receding horizon controller that stabilizes 

the nominal approximate discrete-time model also practically stabilizes the exact 

discrete-time system with small disturbances. This version of the method is motivated 

by recently developed models of the interaction of the HIV virus and the immune system 

of the human body. In this model the drug dose is considered as control input, and 

the uninfected steady state is to be stabilized. Reverse transcriptase inhibitors is used. 

Simulation results are discussed. 

Approximate solutions for non-linear autonomous ODEs on 

the basis of PWL approximation theory 

A. GARCÍA a , S. BIAGIOLA b , J. FIGUEROA c , L. CASTRO d 

O. AGAMENNONI e 

a,b,c,e: Departamento de Ingeniería Eléctrica y de Computadoras, Universidad 

Nacional del Sur, Alem 1253, 8000 Bahía Blanca, Argentina 

{agarcia, biagiola, figueroa, oagamen}@uns.edu.ar 

d: Departamento de Matemática, Universidad Nacional del Sur, Alem 1253, 8000 

Bahía Blanca, Argentina, lcastro@uns.edu.ar 

The present work introduces a new approach to the problem of describing the dynamic 

behavior of nonlinear autonomous systems. A general method to approximate 

the solutions of a set of ODEs is presented. The technique makes use of a PWL approximation 

of the ODEs vector field which describes the dynamics of a system. A measure 

of the dynamics approximation error is used to estimate upper and lower bounds for the 

error between the real and the approximate trajectories (i.e. the real and approximate 

solutions to the ODE). The proposed solution approach can be used in many applications. 

For instance, in the field of physical and (bio) chemical processes that exhibit 

nonlinear behavior. These ones, are good examples since many of them are modelled by 

systems of ODEs in continuous-time domain. Moreover, it could be used in the solution 

of dynamic optimization problems with significant advantages as regards computational 

time consumption. It is important to remark the possibility of the implementation of 

an integrated circuit capable to emulate PWL functions, thus providing a very efficient 

real time application. 

22

A fourth order problem in a thin multidomain 

Antonio GAUDIELLO 

DAEIMI, Università degli Studi di Cassino 

via G. Di Biasio 43, 03043 Cassino (FR), ITALIA 

gaudiell@unina.it 

This is joint work with E. Zappale. 

We consider a thin multidomain of RN , N ≥ 2, consisting (e.g. in a 3D setting) of a 

vertical rod upon a horizontal disk. In this thin multidomain we introduce a bulk energy 

density of the kind W (D2U), where W is a convex function with growth p ∈]1, +∞[, 

and D2U denotes the Hessian tensor of a scalar (or vector valued) function U. By 

assuming that the two volumes tend to zero with same rate, under suitable boundary 

conditions, we prove that the limit model is well posed in the union of the limit domains, 

with dimensions, respectively, 1 and N − 1. Moreover, we show that the limit problem 

is uncoupled if 1 

N−1 

, “partially” coupled if 

2 2 

N − 1 < p. The main result is applied in order to derive the equilibrium configuration 

of two joint beams, T-shaped, clamped at the three endpoints and subject to transverse 

loads. The main result is also applied in order to describe the equilibrium configuration 

of a wire upon a thin film with contact at the origin, when the thin structure is filled 

with a martensitic material. 

Conjugacy and Fenchel duality for almost convex and nearly 

convex functions 

Radu Ioan BOT¸ , Sorin Mihai GRAD, Gert WANKA 


grad@mathematik.tu-chemnitz.de 

A function f : R n → R is called (cf. [2]) almost convex if ¯ f is convex and ri(epi( ¯ f)) ⊆ 

epi(f), nearly convex if epi(f) is nearly convex and closely convex if epi( ¯ f) is convex 

(i.e. ¯ f is convex), where ¯ f denotes the lower-semicontinuous hull of f. 

We show that the classical formulae of the conjugates of the precomposition with a 

linear operator, of the sum of finitely many functions and of the sum between a function 

and the precomposition of another one with a linear operator hold even when the usual 

convexity assumptions (cf. [3]) are replaced by almost convexity or near convexity. 

We also prove that the hypotheses of duality statements due to Fenchel can be weakened 

when the functions involved are almost (nearly) convex. 

References: 

[1] R. I. Bot¸, S. M. Grad, and G. Wanka – Almost convex functions: conjugacy and 

duality, to appear in Proceedings of the 8th International Symposium on Generalized 

Convexity and Generalized Monotonicity, Varese, Italy, 4-8 July, 2005. 

23

[2] W. W. Breckner and G. Kassay – A systematization of convexity concepts for 

sets and functions, Journal of Convex Analysis No. 4 Vol. 1 (1997), 109–127. 

[3] R. T. Rockafellar – Convex analysis, Princeton University Press, 1970. 

Semimonotone stochastic integral equations and explosion time 

Hamideh D. HAMEDANI 

Department of Statistics, Faculty of Mathematical Sciences 

Shahid Beheshti University Tehran, IRAN 

h-hamedani@sbu.ac.ir 

In this talk, we first give an existence and uniqueness result for the following Hilbert 

space valued semimonotone nonlinear integral equation 

� t 

Xt = U(t, 0)X0 + 

� 

0 

U(t, s)F (s, Xs)ds+ 

(1) 

+ U(t, s)G(s−, Xs−)dMs + Vt; t ≤ τ, 

(o,t] 

where U(t, s) is a contraction-type evolution operator, Ft(.) = F (t, ω, .) is a semimonotone 

function, G is a good operator-valued Lipschitz integrand, M is a cadlag martingale, 

V is a cadlag adapted process, X0 is a random variable and τ is a stopping time. 

Then we define the explosion time of (1) and in the lack of linear growth condition, 

we show that the solution will be exploded. 

On the integral and asymptotical representation of singular 

solutions of elliptic equations near boundary 

Nicolae JITARAS¸U 

USM, Chi¸snău, Republic of Moldova 

jitarasu@usm.md 

Let G0 ∈ R n be a bounded domain with (n − 1)-dimensional boundary Γ0 ∈ C ∞ and 

the nk-dimensional manifolds Γk without boundary lying inside of Γ0, 0 ≤ nk ≤ n − 1. 

Assume that Γk ∈ C ∞ and Γk ∩Γj = ∅ for k �= j. Denote by G the domain G0 \∪ χ 

k=1 Γk, 

Γ = ∪ χ 

k=0 Γj the boundary of domain G. In the domain G we consider the elliptic 

boundary value problem (BVP) 

L(x, ∂)u(x) = f(x), ordL = 2m, (1) 

Bj(x, ∂)u(x)|Γ0 = ϕj(x), j = 1, 2, . . . , m, (2) 

in the scale of Sobolev spaces H s (G), s ∈ R 1 . Using the Green function G(x) of BVP 

(1), (2) in G0 [1], we obtain the integral representations of singular solutions u(x) of 

24

BVP (1), (2) in G near variety Γk, u(x) ∈ H −s (G), s > 0. In the model case of 

the operators with constant coefficients we obtain the asymptotical representations of 

singular solutions u(x) ∈ H −s (G) near Γk. These asymptotical representations are used 

for correct formulation of BVP with singular boundary conditions on Γk [2]. 

References: 

[1] Solonnikov V.A., On Green’s matrices for elliptic boundary value problems, Trudy 

Mat. Inst. Acad. Nauk SSSR, v. 110, 1970, p. 107 - 145. 

[2] Jitarasu N., On the Sobolev boundary value problem with singular and regularized 

boundary conditions for elliptic equations, Anal. and Optimiz. of Differential systems, 

Kluwer Acad. Publ., 2003, p.219 - 226. 

