Lecture 3: Controllability of some hyperbolic equations Enrique FERN ´ ANDEZ-CARA Dpto. E.D.A.N. - Univ. of Sevilla Results for the wave equation Other hyperbolic equations and systems Applications . . . E. Fern´ andez-Cara Control and hyperbolic PDEs
Outline Basic results for the wave equation 1 Controllability concepts The main results Other hyperbolic equations and systems 2 Semilinear wave equations Linear elasticity and Lam´ e systems Visco-elasticity and Maxwell fluids E. Fern´ andez-Cara Control and hyperbolic PDEs
Basic results for the wave equation Controllability concepts Consider the controlled linear wave equation: y tt − ∆ y = v 1 ω in Q y = 0 on Σ (1) y ( x , 0 ) = y 0 ( x ) , y t ( x , 0 ) = y 1 ( x ) in Ω (same notation as in Lecture 2: Q = ω × ( 0 , T ) , ω ⊂ Ω , etc.) ∀ ( y 0 , y 1 ) ∈ H 1 0 × L 2 , ∀ v ∈ L 2 ( ω × ( 0 , T )) ∃ ! solution y ∈ C 0 ([ 0 , T ]; H 1 0 ) ∩ C 1 ([ 0 , T ]; L 2 ) Now: the control is v and the state is ( y , y t ) R ( T ; y 0 , y 1 ) := { ( y ( · , T ) , y t ( · , T )) : v ∈ L 2 ( ω × ( 0 , T )) } . Then: 0 × L 2 for all ( y 0 , y 1 ) • ( 1 ) is AC if R ( T ; y 0 , y 1 ) = H 1 0 × L 2 for all ( y 0 , y 1 ) • It is EC if R ( T ; y 0 , y 1 ) = H 1 • It is NC if R ( T ; y 0 , y 1 ) ∋ ( 0 , 0 ) for all ( y 0 , y 1 ) Other spaces where (1) is well posed can also be used E. Fern´ andez-Cara Control and hyperbolic PDEs
Basic results for the wave equation Controllability concepts � y tt − ∆ y = v 1 ω in Ω × ( 0 , T ) ( 1 ) y ( 0 ) = y 0 , y t ( 0 ) = y 1 in Ω , etc. First results: • For AC, EC or NC to hold, T has to be large (finite speed of propagation) • NC and EC are equivalent; ( 1 ) is linear and reversible in time • EC ⇒ AC, but the reciprocal is false (conditions on T and ω ) Very different properties to those satisfied by the heat equation! E. Fern´ andez-Cara Control and hyperbolic PDEs
Basic results for the wave equation The main results Theorem 1 AC holds for ( 1 ) for any ω and any T > T ∗ ( ω ) S KETCH OF THE P ROOF : Fix ω and T > 0 ( 1 ) is AC ⇔ R ( T ; 0 , 0 ) is dense ⇔ R ( T ; 0 , 0 ) ⊥ = { 0 , 0 } Assume ( ϕ 0 , ϕ 1 ) ∈ R ( T ; 0 , 0 ) ⊥ and introduce � ϕ tt − ∆ ϕ = 0 in Ω × ( 0 , T ) (2) ϕ ( T ) = ϕ 0 , ϕ t ( T ) = ϕ 1 in Ω , etc. � � ϕ v dx dt = � ( ϕ 0 , ϕ 1 ) , ( y ( T ) , y t ( T )) � = 0 ∀ v Then: ω × ( 0 , T ) 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 if T > T ∗ ( ω ) (Holmgren’s Theorem) ✷ E. Fern´ andez-Cara Control and hyperbolic PDEs
