Null Controllability for Degenerate Parabolic Operators

Null Controllabilityfor Degenerate Parabolic OperatorsPiermarco CannarsaUniversity of Rome “Tor Vergata” (Italy)http://www.mat.uniroma2.itIFAC WORKSHOP — CDPS 2009Toulouse (France) July 20 – 24, 2009

degenerate parabolic operatorsO ⊂ R nwherewithbounded⎧⎪⎨ u t − Lu = f in O×]0, T [u(x, 0) = u 0 (x) x ∈ O⎪⎩(possibly) + b. c. on ∂O×]0, T [⎧⎪⎨ div(A(x)∇u) + lower order tmsLu = or⎪⎩Tr [A(x)∇ 2 u] + lower order tmsA(x) > 0 in O but A(x) ≥ 0 on ∂O

degenerate parabolic operatorsO ⊂ R nwherewithbounded⎧⎪⎨ u t − Lu = f in O×]0, T [u(x, 0) = u 0 (x) x ∈ O⎪⎩(possibly) + b. c. on ∂O×]0, T [⎧⎪⎨ div(A(x)∇u) + lower order tmsLu = or⎪⎩Tr [A(x)∇ 2 u] + lower order tmsA(x) > 0 in O but A(x) ≥ 0 on ∂O

Outline1 Examples of degenerate parabolic problems2 Null controllability for degenerate parabolic operatorsintroduction to null controllabilityone dimensional case3 Higher dimensional problems


examples of degenerate problems• laminar flow on flat plates(Crocco type equations)• mathematical finance(Black & Scholes type equations)• climate models(Budyko-Sellers)• population genetics(Wrigth-Fisher, Fleming-Viot diffusion processes)• domain invariance for stochastic flow(Friedman & Pinsky, Da Prato & Frankowska)






{ut − ( (1 − x 2 )u x)(1 − x 2 )u x|x=±1 = 0xa climate model= f (x) g(x, u) − h(u) x ∈ (−1, 1)Budyko-Sellers (1969)effect of solar radiation on climate✬✩x = sin α α✫✪• u(t, x) = sea-level zonally averaged temperature• f (x) = solar input• g(x, u) = co-albedo• h(u) = outgoing infrared radiation




Fleming-Viot diffusion processevolution of genetic types in a populationu t − Tr ( A(x)∇ 2 u ) = · · ·in O = {(x 1 , x 2 ) ∈ R 2 | x i ∈]0, 1[ , x 1 + x 2 ≤ 1}❅❅❅❅O ❅❅❅withA(x 1 , x 2 ) =(x1 (1 − x 1 ) −x 1 x 2−x 1 x 2 x 2 (1 − x 2 ))notice det A(x 1 , x 2 ) = x 1 x 2 (1 − x 1 − x 2 ) = 0 on ∂O





• X(·, x)invariant sets for stochastic flowsunique solution{dX(t) = b(X(t))dt + σ(X(t)) dW (t) t ≥ 0X(0) = x ∈ R n• b : R n → R n , σ : R n → L(R n ; R m ) Lipschitz• W (t) m-dimensional Brownian (Ω, F, {F t } t , P)• K ⊂ R n invariantx ∈ K =⇒ X(t, x) ∈ K P − a.s. ∀t ≥ 0✬ K x✩✫✪



two invariance problemsProblemgiven O ⊂ R n open (∂O = Γ) to study invariance of• closed domain• open domainOOKolmogorov operator{D(L 0 ) = { ϕ ∈ C(O) ∣ ∣ ϕ ∈ H 2 loc (O) , L 0ϕ ∈ C(O) }L 0 ϕ(x) := 1 2 Tr [A(x)∇2 ϕ(x)] + 〈b(x), ∇ϕ(x)〉x ∈ Owhere A(x) = σ(x)σ ∗ (x) ≥ 0oriented{distance from Γ❅d Γdist(x, Γ) if x ∈ Od Γ (x) =Γ ❅−dist(x, Γ) if x ∈ O c ❅ O ❅






Theoremconditions for invariance(a) ⇐⇒ (b) ⇐⇒ (c)(a) O invariant{(i) L0 d Γ (x) ≥ 0(b) ∀x ∈ Γ(ii) 〈A(x)∇d Γ (x), ∇d Γ (x)〉 = 0(c) O invariantref Friedman & Pinsky (1975)Da Prato & Frankowska (2004)C & Da Prato & Frankowska (2009)stochastic viability . . .invariance =⇒ L 0 degenerate on Γ







controlled parabolic equationsω ⊂⊂ O T > 0⎧⎪⎨ u t − div(A(x)∇u) = χ ω (x)f (t, x) in Q T := O×]0, T [u f → u(x, 0) = u 0 (x)x ∈ O⎪⎩(possibly + boundary conditions)• f controlχ ω characteristic function of ω• A(x) = ( a ij (x) ) ni,j=1• a ij = a ji ∈ C(O) ∩ C 1 (O)• positive definite in O (not in O)✬✩✓✏ω✒✑✫O✪