Duality in vector optimization 

Frank HEYDE, Andreas LÖHNE and Christiane TAMMER 

Universität Halle-Wittenberg 

FB Mathematik und Informatik, 06099 Halle 

andreas.loehne@mathematik.uni-halle.de 

A new approach to duality theory for convex vector optimization problems is developed. 

We modify a given (set-valued) vector optimization problem such that the image 

space becomes a complete lattice (a sublattice of the power set of the original image 

space), where the corresponding infimum and supremum are sets that are related to 

the usual solution concepts of vector optimization. In doing so we can carry over the 

structures of the duality theory of scalar convex programming. It follows a discussion 

of the special case of multi-objective linear programming, in particular, the possibility 

to develop a dual simplex algorithm based on this duality theory. 

An existence result for a class of nonlinear differential systems 

Rodica LUCA-TUDORACHE 

Department of Mathematics, “Gh. Asachi” Technical University 

11 Bd. Carol I, Ia¸si 700506, ROMANIA 

rluca@hostingcenter.ro 

We investigate the existence, uniqueness and asymptotic behaviour of the strong and 

weak solutions to the nonlinear discrete hyperbolic system 

⎧ 

⎪⎨ 

dun 

(S) 

dt 

⎪⎩ 

(t) + vn(t) − vn−1(t) 

+ cnA(un(t)) ∋ fn(t), 

h 

dvn 

dt (t) + un+1(t) − un(t) 

+ dnB(vn(t)) ∋ gn(t), 

h 

n = 1, 2, . . . , 0 < t < T, in H, 

25

with the extreme condition 

(EC) (v0(t), s1w ′ 1(t), . . . , smw ′ m(t)) T ∈ −Λ((u1(t), w1(t), . . . , wm(t)) T ), 0 < t < T 

and the initial data 

(ID) 

� un(0) = un0, vn(0) = vn0, n = 1, 2, . . . , 

wi(0) = wi0, i = 1, m, 

where H is a real Hilbert space, T > 0, m ∈ IN, h > 0, cn > 0, dn > 0, ∀ n ∈ IN, 

si > 0, ∀ i = 1, m, and A, B are multivalued operators in H and Λ is a multivalued 

operator in H m+1 which satisfy some assumptions. 

This problem is a discrete version with respect to x (with H = IR) of some problems 

which have applications in the theory of integrated circuits. For the proofs of our 

theorems we use some results from the theory of monotone operators and nonlinear 

evolution equations of monotone type in Hilbert spaces. 

Analytical solutions for integral operator’s nonlinear 

optimization in case of the airfoils curvilinear of maximal drag 

in aero hydrodynamics 

Mircea LUPU and Ernest SCHEIBER 

Faculty of Mathematics-Informatics 

Transilvania University of Brasov 

m.lupu@info.unitbv.ro, e.scheiber@info.unitbv.ro 

In the paper there are solved direct and inverse Dirichlet or Riemann - Hilbert 

problems and analytical solution are obtained for optimization problems in the case 

of some nonlinear integral operators. It is modeled the plane potential flow of an inviscid, 

incompressible jet, is eliminated from the channel and encounters a curvilinear 

symmetrical obstacle (airfoils/hydrofoils) which must modeled geometrically obtained 

the deflector of maximal drag. There are delivered integral singular equations, for direct 

and inverse problems and the movement in the auxiliary canonical half-plane is 

obtained. Next, the optimization problem is solved in analytical manner. The mathematical 

model used is “Valcovich–Birkhoff–Popp” in the case of the curvilinear profile; 

it is used integral Jensen’s inequality for increase the integral nonlinear operator - who 

represents the resistance to advancement - determined the global maximum accepted. 

The author established one of the most general velocity distributions on the airfoil, with 

distributed parameters, for getting the optimal forms of the maximal drag airfoil in the 

Brillouin–Villat’s condition. The design of the optimal airfoils is performed, numerical 

computation concerning the drag coefficient. 

In particularly case is obtained the limited jets which encountered the curvilinear 

airfoils (the Cisotti–Iacob’s model) and the case of the obstacle situated and the free 

unlimited flow (the Helmholtz’s model — impermeable parachute). The constructive 

26

techniques have a theoretical and practical importance. The problems of maximal drag 

are very important, in relation with applications to the most reversal devices, or the 

direction control of the reactive vehicles. We notice also other applications to the solving 

by means of fluid jets or to the jet flaps system from the airplane wings or turbine bucket. 

On the method of Lyapunov functionals in inverse problems of 

distributed systems 

Vyacheslav MAKSIMOV 

Institute of Mathematics and Mechanics 

Ural Branch of the Russian Academy of Sciences 

S. Kovalevskoi Str. 16, Ekaterinburg, 620219, RUSSIA 

maksimov@imm.uran.ru 

Inverse problems of dynamical reconstruction of unknown characteristics for distributed 

systems are considered. The role of these characteristics may be played by 

distributed or boundary disturbances, by varying coefficient at higher derivative of elliptic 

operator. Solving algorithms, which are stable with respect to informational noises 

and computational errors and operate in real time mode, are designed. These algorithms 

are based on the method of Lyapunov functions, on the theory of positional differential 

games principle of control with a model. Inaccurate measurements of current phase 

states of systems represent input data for the algorithm, which provide some values 

approximating real (but unknown) characteristics as outputs. The basic elements of the 

algorithms are stabilization procedures (functioning by feedback principle) for appropriate 

Lyapunov functionals. 

Nonlinear Multiobjective Transportation Problem: A Fuzzy 

Goal Programming Approach 

H.R. MALEKI and H. MISHMASTE NEHI 

Department of Basic Sciences, Shiraz University of Technology, Shiraz, IRAN 

maleki@sutech.ac.ir 

Department of Mathematics, Sistan and Baluchestan University, Zahedan, IRAN 

hmnehi@hamoon.usb.ac.ir 

The nonlinear multiobjective transportation problem refers to a special class of nonlinear 

multiobjective problem. In this paper we present a fuzzy goal programming 

approach to solve the nonlinear multiobjective transportation problem. To this end we 

use a special type of nonlinear (hyperbolic) membership functions. A numerical example 

is given to illustrate the efficiency of the proposed approach. 

27

References: 

[1] A.K. Bit, M.P. Biswal and S.S. Alam, Fuzzy programming approach to multiobjective 

solid transportation problem, Fuzzy Sets and Systems 57 (1993), pp. 183-194. 

[2] R.H. Mohamed, The relationship between goal programming and fuzzy programming, 

Fuzzy Sets and Systems 89 (1997), pp. 215–222. 

[3] B.B. Pal, B.N. Moitra, U. Maulik, A goal programming procedure for fuzzy 

multiobjective linear programming problem, Fuzzy Sets and Systems 139 (2003), pp. 

395–405. 

[4] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965), pp. 338–353. 

On Schrödinger operators with multipolar inverse-square 

potentials 

Elsa MARCHINI 

Dipartimento di Matematica e Applicazioni 

Universita di Milano-Bicocca Via R. Cozzi 53 - Edificio U5 - 20125 Milano 

elsa.marchini@unimib.it 

Positivity, essential self-adjointness, and spectral properties of a class of Schrödinger 

operators with multipolar inverse-square potentials are discussed. In particular a necessary 

and sufficient condition on the masses of singularities for the existence of at 

least a configuration of poles ensuring the positivity of the associated quadratic form is 

established. 

