Stabilization and controllability of first-order integro-differential hyperbolic equations Guillaume OLIVE joint work with Jean-Michel CORON and Long HU Nonlinear Partial Differential Equations and Applications – A conference in the honor of Jean-Michel CORON for his 60th birthday – Paris, March 23 2016
The equation We consider � L u t ( t , x ) − u x ( t , x ) = g ( x , y ) u ( t , y ) dy t ∈ (0 , T ) , 0 (1) u ( t , L ) = U ( t ) x ∈ (0 , L ) , u 0 ( x ) , u (0 , x ) = where : x U ( t ) T > 0 is the time of control and L > 0 L is the length of the domain. u 0 is the initial data and u is the state. u 0 ( x ) g ∈ L 2 ((0 , L ) × (0 , L )) is a given kernel. t U ∈ L 2 (0 , T ) is a boundary control. 0 T G. Olive (IMB) Stabilization of integro-differential equations 2 / 20
An application : PDE-ODE systems Example borrowed from A. Smyshlyaev and M. Krstic (2008) : u t ( t , x ) − u x ( t , x ) = v ( t , x ) , v xx ( t , x ) − v ( t , x ) = u ( t , x ) , t ∈ (0 , T ) , u ( t , L ) = U ( t ) , v x ( t , 0) = 0 , x ∈ (0 , L ) . = u 0 ( x ) , u (0 , x ) v ( t , L ) = V ( t ) . Can we find U , V as functions of u , v such that, for some T > 0, u ( T , · ) = v ( T , · ) = 0 ? (remark : u ( T , · ) = 0 = ⇒ v ( T , · ) = 0) . G. Olive (IMB) Stabilization of integro-differential equations 3 / 20
An application : PDE-ODE systems Example borrowed from A. Smyshlyaev and M. Krstic (2008) : u t ( t , x ) − u x ( t , x ) = v ( t , x ) , v xx ( t , x ) − v ( t , x ) = u ( t , x ) , t ∈ (0 , T ) , u ( t , L ) = U ( t ) , v x ( t , 0) = 0 , x ∈ (0 , L ) . = u 0 ( x ) , u (0 , x ) v ( t , L ) = V ( t ) . Can we find U , V as functions of u , v such that, for some T > 0, u ( T , · ) = v ( T , · ) = 0 ? (remark : u ( T , · ) = 0 = ⇒ v ( T , · ) = 0) . First, we solve the ODE : � � L � � x v ( t , x ) = cosh( x ) V ( t ) − u ( t , y ) sinh( L − y ) dy + u ( t , y ) sinh( x − y ) dy . cosh( L ) 0 0 � �� � � �� � Fredholm Volterra G. Olive (IMB) Stabilization of integro-differential equations 3 / 20
An application : PDE-ODE systems Example borrowed from A. Smyshlyaev and M. Krstic (2008) : u t ( t , x ) − u x ( t , x ) = v ( t , x ) , v xx ( t , x ) − v ( t , x ) = u ( t , x ) , t ∈ (0 , T ) , u ( t , L ) = U ( t ) , v x ( t , 0) = 0 , x ∈ (0 , L ) . = u 0 ( x ) , u (0 , x ) v ( t , L ) = V ( t ) . Can we find U , V as functions of u , v such that, for some T > 0, u ( T , · ) = v ( T , · ) = 0 ? (remark : u ( T , · ) = 0 = ⇒ v ( T , · ) = 0) . First, we solve the ODE : � � L � � x v ( t , x ) = cosh( x ) V ( t ) − u ( t , y ) sinh( L − y ) dy + u ( t , y ) sinh( x − y ) dy . cosh( L ) 0 0 � �� � � �� � Fredholm Volterra If we have 2 controls : take V such that v ( t , 0) = 0 : Volterra integral. G. Olive (IMB) Stabilization of integro-differential equations 3 / 20
An application : PDE-ODE systems Example borrowed from A. Smyshlyaev and M. Krstic (2008) : u t ( t , x ) − u x ( t , x ) = v ( t , x ) , v xx ( t , x ) − v ( t , x ) = u ( t , x ) , t ∈ (0 , T ) , u ( t , L ) = U ( t ) , v x ( t , 0) = 0 , x ∈ (0 , L ) . = u 0 ( x ) , u (0 , x ) v ( t , L ) = V ( t ) . Can we find U , V as functions of u , v such that, for some T > 0, u ( T , · ) = v ( T , · ) = 0 ? (remark : u ( T , · ) = 0 = ⇒ v ( T , · ) = 0) . First, we solve the ODE : � � L � � x v ( t , x ) = cosh( x ) V ( t ) − u ( t , y ) sinh( L − y ) dy + u ( t , y ) sinh( x − y ) dy . cosh( L ) 0 0 � �� � � �� � Fredholm Volterra If we have 2 controls : take V such that v ( t , 0) = 0 : Volterra integral. If we have 1 control ( V = 0) : Fredholm integral. G. Olive (IMB) Stabilization of integro-differential equations 3 / 20
Abstract form of (1) Let us rewrite (1) in the abstract form in L 2 (0 , L ) : � d dt u = Au + BU , t ∈ (0 , T ) , u 0 , u (0) = G. Olive (IMB) Stabilization of integro-differential equations 4 / 20
Abstract form of (1) Let us rewrite (1) in the abstract form in L 2 (0 , L ) : � d dt u = Au + BU , t ∈ (0 , T ) , u 0 , u (0) = where the unbounded operator A is � L Au = u x + g ( · , y ) u ( y ) dy , 0 � with domain D ( A ) = � � u ( L ) = 0 � u ∈ H 1 (0 , L ) , and B ∈ L ( C , D ( A ∗ ) ′ ) is � BU , z � D ( A ∗ ) ′ , D ( A ∗ ) = Uz ( L ) . G. Olive (IMB) Stabilization of integro-differential equations 4 / 20