the null controllability problemwant to study null-controllability in time T > 0∀u 0 ∈ L 2 (O) ∃f ∈ L 2 (Q T ) :{u f (·, T ) ≡ 0∫Q T|f | 2 ≤ C T∫O |u 0| 2uniformly parabolic equations∃ m > 0 : A(x) ≥ m I n =⇒ null-controllability ∀ T > 0• Fattorini and Russell (1971), Russell (1978)• Lebeau and Robbiano (1995)• Fursikov and Emanouilov (1996)• Tataru (1997)



oadmap to null controllability• show equivalence with observability inequality{adjoint v t +div(A(x)∇v) = 0 in Q Tproblem v = 0 on Γ×]0, T [=⇒∫O∫ T ∫v 2 (x, 0) dx ≤ C T• prove observability by Carleman’s estimates > 0 large∫∫Q Ts 3 θ 3 (t)v 2} {{ }e 2sφ(x,t) dxdt ≤ C+sθ(t)|Dv| 2 +···0ωv 2 (x, t) dxdt∫ T0∫ωv 2 dxdtφ(x, t) = θ(t) [ e rψ(x) − e 2r‖ψ‖∞] any Dψ(x) ≠ 0 in O \ ω



difficulties in degenerate casefor degenerate parabolic equations:• observability (⇒ null controllability) may fail(for violent degeneracies)• φ in Carleman must be adapted to degeneracy• Carleman’s estimate up to higher order terms• Hardy’s inequality essential α ≠ 1∫∫d α−2Γw 2 dx ≤ C dΓ α |∇w|2 dxOO






one space dimensiona ∈ C([0, 1]) ∩ C 1 (]0, 1]) and a > 0 on ]0, 1]{u t − ( a(x)u x)x = f in Q T =]0, 1[×]0, T [u(x, 0) = u 0 (x)u 0 ∈ L 2 (0, 1) , f ∈ L 2 (Q T )Campiti, Metafune, Pallara (1998)weakly degenerate case: 1/a ∈ L 1 (0, 1)H 1 a(0, 1) ={u ∈ L 2 (0, 1) ∣ ∣∫ 1strongly degenerate case: 1/a /∈ L 1 (0, 1)H 1 a(0, 1) =0{u ∈ L 2 (0, 1) ∣ ∣}aux 2 dx < ∞ & u(0) = 0 = u(1)∫ 10}aux 2 dx < ∞ & u(1) = 0




well-posednessin both cases{D(A) ={u ∈ H1a (0, 1) ∣ ∣ au x ∈ H 1 (0, 1)}Au = ( au x)xgenerates analytic semigroup in L 2 (0, 1)(u• t − `a(x)u x´= f in Q xT =]0, 1[×]0, T [u(x, 0) = u 0 (x)unique solution u ∈ C(0, T ; L 2 (0, 1)) ∩ L 2 (0, T ; Ha(0, 1 1))• maximal regularityu 0 ∈ H 1 a(0, 1) =⇒ u ∈ H 1 (0, T ; L 2 (0, 1))∩L 2 (0, T ; D(A))• strongly degenerate caseu ∈ D(A) =⇒ au x(x→0)−→ 0





the simplest exampleω =]a, b[⊂⊂]0, 1[ a(x) = x α (α > 0)u t − ( x α u x)x = χ ωf , u(0, x) = u 0 (x)C & Martinez & Vancostenoble (2008)⎧⎪⎨ false α ≥ 2 (→{regional null controllability)n. c.any b.c. 0 ≤ α < 1 weak⎪⎩ true 0 ≤ α < 2Neumann b.c. 1 ≤ α < 2 strongTregional 0 ω 1Figure: regional null controllability

the simplest exampleω =]a, b[⊂⊂]0, 1[ a(x) = x α (α > 0)u t − ( x α u x)x = χ ωf , u(0, x) = u 0 (x)C & Martinez & Vancostenoble (2008)⎧⎪⎨ false α ≥ 2 (→{regional null controllability)n. c.any b.c. 0 ≤ α < 1 weak⎪⎩ true 0 ≤ α < 2Neumann b.c. 1 ≤ α < 2 strongTregional 0 ω 1Figure: regional null controllability

why null controllability fails for α ≥ 2• classical change of variable (Courant-Hilbert)y(x) =∫ 1xdss α/2 U(y(x), t) = x −α/4 u(x, t)transforms equation into˜ω =]˜b, ã[ boundedU t − U yy + V α (y)U = χ eω F0 < y < ∞V α (y) = α 2( 34 α − 1) 1[2−(2−α)y] 2 bounded for α ≥ 2• Escauriaza, Seregin, Sverak (2003, 2004)U(·, T ) = 0 ⇐⇒ spt ( U(·, 0) ) ⊂ [0, ã[• u t − ( x α u x)x = χ ωf{no [0, 1]yes if spt(u 0 ) ⊂]a, 1]