Basic results for the wave equation The main results Again, 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 , ϕ 1 ) = 1 � � | ϕ | 2 + ε � ( ϕ 0 , ϕ 1 ) � L 2 × H − 1 − � ( ϕ 0 , ϕ 1 ) , ( y 0 , y 1 ) � 2 ω × ( 0 , T ) QUESTIONS: A more general system: � y tt − ∇ · ( D ( x , t ) ∇ y ) + B ( x , t ) · ∇ y + a ( x , t ) y = v 1 ω ( y , y t ) | t = 0 = ( y 0 , y 1 ) , 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 large T ? Can we then repeat this construction? Unknown; even for D ≡ Id , B ≡ 0 E. Fern´ andez-Cara Control and hyperbolic PDEs
Basic results for the wave equation The main results What about EC? � y tt − ∆ y = v 1 ω � ϕ tt − ∆ ϕ = 0 ( 1 ) ( y , y t )( 0 ) = ( y 0 , y 1 ) , . . . ( 2 ) ( ϕ, ϕ t )( T ) = ( ϕ 0 , ϕ 1 ) , . . . Proposition 1 [J-L Lions, H.U.M. method, 1988] EC holds for ( 1 ) with v ∈ L 2 ( ω × ( 0 , T )) iff ( 2 ) is observable: � � | ϕ | 2 dx dt � ( ϕ, ϕ t )( 0 ) � 2 L 2 × H − 1 ≤ C (3) ω × ( 0 , T ) When ( 3 ) holds, one can minimize in L 2 × H − 1 W ( ϕ 0 , ϕ 1 ) = 1 � � | ϕ | 2 + � ( ϕ, ϕ t )( 0 ) , ( y 0 , y 1 ) � 2 ω × ( 0 , T ) ϕ 0 , ˆ ϕ 1 ) , is the null control with Then: v = ˆ ϕ 1 ω , where ˆ ϕ corresponds to the minimizer ( ˆ minimal L 2 norm E. Fern´ andez-Cara Control and hyperbolic PDEs
Basic results for the wave equation The main results The EC problem is reduced to the analysis of ( 3 ) . We have Theorem 2 Assume: x 0 ∈ R N , ω ⊃ Γ( x 0 ) := { x ∈ Γ : ( x − x 0 ) · n ( x ) > 0 } , T > T ( x 0 ) := 2 � x − x 0 � L ∞ . Then: ( 1 ) is EC at time T S KETCH OF THE P ROOF : First, boundary observability for T > T ( x 0 ) 2 � � � ∂ϕ � � ( ϕ, ϕ t )( 0 ) � 2 � � 0 × L 2 ≤ C d Γ dt (4) � � H 1 ∂ n Γ( x 0 ) × ( 0 , T ) � � [Ho, 1986; J-L Lions, 1988], multipliers techniques This gives EC with controls on the boundary Then: ( 4 ) ⇒ ( 3 ) when ω is a neighborhood of Γ( x 0 ) ✷ The class of sets ω can be enlarged, [Osses, 2001] E. Fern´ andez-Cara Control and hyperbolic PDEs
Basic results for the wave equation The main results More generally: Theorem 3 [Bardos-Lebeau-Rauch, 1992; Burq, 1997] Assume: Ω is of class C 3 . Then: ( 3 ) holds iff ( ω, T ) satisfies the GCC: Every ray that begins to propagate in Ω at time t = 0 and is reflected on Γ enters ω at a time t < T The proof uses microlocal defect measures, [Gerard, 1991] The result also holds for � y tt − ∇ · ( D ( x , t ) ∇ y ) + B ( x , t ) · ∇ y + a ( x , t ) y = v 1 ω ( y , y t ) | t = 0 = ( y 0 , y 1 ) , etc. when D ij , B i , a ∈ C 2 , D ( x , t ) ξ · ξ ≥ α 0 | ξ | 2 QUESTION: Minimal regularity hypotheses for Theorem 3? Other methods exist, in particular Carleman estimates [Zhang, 2000; Puel, 2004; etc.] E. Fern´ andez-Cara Control and hyperbolic PDEs