A mathematical model for the flow in a porous medium with a 

space and time-dependent porosity 

Gabriela MARINOSCHI 

Institute of Mathematical Statistics and Applied Mathematics 

Bucharest, Romania 

gmarino@acad.ro 

The paper deals with the study of the well-posedness of a model describing the 

water flow in a porous medium characterized by a time and space variability of the 

pore structure. The model consists in a parabolic partial differential equation with a 

blowing-up diffusivity and with initial and boundary data. The model leads to a Cauchy 

problem involving a time-dependent multivalued operator, for which an existence and 

uniqueness result is proved. 

28

Petri net toolbox for MATLAB in modeling and simulation of 

discrete-event and hybrid systems 

Mihaela MATCOVSCHI and Octavian PĂSTRĂVANU 

Department of Automatic Control and Applied Informatics 

Technical University “Gh. Asachi” of Iasi 

Blvd. Mangeron 53A, Iasi 700050, Romania 

{opastrav, mhanako}@ac.tuiasi.ro 

The Petri Net Toolbox (PN Toolbox) for MATLAB is a software package that provides 

instruments for the simulation, analysis and design of the discrete-event systems 

modeled via the Petri net formalism. The toolbox is equipped with a user-friendly 

graphical interface and can handle five types of Petri nets (untimed, transition-timed, 

place-timed, stochastic and generalized stochastic) with finite or infinite capacity. Three 

simulation modes, accompanied or not by animation, are available. Dedicated procedures 

cover the key topics of analysis such as behavioral properties, structural properties, 

time-dependent performance indices, max-plus state-space representations. A design 

procedure is also available, based on parameterized models. The Petri Net Simulink 

Block (PNSB) allows the modeling and analysis of hybrid systems whose event-driven 

part(s) is (are) modeled based on the PN formalism. A synchronized Petri net contain 

transitions that are triggered by external events (corresponding to the evolution of the 

continuous part(s) of the hybrid system) and exports internal events, generated by transition 

firings, and its current marking as input signals for the continuous part(s). The 

PN Toolbox is included in the Connections Program of The MathWorks Inc., as a third 

party product. 

Stochastic approach for multivalued Dirichlet–Neumann 

problems 

Lucian MATICIUC and Aurel RĂS¸CANU 

Department of Mathematics, “Gheorghe Asachi” Technical University of Ia¸si 

Bd. Carol I, no. 11, Iasi - 700506, Romania 

lucianmaticiuc@yahoo.com 

Faculty of Mathematics, “Alexandru Ioan Cuza” University of Ia¸si and 

“Octav Mayer” Mathematics Institute of the Romanian Academy 

Bd. Carol I, no. 11, Ia¸si - 700506, Romania 

rascanu@uaic.ro 

We prove the existence and uniqueness of a viscosity solution of the parabolic variational 

inequality with a mixed nonlinear multivalued Neumann-Dirichlet boundary 

29

condition: 

⎧ 

⎪⎨ 

⎪⎩ 

∂u(t, x) 

∂t + Ltu (t, x) + f(t, x, u(t, x), (∇uσ)(t, x)) ∈ ∂ϕ(u(t, x)), 

t ∈ [0, T ], x ∈ D, 

∂u(t, x) 

+ ∂ψ(u(t, x) ∋ g(t, x, u(t, x)), t ∈ [0, T ], x ∈ Bd (D) , 

∂n 

u(T, x) = h(x), x ∈ D, 

where ∂ϕ and ∂ψ are subdifferential operators and Lt is a second differential operator 

given by 

Ltϕ (x) = 1 

d� 

(σσ 

2 

∗ )ij(t, x) ∂2 d� ϕ (x) 

∂ϕ (x) 

+ bi(t, x) . 

∂xi∂xj 

∂xi 

i,j=1 

The result is obtained by a stochastic approach: we study a backward stochastic 

generalized variational inequality 

⎧ 

⎪⎨ 

dYt + F (t, Yt, Zt) dt + G (t, Yt) dAt ∈ ∂ϕ (Yt) dt + ∂ψ (Yt) dAt + ZtdWt , 

0 ≤ t ≤ T , 

⎪⎩ 

YT = ξ 

and we obtain a Feynman-Kaç representation formula for the viscosity solution u of the 

problem (1). 

Reducing a differential game to a pair of nonsmooth optimal 

control problems 

S¸tefan MIRICĂ 

i=1 

Faculty of Mathematics, University of Bucharest 

Academiei 14, 010014 Bucharest, Romania 

mirica@fmi.unibuc.ro 

We are extending to the case of non-autonomous differential games author’s recent 

concepts and results introduced and developed for autonomous (i.e. time-invariant) 

differential games. 

The main idea is to introduce first the concept of an “admissible pair of (multivalued) 

feedback strategies” to which one may associate a value function and which reduces the 

differential game to a pair of symmetric nonsmooth optimal control problems for differential 

inclusions; secondly, one introduces the concepts of bilaterally-optimal pairs of 

feedback strategies and one proves an abstract verification theorem containing necessary 

and sufficient optimality conditions; next, this approach is made more realistic by the 

proof of several “practical verification theorems” containing corresponding differential 

inequalities and regularity hypotheses on the value function that imply the optimality. 

30 

(1)

Moreover, one may prove that under certain conditions, the value function associated 

to a pair of bilaterally-optimal feedback strategies, is a generalized solution and, in 

particular, a viscosity solution, of Isaacs’ main equation and suggest the possibility to 

use suitable extensions of Cauchy’s Method of Characteristics to construct, both, the 

value function and the pair of optimal feedback strategies. Already tested on several 

non-trivial examples, this work may be considered as a ”rehabilitation” on a rigorous 

basis of Isaacs’ (1965) original approach. 

A polytope approach for quadratic assignment problem 

H. MISHMASTE NEHI and H.R. MALEKI 

Department of Mathematics, Sistan and Baluchestan University 

Zahedan, IRAN 

hmnehi@hamoon.usb.ac.ir 

Department of Basic Sciences, Shiraz University of Technology, Shiraz, IRAN 

maleki@sutech.ac.ir 

The Quadratic Assignment Problem (QAP) is one of the classical optimization problem 

and is widely regarded as one of the most difficult problem in this class. Many 

researchers proposed different formulation for this problem based on a special approach. 

Given a set of N = 1, 2, ..., n, and n×n matrices F = {fij}, called flow matrix, D = {dij}, 

as distance matrix, and C = {cij}, as setup cost. The QAP is to find a permutation φ 

of the set N which minimizes: 

z = 

n� 

i=1 

n� 

fijdφ(i)φ(j) + 

j=1 

n� 

i=1 

ciφ(i). 

The quadratic assignment polytope will be defined in section 2 as the convex hull of 

the feasible solutions to a suitable linearization of QAP. In order to profit from the 

convenient notions of graph theory we formulate the quadratic assignment problem as 

a graph problem before starting its polyhedral investigations in Section 3. In Section 

4 connections between the quadratic assignment polytope and polytopes like the linear 

ordering polytope and the traveling salesman polytope via certain projections are considered. 

The quadratic assignment polytope will be proved to be a face of the boolean 

quadric as well as of the cut polytope. Finally we present an integer linear programming 

formulation in Section 5. 

31

A delayed prey-predator system with parasitic infection 

Debasis MUKHERJEE 

Department of Mathematics, Vivekananda College 

Thakurpukur, Kolkata-700 063, INDIA 

debasis mukherjee2000@yahoo.co.in 

This paper analyzes a prey-predator system in which some members of the prey 

population and all predators are subjected to infection by a parasite. The predator 

functional response is a function of weighted sum of prey abundances. Persistence 

and extinction criteria are derived. The stability of the interior equilibrium point is 

discussed. The role of delay is also addressed. Lastly the results are verified through 

computer simulation. Numerical simulation suggests that the delay has a destabilizing 

effect. 