Carleman’s estimate 0 < α < 2w t + ( x α w x)x= f in ]0, 1[×]0, T [ + b. c.let φ(t, x) = θ(t)ψ(x) where(θ(t) =1t(T − t)) 4ψ(x) = x 2−α − 2(2 − α) 2Theorem∃ s 0 , C > 0 such that ∀s ≥ s 0∫∫Q T(sθx α w 2 x + s 3 θ 3 x 2−α w 2) e 2sφ dxdt∫∫∫ T≤ C |f | 2 e 2sφ {dxdt + C sθw2x e 2sφ} dt| x=1Q T 0




adjoint problem for 0 < α < 2 (⋆){v t + ( x α v x)+ b. c.x= 0 in ]0, 1[×]0, T [Carleman’s estimate → localization of v(sθx∫∫Q α vx 2 + s 3 θ 3 x} 2−α{{ } v 2) ∫ T ∫e 2sφ dxdt ≤ C TT 0→0• zero order estimate too weak• need Hardy’s inequality (α ≠ 1) ∀w ∈ H 1 α(0, 1)ω|v| 2 dxdt∫ 10x α−2 w 2 dx ≤∫4 1(α − 1) 2 x α wx 2 dx0

adjoint problem for 0 < α < 2 (⋆){v t + ( x α v x)+ b. c.x= 0 in ]0, 1[×]0, T [Carleman’s estimate → localization of v(sθx∫∫Q α vx 2 + s 3 θ 3 x} 2−α{{ } v 2) ∫ T ∫e 2sφ dxdt ≤ C TT 0→0• zero order estimate too weak• need Hardy’s inequality (α ≠ 1) ∀w ∈ H 1 α(0, 1)ω|v| 2 dxdt∫ 10x α−2 w 2 dx ≤∫4 1(α − 1) 2 x α wx 2 dx0

observability v t + ( x α v x)x = 0 (⋆)• t ↦→ ∫ 10 x α v 2 x dx increasing• integrate & use Carleman’s estimate∫ 10x α v 2 x (x, 0) dx ≤ 2 T∫ 3T /4 ∫ 1T /4 0x α v 2 x (x, t) dxdt• use Hardy’s inequality∫ 10≤ C T∫ ∫Q Tθ(t)x α v 2 x (x, t)e 2sφ(x,t) dxdtx α−2 v 2 (x, 0) dx ≤∫ 10∫ T ∫≤ C T0x α v 2 x (0, x) dxωv 2 (x, t) dxdt∫ T ∫≤ C v 2 (x, t)dxdt0 ω

observability v t + ( x α v x)x = 0 (⋆)• t ↦→ ∫ 10 x α v 2 x dx increasing• integrate & use Carleman’s estimate∫ 10x α v 2 x (x, 0) dx ≤ 2 T∫ 3T /4 ∫ 1T /4 0x α v 2 x (x, t) dxdt• use Hardy’s inequality∫ 10≤ C T∫ ∫Q Tθ(t)x α v 2 x (x, t)e 2sφ(x,t) dxdtx α−2 v 2 (x, 0) dx ≤∫ 10∫ T ∫≤ C T0x α v 2 x (0, x) dxωv 2 (x, t) dxdt∫ T ∫≤ C v 2 (x, t)dxdt0 ω

More general 1-d problems• divergence form• Martinez & Vancostenoble (2006) u t − ( a(x)u x)x = χ ωf• Alabau & C Fragnelli (2006) u t − ( a(x)u x)x + g(u) = χ ωf• non-divergence C & Fragnelli & Rocchetti (2007, 2008)u t − a(x)u xx − b(x)u x = χ ω f• using (invariant) measure ρ(x)dx = 1Ra(x) e x b(s)1/2 a(s) ds dx[ ]operator transforms a(x)u xx + b(x)u x = 1ρ(x) ρ(x)a(x)ux( {self-adjoint L 2 ρ 0, 1) := u ∈ L 2 (0, 1) ∣ ∫ 10 u2 ρ dx < ∞ }• degenerate/singular problemsVancostenoble & Zuazua (2008), Vancostenoble (2009)u t − ( x α u x)x −λx β u = χ ωfx




the simplest problem in 2dn = 2⎧⎪⎨ u t − div(A(x)∇u) = χ ω (x)f (t, x) in Q Tu(x, 0) = u 0 (x)x ∈ O⎪⎩(possibly) + boundary conditions on Γσ(A(x)) = {λ 1 (x), λ 2 (x)} eigenvectors ε 1 (x), ε 2 (x){λ 1 (x) = d Γ (x) α , ε 1 (x) = −Dd Γ (x) = ν Γ (Π Γ (x)) near Γλ 2 (x) ≥ m > 0✬✩∀x ∈ Oε 2 (x) O❅■❅ x✫ε 1 (x)✠ Π Γ (x)Γ✪




concluding remarks• for n = 1 null controllability yields strong Feller property oftransition semigroup associated to SPDEdu(t) = (au x (t)) x dt + χ ω dW (t)needed to study invariant measures• application to inverse problemsu t − ( x α u x)x = r(t, x)f (x) |r(t 0, ·)| > 0f (·)←−Tu(t 0 , ·)u x (·, 1)ref0 1C & Tort & Yamamoto (in progress)




Thank you

Null Controllability for Degenerate Parabolic Operators

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?