Other hyperbolic equations and systems Semilinear wave equations The semilinear wave equation: y tt − ∆ y + f ( y ) = v 1 ω in Q y = 0 on Σ (5) y ( x , 0 ) = y 0 ( x ) , y t ( x , 0 ) = y 1 ( x ) in Ω Theorem 4 [Zuazua, 1993] f ( s ) Assume: N = 1, f ∈ C 1 ( R ) , lim | s |→∞ s log 2 ( 1 + | s | ) = 0 Then: EC holds for ( 5 ) for any ω = ( a , b ) , T > max ( a , 1 − b ) Theorem 5 [Zhang, 2000] f ( s ) Assume: f ∈ C 1 ( R ) , lim | s |→∞ s log 1 / 2 ( 1 + | s | ) = 0 Then: EC holds for ( 5 ) for ω ⊃ Γ( x 0 ) , T > T ∗ (Ω , x 0 ) QUESTION: N ≥ 2, f ∈ C 1 ( R ) , ( ω, T ) satisfying GCC Minimal hypotheses on f to have EC? E. Fern´ andez-Cara Control and hyperbolic PDEs
Other hyperbolic equations and systems Linear elasticity and Lam´ e systems The Lam´ e system (linear elasticity + isotropy) y tt − λ ∆ y − µ ∇ ( ∇ · y ) = v 1 ω in Q y = 0 on Σ (6) y ( x , 0 ) = y 0 ( x ) , y t ( x , 0 ) = y 1 ( x ) in Ω with λ, µ > 0 (for instance) y = ( y 1 , . . . , y N ) : the displacement v 1 ω : a force field Theorem 6 [Imanuvilov-Yamamoto, 2005] EC holds for ( 6 ) for any ( ω, T ) satisfying GCC E. Fern´ andez-Cara Control and hyperbolic PDEs
Other hyperbolic equations and systems Visco-elasticity and Maxwell fluids The linearized Maxwell system: y t + ∇ π = ∇ · τ + v 1 ω , ∇ · y = 0 in Q τ t + a τ = 2 bDy in Q (7) y = 0 on Σ y ( 0 ) = y 0 , τ ( 0 ) = τ 0 in Ω y , π , τ : the velocity, pressure, extra-stress elastic tensor, resp. v 1 ω : a force field The linearization at zero of the true Maxwell model: y t + ( y · ∇ ) y + ∇ π = ∇ · τ + v 1 ω , ∇ · y = 0 in Q τ t + ( y · ∇ ) τ + a τ + g ( τ, ∇ y ) = 2 bDy in Q (8) y = 0 on Σ y ( 0 ) = y 0 , τ ( 0 ) = τ 0 in Ω Very difficult to solve and analyze . . . E. Fern´ andez-Cara Control and hyperbolic PDEs
Other hyperbolic equations and systems Visco-elasticity and Maxwell fluids � y t + ∇ π = ∇ · τ + v 1 ω , ∇ · y = 0 ( 7 ) τ t + a τ = 2 bDy , etc. Observe: (7) can be rewritten in the form z tt − az t − b ∆ z + ∇ Z = u 1 ω + e − at ∇ · τ 0 , ∇ · z = 0 in Q z = 0 on Σ ( z , z t )( 0 ) = ( 0 , y 0 ) in Ω � t 0 e as y ( s ) ds , Z = e at π with z = Results from [EFC, Boldrini, Doubova, Gonz´ alez-Burgos]: Theorem 7 AC holds for ( 7 ) for any ω and any T > T ∗ (Ω , ω, a , b ) Theorem 8 Assume: x 0 ∈ R N , ω ⊃ Γ( x 0 ) := { x ∈ Γ : ( x − x 0 ) · n ( x ) > 0 } , � λ 1 b , T > T ∗ (Ω , x 0 , a , b ) 0 < a < 2 Then: ( 7 ) is EC at time T E. Fern´ andez-Cara Control and hyperbolic PDEs
Other hyperbolic equations and systems Visco-elasticity and Maxwell fluids � y t + ∇ π = ∇ · τ + v 1 ω , ∇ · y = 0 ( 7 ) τ t + a τ = 2 bDy , etc. � QUESTIONS: EC for 0 < a < 2 λ 1 b if ( ω, T ) satisfies GCC? EC for general a , b > 0, ω ⊃ Γ( x 0 ) and large T ? E. Fern´ andez-Cara Control and hyperbolic PDEs
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