Numerical aspects on computational electrocardiology 

Marilena MUNTEANU 

University of Milan 

via C. Saldini, 50 CAP 20133, Milan, Italy 

munteanu@mat.unimi.it 

We present numerical results regarding the stability and parallel scalability of semiimplicit 

and implicit discretizations of Fitz Hugh-Nagumo, monodomain and bidomain 

systems. 

On the convergent solutions of a class of nonlinear ordinary 

differential equations 

Octavian G. MUSTAFA 

Department of Mathematics, University of Craiova 

Al. I. Cuza 13, Craiova, Romania 

octaviangenghiz@yahoo.com 

Via a special integral transformation, asymptotic integration results for ordinary 

differential equations are used to establish accurate asymptotic developments for radial 

solutions of the elliptic equation ∆u + K(|x|)e u = 0, |x| > x0 > 0, in the bidimensional 

case. 

32

Time-dependent invariant sets in system dynamics 

Octavian PĂSTRĂVANU, Mihaela-Hanako MATCOVSCHI and 

Mihail VOICU 

Department of Automatic Control and Applied Informatics 

Technical University “Gh. Asachi” of Iasi 

Blvd. Mangeron 53A, Iasi 700050, Romania 

{mvoicu, opastrav, mhanako}@ac.tuiasi.ro 

A general framework has been developed to explore the positively (flow) invariant 

sets, for a large class of time-variant, nonlinear dynamical systems. We introduce the 

concept of “diagonal invariance” defined by time-dependent diagonal matrices and for 

arbitrary Hölder norms. The flow invariance results are formulated as necessary and 

sufficient conditions for linear systems. The approach to nonlinear systems relies on sufficient 

conditions that allow formulating a comparison theorem where the comparison 

system has linear dynamics. The time-dependence of the invariant sets is considered 

either arbitrary, or constrained by requirements such as: boundedness, approaching the 

equilibrium point in accordance with a certain law, in particular decreasing exponentially. 

The special forms of the time-dependence exhibited by the invariant sets induce 

stability properties stronger than the standard ones known from the qualitative analysis. 

These properties are called “diagonally invariant stability”, “diagonally invariant asymptotic 

stability” and “diagonally invariant exponential stability”, and for their study we 

propose a methodology based on comparison principles. Thus, we also point out the 

role of essentially nonnegative matrices and of M matrices in defining the dynamics of 

the comparison system. We illustrate the applicability of our results for the nonlinear 

systems defined by Hopfield neural networks. 

Range condition in optimization 

N. H. PAVEL 

Department of Mathematics, Ohio University 

Athens OH 45701, USA 

npavel@bing.math.ohiou.edu 

In the early 1996 I have started the investigation of the problem: 

Minimize the cost functional L(y, u) 

subject to the constraints Ay = Bu + f. 

I have observed that if the linear operators A (unbounded) and B are densely defined 

on a Hilbert space, with closed range and if the range condition: 

(RC) R(A) ⊆ R(B), or vice versa: R(B) ⊆ R(A), 

holds, then one can obtain maximum principles of the form: 

If (y o , u o ) is an optimal pair, then there is p such that: 

A ∗ p ∈ −∂yL(y o , u o ), B ∗ p ∈ ∂uL(y o , u o ). 

33

Here ∂L could mean Fréchet derivative, or subdifferential or Clarke’s generalized gradients, 

depending on the conditions on L. Such maximum principles have applications 

to optimal control of some differential systems including some PDE. Moreover, this abstract 

scheme includes the essentials of many existing results on optimal control, i.e. 

it has a unifying effect, too. This idea was extensively extended to more general cases 

by Pavel and in some joint papers by Pavel, Aizicovici, Motreanu, and by Voisei in his 

Ph.D thesis. 

We are currently investigating some cases in which the above (RC) is not necessary. 

Limits of solutions to the initial boundary Dirichlet problem 

for semilinear hyperbolic equation with small parameter 

Andrei PERJAN 

USM, Chi¸snău, Republic of Moldova 

perjan@usm.md 

Let Ω ∈ Rn be an open and bounded set with the smooth boundary ∂Ω. Consider 

the following problems which will called (Pε) and (P0) respectively: 

� 

εutt(x, t) + ut(x, t) − ∆u(x, t) + |u(x, t)| pu(x, t) = f(x, � t), x ∈ Ω, t > 0, 

� 

u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω, u(x, t) � = 0, t ≥ 0, 

x∈∂Ω 

� 

vt(x, t) − ∆v(x, t) + |v(x, t)| pv(x, t) � = f(x, t), x ∈ Ω, t > 0, 

� 

v(x, 0) = u0(x), x ∈ Ω, v(x, t) � = 0, t ≥ 0. 

x∈∂Ω 

Theorem. Suppose that p ∈ [0, 2/(n − 2)] if n ≥ 3 and p ∈ [0, ∞) if n = 1, 2. 

If f ∈ W 2,1 (0, T ; L 2 (Ω)), u0, u1, α ∈ H 1 0(Ω) ∩ H 2 (Ω), then for any strong solution of 

the problem (Pε) the following relationships: u → v in C([0, T ]; L 2 (Ω)), u → v in 

L ∞ (0, T ; H 1 0(Ω)) and u ′ −v ′ −αe −t/ε → 0 in L ∞ (0, T ; L 2 (Ω)), as ε → 0, are valid, where 

α = f(0) − u1 + ∆u0 − |u0| p u0. 

Well-posedness and fixed point problems 

Adrian PETRUS¸EL and Ioan A. RUS 

Department of Applied Mathematics, Babe¸s-Bolyai University Cluj-Napoca 

400084, Cluj-Napoca, ROMANIA 

{petrusel, iarus}@math.ubbcluj.ro 

Definition 1. Let (X.d) be a metric space, Y ∈ P (X) and T : Y → Pcl(X) be a 

multivalued operator. The fixed point problem is well-posed for T with respect to Dd 

iff: 

(a1) FT = {x ∗ } 

34

(b1) If xn ∈ Y , n ∈ N and Dd(xn, T (xn)) → 0, as n → +∞ then xn → x ∗ , as 

n → +∞. 

Definition 2. Let (X.d) be a metric space, Y ∈ P (X) and T : Y → Pcl(X) be a 

multivalued operator. The fixed point problem is well-posed for T with respect to Hd 

iff: 

(a2) (SF )T = {x ∗ } 

(b2) If xn ∈ Y , n ∈ N and Hd(xn, T (xn)) → 0, as n → +∞ then xn → x ∗ , as 

n → +∞. 

The purpose of this paper is to define the concept of well-posed fixed point problem 

an to study it for the case of multi-valued operators. Several examples of well-posed 

fixed point problem are given. 

Numerical solutions of two-point boundary value problems for 

ordinary differential equations using particular Newton 

interpolating series 

Ghiocel GROZA and Nicolae POP 

Department of Mathematics, Technical University of Civil Engineering 

Lacul Tei 124, Sect. 2, 020396-Bucharest, Romania 

North University, Baia Mare, Department of Mathematics and Computer Science 

Victoriei 76, 43012-Baia Mare, Romania 

nic pop2002@yahoo.com 

Let {xk} k≥1 be a sequence of real numbers. We construct the polynomials 

u0(x) = 1, ui(x) = 

i� 

(x − xk), i = 1, 2, ..., 

k=1 

where x is a real variable. We call an infinite series of the form 

∞� 

aiui(x), (1) 

i=0 

where ai ∈ R, a Newton interpolating series with real coefficients ai at {xk} k≥1 . These 

series are useful generalization of power series which, in particular forms, were used in 

number theory to prove the transcendence of some values of exponential series. Taking 

into account the importance of power series in the theory of initial value problems for 

differential equations, it seems to be very useful to study Newton interpolating series 

in order to find the solution of the multipoint boundary value problems for differential 

equations. If we consider a function f : [0, 1] → R and the points xk ∈ [0, 1], then 

say that the function f is representable into a Newton interpolating series at {xk} k≥1 if 

there exists a series of the form (1), which converges uniformly to f on [0, 1]. 

