Lecture 2: Controllability of parabolic equations Enrique FERN ´ ANDEZ-CARA Dpto. E.D.A.N. - Univ. of Sevilla Controllability concepts Results for heat-like equations and systems Further comments and applications E. Fern´ andez-Cara Controllability and parabolic PDEs
Outline Generalities 1 Basic results for the heat equation 2 Controllability concepts The main results Additional comments Controllability of other parabolic systems 3 Non-scalar coupled systems Stochastic controllability E. Fern´ andez-Cara Controllability and parabolic PDEs
Generalities Controllability of time-dependent systems An abstract problem: � y t − Ay = Bv , t ∈ ( 0 , T ) y ( 0 ) = y 0 • A : D ( A ) ⊂ H �→ H , B : D ( B ) ⊂ U �→ H are linear • v = v ( t ) is the control, y = y ( t ) is the state Exact controllability problem: Choose y 0 , y 1 ∈ H (the space of states) ∃ v such the state associated to y 0 satisfies y ( T ) = y 1 ? Relaxing y ( T ) = y 1 : other controllability problems Solvability can depend on time reversibility, regularity, structure of the control set, size of T , etc. Many contributions: [Fattorini, Russell, J-L Lions, . . . ] ; more recently, [Fursikov, Imanuvilov, Lasiecka, Triggiani, Lebeau, Zuazua, Coron, . . . ] E. Fern´ andez-Cara Controllability and parabolic PDEs
Basic results for the heat equation Controllability concepts Ω ⊂ R N , T > 0 ( N ≥ 1), regular Γ = ∂ Ω ; ω ⊂ Ω (small) The state equation: y t − ∆ y = v 1 ω in Ω × ( 0 , T ) y = 0 on Γ × ( 0 , T ) (1) y ( 0 ) = y 0 in Ω We assume: y 0 ∈ L 2 (Ω) , v ∈ L 2 ( ω × ( 0 , T )) ∃ ! solution y ∈ C 0 ([ 0 , T ]; L 2 (Ω)) Set R ( T ; y 0 ) = { y ( · , T ) : v ∈ L 2 ( ω × ( 0 , T )) } . Then: • ( 1 ) is approximately controllable if R ( T ; y 0 ) = L 2 (Ω) for all y 0 • It is exactly controllable if R ( T ; y 0 ) = L 2 (Ω) for all y 0 • It is null controllable if R ( T ; y 0 ) ∋ 0 for all y 0 E. Fern´ andez-Cara Controllability and parabolic PDEs
Basic results for the heat equation Controllability concepts � y t − ∆ y = v 1 ω in Ω × ( 0 , T ) y ( 0 ) = y 0 in Ω , etc. First results: • EC cannot hold, except possibly if ω = Ω : y ( T ) is always smooth in Ω \ ω • NC is equivalent to the EC to the states in S ( T )( L 2 (Ω)) , i.e. R ( T ; y 0 ) ∋ 0 ∀ y 0 ⇔ R ( T ; y 0 ) ∋ S ( T )( L 2 (Ω)) ∀ y 0 (write y = S ( t ) y 0 + z and work with z ) Consequence: NC ⇒ AC • We will see: AC and NC hold. Thus, AC �⇔ EC for PDEs! (contrarily to ODEs) E. Fern´ andez-Cara Controllability and parabolic PDEs
Basic results for the heat equation The main results Theorem 1 AC holds for ( 1 ) for any ω and any T > 0 S KETCH OF THE P ROOF : Fix ω and T > 0 ( 1 ) is AC ⇔ R ( T ; 0 ) is dense in L 2 (Ω) ⇔ R ( T ; 0 ) ⊥ = { 0 } ( R ( L ) is dense iff L ∗ is one-to-one, uniqueness) Assume ϕ 0 ∈ R ( T ; 0 ) ⊥ and introduce � − ϕ t − ∆ ϕ = 0 in Ω × ( 0 , T ) (2) ϕ ( T ) = ϕ 0 in Ω , etc. � � � ϕ 0 y ( T ) dx = 0 ∀ v Then: ϕ v dx dt = ω × ( 0 , T ) Ω Hence: AC holds iff the following uniqueness property is true: ϕ solves ( 2 ) , ϕ = 0 in ω × ( 0 , T ) ⇒ ϕ ≡ 0 But this is true: the solutions to ( 2 ) are analytic in space! ✷ E. Fern´ andez-Cara Controllability and parabolic PDEs