Abstract form of (1) Let us rewrite (1) in the abstract form in L 2 (0 , L ) : � d dt u = Au + BU , t ∈ (0 , T ) , u 0 , u (0) = where the unbounded operator A is � L Au = u x + g ( · , y ) u ( y ) dy , 0 � with domain D ( A ) = � � u ( L ) = 0 � u ∈ H 1 (0 , L ) , and B ∈ L ( C , D ( A ∗ ) ′ ) is � BU , z � D ( A ∗ ) ′ , D ( A ∗ ) = Uz ( L ) . We can show that, there exists a unique solution (by transposition) u ∈ C 0 ([0 , T ]; L 2 (0 , L )) . G. Olive (IMB) Stabilization of integro-differential equations 4 / 20
Notions of controllability u ( T ; 0) u 1 u 0 Figure – Uncontrolled trajectory u 0 : initial state, u 1 : target, u ( T ; U ) : value of the solution to (1) at time T with control U . G. Olive (IMB) Stabilization of integro-differential equations 5 / 20
Notions of controllability u ( T ; 0) u 1 = u ( T ; U ) u 0 Figure – Trajectory controlled exactly u 0 : initial state, u 1 : target, u ( T ; U ) : value of the solution to (1) at time T with control U . G. Olive (IMB) Stabilization of integro-differential equations 5 / 20
Notions of controllability u ( T ; 0) 0 = u ( T ; U ) u 0 Figure – Trajectory controlled to 0 u 0 : initial state, u 1 : target, u ( T ; U ) : value of the solution to (1) at time T with control U . G. Olive (IMB) Stabilization of integro-differential equations 5 / 20
Notions of controllability u ( T ; 0) u 1 ε u ( T ; U ) u 0 Figure – Trajectory controlled approximatively u 0 : initial state, u 1 : target, u ( T ; U ) : value of the solution to (1) at time T with control U . G. Olive (IMB) Stabilization of integro-differential equations 5 / 20
Notions of stabilization Stability ( U ( t ) = 0 ) : We say that (1) is exp. stable if the solution u with U ( t ) = 0 satisfies � u ( t ) � L 2 ≤ M ω e − ω t , ∀ t ≥ 0 , (2) for some ω > 0 and M ω > 0. G. Olive (IMB) Stabilization of integro-differential equations 6 / 20
Notions of stabilization Stability ( U ( t ) = 0 ) : We say that (1) is exp. stable if the solution u with U ( t ) = 0 satisfies � u ( t ) � L 2 ≤ M ω e − ω t , ∀ t ≥ 0 , (2) for some ω > 0 and M ω > 0. stable in finite time T if the solution u with U ( t ) = 0 satisfies u ( t ) = 0 , ∀ t ≥ T . (3) G. Olive (IMB) Stabilization of integro-differential equations 6 / 20
Notions of stabilization Stability ( U ( t ) = 0 ) : We say that (1) is exp. stable if the solution u with U ( t ) = 0 satisfies � u ( t ) � L 2 ≤ M ω e − ω t , ∀ t ≥ 0 , (2) for some ω > 0 and M ω > 0. stable in finite time T if the solution u with U ( t ) = 0 satisfies u ( t ) = 0 , ∀ t ≥ T . (3) Stabilization ( U ( t ) = Fu ( t ) ) : We say that (1) is exp. stabilizable if (1) with U ( t ) = F ω u ( t ) is exp. stable. G. Olive (IMB) Stabilization of integro-differential equations 6 / 20
Notions of stabilization Stability ( U ( t ) = 0 ) : We say that (1) is exp. stable if the solution u with U ( t ) = 0 satisfies � u ( t ) � L 2 ≤ M ω e − ω t , ∀ t ≥ 0 , (2) for some ω > 0 and M ω > 0. stable in finite time T if the solution u with U ( t ) = 0 satisfies u ( t ) = 0 , ∀ t ≥ T . (3) Stabilization ( U ( t ) = Fu ( t ) ) : We say that (1) is exp. stabilizable if (1) with U ( t ) = F ω u ( t ) is exp. stable. rap. stabilizable if this holds for every ω > 0. G. Olive (IMB) Stabilization of integro-differential equations 6 / 20
Notions of stabilization Stability ( U ( t ) = 0 ) : We say that (1) is exp. stable if the solution u with U ( t ) = 0 satisfies � u ( t ) � L 2 ≤ M ω e − ω t , ∀ t ≥ 0 , (2) for some ω > 0 and M ω > 0. stable in finite time T if the solution u with U ( t ) = 0 satisfies u ( t ) = 0 , ∀ t ≥ T . (3) Stabilization ( U ( t ) = Fu ( t ) ) : We say that (1) is exp. stabilizable if (1) with U ( t ) = F ω u ( t ) is exp. stable. rap. stabilizable if this holds for every ω > 0. stabilizable in finite time T if (1) with U ( t ) = Fu ( t ) is stable in finite time T . G. Olive (IMB) Stabilization of integro-differential equations 6 / 20
References on the stabilization of (1) We know that : In general, (1) is not stable. G. Olive (IMB) Stabilization of integro-differential equations 7 / 20
References on the stabilization of (1) We know that : In general, (1) is not stable. (1) is stabilizable in finite time T = L , if � x g ( x , y ) = 0 , x ≤ y ( Volterra Integral dy ) . 0 A. Smyshlyaev and M. Krstic (2008) G. Olive (IMB) Stabilization of integro-differential equations 7 / 20
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