35

We obtain results concerning Newton interpolating series and their derivatives and 

sufficient conditions for a function to be representable into a Newton interpolating series. 

The representation of solutions of particular differential equations through Newton 

interpolating series are also considered. 

Pareto reducibility and contractibility in vector optimization 

Nicolae POPOVICI 

Babe¸s-Bolyai University of Cluj, Romania 

popovici@math.ubbcluj.ro 

A multicriteria optimization problem is said to be Pareto reducible if the set of 

weakly efficient solutions can be represented as the union of the sets of (Pareto) efficient 

solutions of its subproblems (i.e. optimization problems obtained from the original 

one by selecting certain criteria). The principal aim of this presentation is to provide 

sufficient conditions for Pareto reducibility, under appropriate generalized convexity 

assumptions. We will also show that Pareto reducibility is intimately related to the 

contractibility of efficient sets. 

References: 

[1] J. Benoist, Contractibility of efficient frontier of simply shaded sets, Journal of Global 

Optimization 25, (2003), 321–335. 

[2] N. Popovici, Pareto reducible multicriteria optimization problems, Optimization 54, 

(2005), 253–263. 

[3] N. Popovici, Structure of efficient sets in lexicographic quasiconvex multicriteria 

optimization, Operations Research Letters 34, (2006), 142–148. 

Singular phenomena in nonlinear elliptic problems 

Vicent¸iu RĂDULESCU 

Department of Mathematics, University of Craiova 

200585 Craiova, Romania 

vicentiu.radulescu@math.cnrs.fr 

We consider two classes of nonlinear elliptic equations and we are concerned with 

the existence and the uniqueness of solutions, as well as with the study of the growth of 

solutions near the boundary. We first consider singular solutions of the logistic equation 

in anisotropic media and we discuss the blow-up rate of solutions in terms of Karamata’s 

regular variation theory. Next, we establish several bifurcation results for the Lane- 

Emden-Fowler equation with singular nonlinearity and convection term. 

36

Functional monotone VP and normed coercivity 

Mihai Turinici 

“A. Myller” Mathematical Seminar; “Al. I. Cuza” University 

11, Copou Boulevard; 700506 Ia¸si, Romania 

mturi@uaic.ro 

A functional extension is given for the monotone variational principle in Turinici [An. 

St. UAIC Iasi, 36 (1990), 329–352]. The obtained facts are then applied to establish 

(via conical Palais–Smale techniques) a monotone functional version of the coercivity 

result in Zhong [Nonlinear Analysis, 29 (1997), 1421–1431]. 

Viability for Nonlinear Reaction-Diffusion Systems 

Mihai NECULA and Ioan I. VRABIE 

Faculty of Mathematics, “Al. I. Cuza” University of Ia¸si, Romania 

necula@uaic.ro 

Faculty of Mathematics, “Al. I. Cuza” University of Ia¸si 

“O. Mayer” Mathematics Institute of the Romanian Academy, Ia¸si, Romania 

ivrabie@uaic.ro 

We prove several necessary and/or sufficient conditions for viability for certain classes 

of nonlinear reaction-diffusion systems governed by continuous perturbations of mdissipative 

operators. 

Integro-differential hamilton-jacobi-bellman equations 

associated to SDES driven by stable processes 

A. ZĂLINESCU 

Laboratoire de Mathématiques et Applications 

Université de La Rochelle, Avenue Michel Crépeau 

17042 La Rochelle, France 

azalines@univ-lr.fr 

We are interested in the way in which (nonlinear) Hamilton–Jacobi–Bellman equations 

(or variational inequalities) involving an integro-differential operator relate to jump 

diffusion processes via optimal stochastic control (and optimal stopping) problems. 

Our main domain of interest lies in the case where the dynamics has infinite variance, 

especially in the case where the jump diffusion process is a solution of a SDE driven by 

stable processes. 

We prove that the value function of the optimal control problem is a viscosity solution 

of the integro-differential variational inequality arising from the associated dynamic 

37

programming. We also establish comparison principles in the class of semi-continuous 

functions with polynomial growth of a given order. 

Regularity and qualitative properties for models of complex 

non-newtonian fluids 

Arghir ZARNESCU 

University of Chicago, Chicago, USA 

5480 S. Cornell Ave. Apt. 517, Chicago, IL 60615, USA 

zarnescu@math.uchicago.edu 

I will discuss models describing nematic liquid crystalline polymers. The models 

consist of a coupling between a nonlinear Fokker-Planck equation and a Navier-Stokes 

system. In the first part of the talk I will present regularity results for the systems, whose 

proof involve new estimates for the 2D Navier-Stokes equations with nearly singular 

forcings. In the second part of the talk I will discuss qualitative properties of solutions 

for simplified models. 

The first part is joint work with P. Constantin, C. Fefferman and E. S. Titi. 

38

Posters 

On the convergence of solutions to a certain fifth order 

nonlinear differential equation 

Olufemi Adeyinka ADESINA 

Department of Mathematics, Obafemi awolowo University, Ile-Ife, Nigeria 

oadesina@oauife.edu.ng 

In this paper, sufficient conditions for convergence of solutions to the fifth order 

nonlinear differential equation: 

x (v) + ax (iv) + bx ′′′ + cx ′′ + g(x ′ ) + h(x) = p(t, x, x ′ , x ′′ , x ′′′ , x (iv) ), 

in which a, b and c are positive constants, functions h(x) and p(t, x, x ′ , 

x ′′ , x ′′′ , x (iv) ) are real valued and continuous in their respective arguments are obtained. 

The function h(x) is not necessarily differentiable but satisfies an incrementally ratio 

h(ζ + η) − h(ζ) 

η 

where I0 is a closed Routh–Hurwitz interval. 

∈ I0, η �= 0, 

Competition in patchy space with cross diffusion and toxic 

substances 

Shaban ALY 

Department of Mathematics, Faculty of Science, Al-Azhar University 

Assiut 71511, Egypt 

shhaly12@yahoo.com 

In this paper we formulate a Lotka-Volterra competitive system affected by toxic 

substances in two patches in which the per capita migration rate of each species is 

in.uenced not only by its own but also by the other one.s density, i.e. there is cross 

diffusion present. Numerical studies show that at a critical value of the bifurcation 

parameter the system undergoes a Turing bifurcation and the cross migration response 

is an important factor that should not be ignored when pattern emerges. 

39

A Stability Theorem for Functional Differential Equations with 

Impulse Effect 

J. O. ALZABUT 

Department of Mathematics and Computer Science 

Çankaya University, 06530 Ankara, Turkey 

jehad@cankaya.edu.tr 

In this paper, we are concerned with functional differential equation with impulse 

effect of the form � x ′ (t) = A(t)x(t) + B(t)x(t − τ), t �= θi, 

∆x(θi) = Cix(θi) + Dix(θi−j), i ∈ N, 

where A, B are n × n continuous bounded matrices, τ > 0 is a positive real number, 

Ci, Di are n × n bounded matrices and j ∈ N is fixed. It is shown that if a functional 

differential equation with impulse effect of the above form verifies a Perron condition 

then its trivial solution is uniformly asymptotically stable. 