Basic results for the heat equation The main results Also, possible to construct the “best” control: ϕ 0 and ˆ ϕ 0 minimizing u = ˆ ϕ | ω × ( 0 , T ) , with ˆ ϕ associated to ˆ J ε ( ϕ 0 ) = 1 � � � | ϕ | 2 dx dt + ε � ϕ 0 � L 2 − ϕ 0 y 1 dx , (3) 2 ω × ( 0 , T ) Ω QUESTIONS: A general linear system: � y t − ∇ · ( D ( x , t ) ∇ y ) + B ( x , t ) · ∇ y + a ( x , t ) y = v 1 ω y | t = 0 = y 0 , etc. with D ij , B i , a ∈ L ∞ , D ( x , t ) ξ · ξ ≥ α 0 | ξ | 2 a.e. Minimal regularity hypotheses to have AC for all ω and T ? Can we do again the same? (unknown for N ≥ 2) E. Fern´ andez-Cara Controllability and parabolic PDEs
Basic results for the heat equation The main results Theorem 2 [Fursikov-Imanuvilov, 1991 . . . ] NC holds for ( 1 ) for any ω and any T > 0 S KETCH OF THE P ROOF : Fix again ω and T > 0 ( 1 ) is NC ⇔ R ( T ; y 0 ) ∋ 0 for all y 0 ⇔ Observability of ( 2 ) , i.e. �� | ϕ | 2 dx dt ∀ ϕ 0 ∈ L 2 (Ω) � ϕ ( 0 ) � 2 L 2 ≤ C (4) ω × ( 0 , T ) ( R ( M ) ⊂ R ( L ) iff � M ∗ ϕ 0 � ≤ C � L ∗ ϕ 0 � for all ϕ 0 ) (4) is a consequence of Carleman: �� �� ρ − 2 | ϕ | 2 dx dt ≤ C ρ − 2 | ϕ | 2 dx dt (5) Ω × ( 0 , T ) ω × ( 0 , T ) This holds for appropriate ρ = ρ ( x , t ) , with ρ ∼ e 1 / ( T − t ) The proof of (5) is complicate E. Fern´ andez-Cara Controllability and parabolic PDEs
Basic results for the heat equation The main results From (5) we get: � 3 T / 4 Ω × ( T / 4 , 3 T / 4 ) ρ − 2 | ϕ | 2 dx dt � ϕ ( 0 ) � 2 � ϕ ( t ) � 2 �� L 2 ≤ C L 2 ≤ C T / 4 ω × ( 0 , T ) ρ − 2 | ϕ | 2 dx dt ≤ C ω × ( 0 , T ) | ϕ | 2 dx dt �� �� ≤ C ✷ O THER PROOF : Construct directly v such that y ( T ) = 0: y ( t ) = � i y i ( t ) ϕ i , with − ∆ ϕ i = λ i ϕ i in Ω , ϕ i = 0 on ∂ Ω , etc. • First, v | ( 0 , T / 2 ) such that y i ( T / 2 ) = 0 for λ i ≤ µ ; then v | ( T / 2 , 3 T / 4 ) ≡ 0 • Then, v | ( 3 T / 4 , 7 T / 8 ) such that y i ( 7 T / 8 ) = 0 for λ i ≤ 2 µ and v | ( 7 T / 8 , 15 T / 16 ) ≡ 0, etc. . . . The key point: a finite dimensional observability inequality √ µ � 2 � � � � a 2 i ≤ e C 0 a i ϕ i dx (LR) � � � � ω λ i ≤ µ λ i ≤ µ [Lebeau-Robbiano, 1995] E. Fern´ andez-Cara Controllability and parabolic PDEs
Basic results for the heat equation The main results QUESTIONS: Again the general linear system: � y t − ∇ · ( D ( x , t ) ∇ y ) + B ( x , t ) · ∇ y + a ( x , t ) y = v 1 ω y | t = 0 = y 0 , etc. with D ij , B i , a ∈ L ∞ , D ( x , t ) ξ · ξ ≥ α 0 | ξ | 2 a.e. Minimal regularity hypotheses to have NC for all ω and T ? For time-independent D , B , a , minimal hypotheses for (LR)? (again unknown for N ≥ 2) Carleman, observability and NC holds if D ∈ W 1 , ∞ , but . . . E. Fern´ andez-Cara Controllability and parabolic PDEs