On nonlinear diffusion problems with strong degeneracy 

Kaouther AMMAR 

TU Berlin, Institut für Mathematik 

Strasse des 17 juni 135 MA 6-4 10623 Berlin, Germany 

ammar@math.tu-berlin.de 

In this paper we prove existence of a unique weak entropy solution for a degenerate 

problem of the type 

⎧ 

⎪⎨ b(v)t − ∆g(v) + div Φ(v) = f on Q :=]0, T [×Ω 

(Pb,g)(v0, a, f) g(v) = g(a) 

⎪⎩ 

b(v)(0, ·) = b(v0) 

on Σ :=]0, T [×∂Ω 

on Ω 

where b, g : R → R are Lipschitz continuous, non-decreasing such that b(0) = g(0) = 0 

and R(g + b) = R. 

Deterministic multivariate model for simulation of downstream 

BIVAL automatic controller in irrigation systems 

L. ROSU, A. BARBULESCU, S. HANCU and A. DUMITRU 

Department of Mathematics, Ovidius University, Constanta, Romania 

abarbulescu@univ-ovidius.ro 

40

Providing irrigation canals with automatic controllers leads to increase the exploitation 

performance. 

The major part of the mathematical simulation models are based on the numerical or 

analytical solution of the nonlinear equations with partial derivatives of hyperbolic type 

that govern the unsteady flow in the canals equipped with the automatic regulators. 

In this paper we offer an answer on how to conceive and use an analytical model 

for the design of the automatic irrigation canals for practical important situations. The 

model was obtained by analytical integration of the linearized equations based on the 

hypothesis of small oscillation theory and the properties of the Fourier transforms. It 

can be used to predict the behavior of the system under the perturbation factors. 

Semi-cardinal models for multivariable interpolation 

Aurelian BEJANCU 

Kuwait University 

abejancu@yahoo.co.uk 

Schoenberg’s ‘semi-cardinal interpolation’ (SCI) model in univariate spline theory 

constructs a polynomial spline function that interpolates values given on the grid Z+ of 

non-negative integers. We present an overview of recent multivariable extensions of the 

SCI model, focusing on the complete results obtained for interpolation on the semi-plane 

grid Z+ × Z from a space of triangular box-splines. The box-spline SCI schemes employ 

boundary conditions that extend the ‘natural’ and ‘not-a-knot’ end-point conditions of 

cubic spline interpolation. The analysis of the localization and polynomial reproduction 

properties of the bivariate SCI schemes proves that the ‘natural’-type boundary conditions 

induce a halving effect in accuracy, while the ‘not-a-knot’-type conditions achieve 

maximal accuracy (Bejancu A., J. Comput. Appl. Math., to appear; Bejancu A., Sabin 

M.A., Adv. Comput. Math. 22 (2005), 275-298). 

A Viability Result for Semilinear Reaction-Diffusion Sytems 

Monica BURLICĂ and Daniela ROS¸U 

Chair of Mathematics, “Gheorghe Asachi” Technical University of Ia¸si, Romania 

monicaburlica@yahoo.com, rosudaniela100@yahoo.com 

Using some some topological assumptions, expressed by the Kuratowski measure of 

noncompactness, we establish several necessary and sufficient conditions for viability for 

various classes of semilinear reaction-diffusion systems. 

41

Some new regularity conditions for Fenchel duality in real 

linear spaces 

Radu Ioan BOT¸ , Ernö Robert CSETNEK, Gert WANKA 


robert.csetnek@mathematik.tu-chemnitz.de 

We consider a convex optimization problem in a real linear space. Using the theory 

of conjugate functions we attach to it the Fenchel dual problem. Then we give a new 

regularity condition which guarantees strong duality between these two problems. To 

this end, we employ some abstract convexity notions. In contrast to other conditions 

given in the literature by Elster and Nehse ([1]) and Lassonde ([2]), written in terms 

of some algebraic interior point conditions of the effective domains of the functions 

involved, the one given by us is formulated by using the epigraphs of their conjugates. 

We prove that our condition is weaker than the aforementioned regularity conditions. 

We treat some particular cases of these convex optimization problems and also we give 

a sufficient condition for the subdifferential sum formula of a convex function with the 

precomposition of another convex function with a linear mapping. 

References: 

[1] Elster, K.H., Nehse, R. Zum Dualitätssatz von Fenchel, Mathematische Operationsforschung 

und Statistik 5, vol. 4/5, (1974), 269–280. 

[2] Lassonde, M. Hahn-Banach theorems for convex functions, Minimax theory and 

applications, 135–145, Nonconvex Optimization and its Applications 26, Kluwer Academic 

Publishers, Dordrecht, (1998). 

Study of a carrier dependent infectious disease-cholera 

Prasenjit DAS 

Research Scholar, Department of Mathematics, Jadavpur University 

Kolkata-700 032, India 

jit das2000@yahoo.com 

This paper analyzes an epidemic model for carrier dependent infectious disease - 

cholera. Existence criteria of carrier-free equilibrium point and endemic equilibrium 

point (unique or multiple) are discussed. Some threshold conditions are derived for 

which disease-free, carrier-free as well as endemic equilibrium become locally stable. 

Further global stability criteria of the carrier-free equilibrium and endemic equilibrium 

are achieved. Conditions for survival of all populations are also determined. Lastly 

numerical simulations are performed to validate the results obtained. 

42

Output feedback stabilization of sampled-data nonlinear 

systems by receding horizon control via discrete-time 

approximations 

A. M. ELAIW 

Al-Azhar University (Assiut), Faculty of Science 

Department of Mathematics, Assiut, EGYPT 

a m elaiw@yahoo.com 

This paper is devoted to the stabilization problem of sampled-data nonlinear systems 

by output feedback receding horizon control. The observer is designed via an approximate 

discrete-time model of the plant. We investigate under what conditions this design 

achieve practical stability for the exact discrete-time model. 

On the Crossing-Time for Two-dimensional Piecewise Linear 

Systems 

A. GARCÍA a , S. BIAGIOLA b , J. FIGUEROA c , L. CASTRO d 

O. AGAMENNONI e 

a,b,c,e: Departamento de Ingeniería Eléctrica y de Computadoras, Universidad 

Nacional del Sur, Alem 1253, 8000 Bahía Blanca, Argentina 

{agarcia, biagiola, figueroa, oagamen}@uns.edu.ar 

d: Departamento de Matemática, Universidad Nacional del Sur, Alem 1253, 8000 

Bahía Blanca, Argentina, lcastro@uns.edu.ar 

Piecewise Linear (PWL) models have proved to be a useful tool for the approximation 

of nonlinear dynamical systems. The dynamics of the real physical system is 

approximated in a region of interest. For this goal, a finite number of simplices is used 

when a PWL model is chosen. The PWL systems is given by a linear time invariant 

system, which varies from one simplex to the other. A relevant topic when using PWL 

models is the determination of the state of the system when the trajectory crosses from 

one simplex to another one. An equivalent dilemma is the calculus of the time value 

at which this crossing takes place. In this paper, we provide a formula for determining 

the crossing time (CT) when a given linear time invariant system reaches a frontier 

between simplices. Explicit expressions for the R 2 case are derived and some examples 

are presented. 

43

Existence and uniqueness results in the micropolar mixture 

theory of porous media 

Ionel-Dumitrel GHIBA 

“Octav Mayer” Mathematics Institute, Romanian Academy of Science 

Ia¸si Branch, 8 Bd. Carol I, 700506-Ia¸si, Romania 

ghiba dumitrel@yahoo.com 

In this paper we study the existence and uniqueness of solutions for the initial– 

boundary value problem associates with a mixture of a micropolar elastic solid and an 

incompressible micropolar viscous fluid. We use some results of the semigroup operators 

to obtain an existence and uniqueness theorem for the initial value problem with 

homogeneous Dirichlet boundary conditions. Continuous dependence of the solutions 

upon the initial data and supply terms is also established. 

Compactness for Linear Evolution Equations with Measures 

Gabriela GROSU 

Chair of Mathematics, “Gheorghe Asachi” Technical University of Ia¸si, Romania 

gcojan@yahoo.com 

We prove a compactness result for the solution operator g ↦→ u attached to a linear 

evolution equation with measures, du = Adt + dg, u(0) = ξ, where A generates a C0semigroup 

in a real Banach space X and g belongs to a certain family of functions, with 

bounded variation, from [ 0, T ] to X. 

A sufficient condition for null controllability of nonlinear 

control systems 

A. HEYDARI and A.V. KAMYAD 

Payame Noor Univ., Fariman, IRAN 

aghileheydari@yahoo.com 

Ferdowsi Univ., Mashhad, IRAN 

Classic control methods such as Minimum principle of Pontyagin, Bang-Bang principle 

and other methods aren’t useful for solving optimal control systems (OCS) specially 

optimal control of nonlinear systems (OCNS). 

In this paper we introduce a new approach for ONCS, by using a combination of 

atomic measures. We define a criterion for controllability of lumped nonlinear control 

systems and when the system is nearly null controllable, we determine controls and 

states. 

At last we use this criterion to solve some numerical examples. 

44

Farkas-type results for fractional programming problems 

Radu Ioan BOT¸ , Ioan Bogdan HODREA, Gert WANKA 


hio@mathematik.tu-chemnitz.de 

Considering a constrained fractional programming problem, we present some necessary 

and sufficient conditions which ensure that the optimal objective value of the 

considered problem is greater than or equal to a given real constant. More precisely, 

we give necessary and sufficient conditions which ensure that x ∈ X, h(x) ≤ 0 ⇒ 

f(x)/g(x) ≥ λ, where the nonempty convex set X ⊂ R n , the functions f : R n → R, 

g : R n → R and h : R n → R m and the real constant λ are given. As usual for a fractional 

programming problem, we assume g(x) > 0 for all x feasible. The desired results 

are obtained using the Fenchel-Lagrange duality approach applied to some optimization 

problems with convex, respectively, difference of convex (DC) objective functions and 

finitely many convex inequality constraints. Recently, Bot¸ and Wanka ([1]) have presented 

some Farkas-type results for inequality systems involving finitely many convex 

functions using an approach based on the theory of conjugate duality for convex optimization 

problems. Their results are naturally extended to the problem we treat and, 

moreover, it is shown that some other recent statements can be derived as special cases 

of our general result. 

References: 

[1] R. I. Bot¸ and G. Wanka Farkas-type results with conjugate functions, SIAM 

Journal on Optimization No. 15 Vol. 2 (2005), 540–554. 

[2] W. Dinkelbach On nonlinear fractional programming, Management Science No. 

13 Vol. 7 (1967), 492–498. 

[3] J. E. Martinez-Legaz and M. Volle Duality in D. C. programming: The case of 

several D. C. constraints, Journal of Mathematical Analysis and Applications No. 237 

Vol 2 (1999), 657–671. 

The passage of long waves above a vertical barrier: method of 

complex variable for calculation of the local disturbances 

A. LAOUAR and A. GUERZIZ 

Department of Mathematics, Faculty of Sciences 

University of Annaba, P. O. Box 12, 23000 Annaba, ALGERIA 

laouarhamid@yahoo.com 

Department of Physics, Faculty of Sciences 

University of Annaba, P. O. Box 12, 23000 Annaba, ALGERIA 

The generalized theory of the shallow-water [1, 2] is applied to the first order approximation 

for the calculation of the local disturbances caused by the presence of an 

immersed vertical barrier. By using the appropriate complex variable theory [3], the 

45

flow is entirely given. This method gives a new physical interpretation of calculations. 

For the certain forms of obstacles, the results can be obtained directly by the knowledge 

of the flows at adapted classical potential. 

References: 

[1] Barthelemy E., Kabbaj A. & Germain J.P, Long surface wave scattered by a step 

in a two-layer fluid. Fluid Dyn. Res., 26, pp 235-255, 2000. 

[2] Dean W.R., On the reflexion of surface waves by a submerged plane barrier. Proc. 

Cam. Phil. Soc. vol 41 pp 231-238, 1945. 

[3] Guerziz A., Etude théorique et expérimentale de la cinématique fine des ondes 

longues de gravité au voisinage des obstacles. Thèse de doctorat U.J.F Grenoble, 1992 

[4] Laouar A., The method of lines and invariant imbedding for free boundary problems 

of Hel-Shaw (submetted 2006), Inter. J. An. Num. and Modeling. 

Parallel backward shooting method for solving stiff IVPs 

Dr. Abdulhabib A. A. MURSHED 

Faculty of Applied Science (Vice Dean) Thamar University Yemen, 

habib12004@yahoo.com 

This paper investigate the effectiveness of parallel shooting technique for solving stiff 

initial value problems in such a way it can be computed in parallel. These techniques 

have the important capacity for controlling the numerical stability of the numerical 

integration methods. The overall performance of these techniques can be improved 

by increasing the number of the processors and corresponding matching points. The 

efficiency of this algorithm is illustrated by means of some numerical examples of stiff 

systems, and a comparison with Gear’s method is made. 

Integral inclusions in Banach spaces using Henstock-type 

integrals 

Bianca SATCO 

Facultatea de Inginerie Electrică, Univ. “S¸tefan cel Mare” 

Universităt¸ii 13, 720229 Suceava 

bisatco@eed.usv.ro 

We consider an integral inclusion, where the set-valued integral involved is of Henstocktype. 

An existence result is obtained via Monch’s fixed point theorem. A condition using 

a measure of noncompactness, as well as uniform integrability conditions appropriate to 

Henstock integral are required. Since the vector Henstock integral is more general than 

Bochner and Pettis integrals, our result extends many of previously obtained existence 

results, in single- or set-valued setting. 

46

Regularization of a solution to the cauchy problem for 

generalized system of Cauchy–Riemann in infinite domains 

Ermamat SATTOROV 

Norkulovich State University, Samarkand, UZBEKISTAN 

sattorov-e@rambler.ru 

We consider the problem of analytic continuation of the solution of the generalized 

system of Cauchy–Riemann in infinite domains through known values of the solution on 

a part of the boundary, i.e., the Cauchy problem. The generalized system of Cauchy- 

Riman is elliptic. The Cauchy problem for elliptic equations is well known to be illposed; 

a solution is unique but unstable (Hadamard’s example). To make the statement 

well-posed, we have to restrict the class of solutions. 

During the last decades the classical ill-posed problems of mathematical physics have 

been of constant interest. This direction in studying the properties of solutions to the 

Cauchy problem for the Laplace equation was originated in the fifties in the articles by 

M. M. Lavrent’ev (1956 y.) and S. N. Mergelyan (1956 y.) and was developed by V. K. 

Ivanov (1965 y.), Sh. Ya. Yarmukhamedov (1977 y.), et al. 

Good intermediate-rank lattice rules based on the L∞ weighted 

star discrepancy 

Vasile SINESCU 

Department of Mathematics, University of Waikato 

Private Bag 3105, Hamilton, New Zealand 

vs27@waikato.ac.nz 

Good lattice rules for numerical multiple integration may be constructed by using 

a ‘component-by-component’ technique, which is a ‘greedy-type’ algorithm based on 

successive 1-dimensional searches. 

Here, we assume that variables are arranged in the decreasing order of their importance. 