Basic results for the heat equation Additional comments Boundary NC: Ω , T as before; γ ⊂ ∂ Ω open, non-empty (small) The state equation: y t − ∆ y = 0 in Ω × ( 0 , T ) on ∂ Ω × ( 0 , T ) y = h 1 ω (6) y ( 0 ) = y 0 in Ω Theorem 3 AC and NC hold for ( 6 ) for any γ and any T > 0 Very easy: “extend” Ω and (6) to a problem in ˜ y t − ∆ y = v 1 ω Ω × ( 0 , T ) on ∂ ˜ y = 0 Ω × ( 0 , T ) y 0 in ˜ y ( 0 ) = ˜ Ω with ω ⊂ ˜ Ω \ Ω , apply Theorems 1 and 2 and then restrict to Ω E. Fern´ andez-Cara Controllability and parabolic PDEs
Basic results for the heat equation Additional comments The semilinear heat equation: y t − ∆ y + f ( y ) = v 1 ω in Q y = 0 on Σ (7) y ( 0 ) = y 0 in Ω Theorem 4 [EFC-Zuazua, 2000] f ( s ) Assume: f ∈ C 1 ( R ) , lim | s |→∞ s log 3 / 2 ( 1 + | s | ) = 0 Then: NC and AC hold for ( 7 ) for any ω and any T > 0 QUESTION: Same results with 3 / 2 replaced by some p ∈ ( 3 / 2 , 2 ) ? E. Fern´ andez-Cara Controllability and parabolic PDEs
Basic results for the heat equation Additional comments Algorithms devised to construct “good” controls: Optimal control + penalty [Glowinski et al. 1995 . . . ] Fix y 0 = 0 and y 1 ∈ L 2 (Ω) and set F k ( v ) = 1 � � | v | 2 dx dt + k 2 � y ( · , T ) − y 1 � 2 L 2 2 ω × ( 0 , T ) Results from [J-L Lions, 1990]: F k has a unique minimizer v k for all k > 0 and the y k satisfy y k ( T ) → y 1 in L 2 (Ω) as k → ∞ Hence: it suffices to take v = v k for large k = k ( ε ) Theorem 5 [EFC-Zuazua, 2000] √ � y k ( T ) − y 1 � L 2 ≤ C k log k and � v k � L 2 ( Q ) ≤ C log k Logarithmic (and therefore very slow) convergence rates In agreement with the high cost of AC: O ( e 1 /ε ) E. Fern´ andez-Cara Controllability and parabolic PDEs
Controllability of other parabolic systems Non-scalar coupled systems ∃ many generalizations and variants of these arguments: Time and space dependent (regular) coefficients Stokes-like systems: y t − ∆ y + ( a · ∇ ) y + ( y · ∇ ) b + ∇ p = v 1 ω ∇ · y = 0 and y t − ∆ y + ( a · ∇ ) y + ( y · ∇ ) b + ∇ p = θ k + v 1 ω ∇ · y = 0 θ t − κ ∆ θ + c · ∇ θ = w 1 ω where a , b , c ∈ L ∞ Other boundary conditions Other linear parabolic non-scalar systems E. Fern´ andez-Cara Controllability and parabolic PDEs
Controllability of other parabolic systems Non-scalar coupled systems Consider: y t − D ∆ y = My + Bv 1 ω , ( x , t ) ∈ Ω × ( 0 , T ) , y ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , T ) , (8) y ( x , 0 ) = y 0 ( x ) , x ∈ Ω , Here: y = ( y 1 , . . . , y n ) , v = ( v 1 , . . . , v m ) , D , M and B are constant matrices ( n ≥ 2) and D is definite positive Notation: [ H ; B ] := [ B | HB | · · · | H n − 1 B ] Theorem 6 [Ammar-Khodja et al. 2008] Assume: D is diagonal. Then: (8) is NC (for all ω and T ) iff Rank [( − λ i D + M ); B ] = n for all i (Kalman-like condition) QUESTIONS: Conditions on D , M and B that ensure NC in the general case? What happens when M = M ( x , t ) ? E. Fern´ andez-Cara Controllability and parabolic PDEs
Recommend
More recommend