Since the first variables are in some sense more important than the rest, there is 

an interest in considering ‘intermediate-rank’ lattice rules. The ‘goodness’ of a lattice 

rule is assessed here by a ‘weighted star discrepancy’, which is based on a L∞ maximum 

error. 

The talk intends to present a survey of results regarding the construction of intermediaterank 

lattice rules under a ‘product-weighted’ setting. The existence and the construction 

of ‘good’ intermediate-rank lattice rules is proved by using an averaging argument (if 

the average is good, there must be a good one). The weighted star discrepancy for the 

lattice rules constructed here has order of magnitude of O(n −1 (ln n) d ). Under appropriate 

conditions over the weights, the weighted star discrepancy will have the optimal 

bound of O(n −1+δ ) for any δ > 0, where the involved constant is independent of d and 

n. Subsequent results for the Lp star discrepancy can be also deduced from the results 

on the L∞ weighted star discrepancy. 

47

Joint work with Stephen Joe (University of Waikato). 

On the minimum energy problem for linear systems 

Alina VIERU 

Department of Mathematics, “Gh. Asachi” Technical University, Ia¸si, ROMANIA 

vieru alina@yahoo.com 

This paper is concerned with some problems of controllability with minimum energy 

associated with linear systems in Banach spaces. First, we give conditions to have the 

controllability of a given pair of elements of the Banach space. Also, we find alternate 

ways of obtaining the minimum norm control by which a given initial state of the 

system can be steered to a given final state, within a given time interval. We treat these 

problems as a special case of an abstract minimum norm problem described by a linear 

mapping. We give examples in order to illustrate the theory. 

48

Contributing participants 

Adesina, O. A., 39 oadesina@oauife.edu.ng 

Aly, S., 39 shhaly12@yahoo.com 

Alzabut, J. O., 40 jehad@cankaya.edu.tr 

Ammar, K., 40 ammar@math.tu-berlin.de 

Anit¸a, S., 12 sanita@uaic.ro 

Azé, D., 1 aze@mip.ups-tlse.fr 

Bîrsan, M., 14 bmircea@uaic.ro 

Baker, A. A., 12 a a baker@yahoo.com 

Barbulescu, A., 40 abarbulescu@univ-ovidius.ro 

Beceanu, M., 13 mbeceanu@uchicago.edu 

Bejancu, A., 41 abejancu@yahoo.co.uk 

Bereanu, C., 14 bereanu@math.ucl.ac.be 

Beznea, L., 14 lucian.beznea@imar.ro 

Blizorukova, M., 15 msb@imm.uran.ru 

Bot¸, R. I., 16, 23, 42, 45 bot@mathematik.tu-chemnitz.de 

Bostan, M., 16 mihai.bostan@math.univ-fcomte.fr 

Bouremani, T., 17 bouremani@yahoo.com 

Buckdahn, R., 1 buckdahn@univ-brest.fr 

Burlică, M., 41 monicaburlica@yahoo.com 

Cernea, A., 18 acernea68@yahoo.com 

Chalishajar, D. N., 19 dipu17370@yahoo.com 

Chalishajar, H. D., 18 heena14672@yahoo.co.in 

Consiglieri, L., 19 lcconsiglieri@fc.ul.pt 

Corduneanu, C., 20 concord@uta.edu 

Csetnek, E. R., 42 robert.csetnek@mathematik.tu-chemnitz.de 

Das, P., 42 jit das2000@yahoo.com 

Dhakne, M. B., 20 mbdhakne@yahoo.com 

Donchev, T., 21 tdd51us@yahoo.com 

Dontchev, A. L., 2 ald@ams.org 

Elaiw, A. M., 12, 22, 43 a m elaiw@yahoo.com 

Engelbert, H.-J., 2 engelbert@minet.uni-jena.de 

Fattorini, H. O., 3 hof@math.ucla.edu 

Fursikov, A.V., 4 fursikov@mtu-net.ru 

García, A., 22, 43 agarcia@uns.edu.ar 

Gaudiello, A., 23 gaudiell@unina.it 

Ghiba, I.-D., 44 ghiba dumitrel@yahoo.com 

Grad, S. M., 23 grad@mathematik.tu-chemnitz.de 

Grosu, G., 44 gcojan@yahoo.com 

Hamedani, H. D., 24 h-hamedani@sbu.ac.ir 

Heydari, A., 44 aghileheydari@yahoo.com 

Hodrea, I. B., 45 hio@mathematik.tu-chemnitz.de 

Iannelli, M., 5 iannelli@science.unitn.it 

Jitara¸su, N., 24 jitarasu@usm.md 

Löhne, A., 25 andreas.loehne@mathematik.uni-halle.de 

Laouar, A., 45 laouarhamid@yahoo.com 

49

Luca-Tudorache, R., 25 rluca@hostingcenter.ro 

Lupu, M., 26 m.lupu@info.unitbv.ro 

Maksimov, V., 27 maksimov@imm.uran.ru 

Maleki, H. R., 27 maleki@sutech.ac.ir 

Marchini, E., 28 elsa.marchini@unimib.it 

Marinoschi, G., 28 gmarino@acad.ro 

Matcovschi, M., 29, 33 mhanako@ac.tuiasi.ro 

Maticiuc, L., 29 lucianmaticiuc@yahoo.com 

Mirică, S., 30 mirica@fmi.unibuc.ro 

Mishmaste Nehi, H., 31 hmnehi@hamoon.usb.ac.ir 

Mordukhovich, B. S., 6 boris@math.wayne.edu 

Mukherjee, D., 32 debasis mukherjee2000@yahoo.co.in 

Munteanu, M., 32 munteanu@mat.unimi.it 

Murshed, A. A., 46 habib12004@yahoo.com 

Mustafa, O. G., 32 octaviangenghiz@yahoo.com 

Nashed, Z., 6 znashed@mail.ucf.edu 

Necula, M., 37 necula@uaic.ro 

Păstrăvanu, O., 29, 33 opastrav@ac.tuiasi.ro 

Pardoux, E., 7 pardoux@latp.univ-mrs.fr 

Pavel, N. H., 33 npavel@bing.math.ohiou.edu 

Perjan, A., 34 perjan@usm.md 

Petrusel, A., 34 petrusel@math.ubbcluj.ro 

Pop, N., 35 nic pop2002@yahoo.com 

Popovici, N., 36 popovici@math.ubbcluj.ro 

Quincampoix, M., 9 marc.quincampoix@univ-brest.fr 

Ră¸scanu, A., 29 rascanu@uaic.ro 

Rădulescu, V., 36 vicentiu.radulescu@math.cnrs.fr 

Ro¸su, D., 41 rosudaniela100@yahoo.com 

Ro¸su, L., 40 lrosu@univ-ovidius.ro 

Roeckner, M. G., 9 roeckner@math.uni-bielefeld.de 

Satco, B., 46 bisatco@eed.usv.ro 

Sattorov, E., 47 sattorov-e@rambler.ru 

Scheiber, E., 26 e.scheiber@info.unitbv.ro 

Sinescu, V., 47 vs27@waikato.ac.nz 

Théra, M., 10 michel.thera@unilim.fr 

Tiba, D., 10 dan.tiba@imar.ro 

Turinici, G., 10 gabriel.turinici@dauphine.fr 

Turinici, M., 37 mturi@uaic.ro 

Ursescu, C., 11 corneliuursescu@yahoo.com 

Vieru, A., 48 vieru alina@yahoo.com 

Voicu, M., 33 mvoicu@ac.tuiasi.ro 

Vrabie, I. I., 37 ivrabie@uaic.ro 

Zălinescu, A., 37 azalines@univ-lr.fr 

Zarnescu, A., 38 zarnescu@math.uchicago.edu 

50

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