Some results on the stabilization and on the controllability of - PowerPoint PPT Presentation

Some results on the stabilization and on the controllability of nonlinear wave equations Jean-Michel Coron http://www.ann.jussieu.fr/~coron/ CESAME, Louvain-la-Neuve, October 21, 2008

Outline Stabilization: Dissipative boundary conditions for hyperbolic equations The equations Main result Comparaison with prior results Proof of the exponential stability Application to the control of open channels

Outline Stabilization: Dissipative boundary conditions for hyperbolic equations The equations Main result Comparaison with prior results Proof of the exponential stability Application to the control of open channels Controllability of hyperbolic systems The control problem Controllability theorem

Outline Stabilization: Dissipative boundary conditions for hyperbolic equations The equations Main result Comparaison with prior results Proof of the exponential stability Application to the control of open channels Controllability of hyperbolic systems The control problem Controllability theorem Controllability of the Korteweg-de Vries (KdV) equation Local controllability Global controllability

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Commercial break JMC, Control and nonlinearity, Mathematical Surveys and Monographs, 136, 2007, 427 pp.

y t + A ( y ) y x = 0 , y ∈ R n , x ∈ [0 , 1] , t ∈ [0 , + ∞ ) , (1)

y t + A ( y ) y x = 0 , y ∈ R n , x ∈ [0 , 1] , t ∈ [0 , + ∞ ) , (1) • Assumptions on A

y t + A ( y ) y x = 0 , y ∈ R n , x ∈ [0 , 1] , t ∈ [0 , + ∞ ) , (1) • Assumptions on A A (0) = diag (Λ 1 , Λ 2 , . . . , Λ n ) ,

y t + A ( y ) y x = 0 , y ∈ R n , x ∈ [0 , 1] , t ∈ [0 , + ∞ ) , (1) • Assumptions on A A (0) = diag (Λ 1 , Λ 2 , . . . , Λ n ) , Λ i > 0 , ∀ i ∈ { 1 , . . . , m } , Λ i < 0 , ∀ i ∈ { m + 1 , . . . , n } ,

y t + A ( y ) y x = 0 , y ∈ R n , x ∈ [0 , 1] , t ∈ [0 , + ∞ ) , (1) • Assumptions on A A (0) = diag (Λ 1 , Λ 2 , . . . , Λ n ) , Λ i > 0 , ∀ i ∈ { 1 , . . . , m } , Λ i < 0 , ∀ i ∈ { m + 1 , . . . , n } , Λ i � = Λ j , ∀ ( i , j ) ∈ { 1 , . . . , n } 2 such that i � = j .

• Boundary conditions on y : � y + ( t , 0) � � y + ( t , 1) � = G , t ∈ [0 , + ∞ ) , (2) y − ( t , 1) y − ( t , 0) where

• Boundary conditions on y : � y + ( t , 0) � � y + ( t , 1) � = G , t ∈ [0 , + ∞ ) , (2) y − ( t , 1) y − ( t , 0) where (i) y + ∈ R m and y − ∈ R n − m are defined by � y + � y = , y −

• Boundary conditions on y : � y + ( t , 0) � � y + ( t , 1) � = G , t ∈ [0 , + ∞ ) , (2) y − ( t , 1) y − ( t , 0) where (i) y + ∈ R m and y − ∈ R n − m are defined by � y + � y = , y − (ii) the map G : R n → R n vanishes at 0.

Notations For K ∈ M n , m ( R ), � K � := max {| Kx | ; x ∈ R n , | x | = 1 } .

Notations For K ∈ M n , m ( R ), � K � := max {| Kx | ; x ∈ R n , | x | = 1 } . If n = m , ρ 1 ( K ) := Inf {� ∆ K ∆ − 1 � ; ∆ ∈ D n , + } , where D n , + denotes the set of n × n real diagonal matrices with strictly positive diagonal elements.

Theorem 1.1 (JMC-G. Bastin-B. d’Andr´ ea-Novel (2008)). If ρ 1 ( G ′ (0)) < 1 , then the equilibrium ¯ y ≡ 0 of the quasi-linear hyperbolic system y t + A ( y ) y x = 0 , with the above boundary conditions, is exponentially stable for the Sobolev H 2 -norm.

Estimate on the exponential decay rate For every ν ∈ (0 , − min {| Λ 1 | , . . . , | Λ n |} ln( ρ 1 ( G ′ (0)))), there exist ε > 0 and C > 0 such that, for every y 0 ∈ H 2 ((0 , 1) , R n ) satisfying | y 0 | H 2 ((0 , 1) , R n ) < ε (and the usual compatibility conditions) the classical solution y to the Cauchy problem y t + A ( y ) y x = 0 , y (0 , x ) = y 0 ( x ) + boundary conditions is defined on [0 , + ∞ ) and satisfies | y ( t , · ) | H 2 ((0 , 1) , R n ) � Ce − ν t | y 0 | H 2 ((0 , 1) , R n ) , ∀ t ∈ [0 , + ∞ ) .

The Ta-tsien Li condition n � R 2 ( K ) := Max { | K ij | ; i ∈ { 1 , . . . , n }} , j =1 ρ 2 ( K ) := Inf { R 2 (∆ K ∆ − 1 ); ∆ ∈ D n , + } . Theorem 1.2 (Ta-tsien Li (1994)). If ρ 2 ( G ′ (0)) < 1 , then the equilibrium ¯ y ≡ 0 of the quasi-linear hyperbolic system y t + A ( y ) y x = 0 , with the above boundary conditions, is exponentially stable for the C 1 -norm.

The Ta-tsien Li approach

The Ta-tsien Li approach The Ta-tsien Li proof relies mainly on the use of direct estimates of the solutions and their derivatives along the characteristic curves.

The Ta-tsien Li approach The Ta-tsien Li proof relies mainly on the use of direct estimates of the solutions and their derivatives along the characteristic curves. Robustness to small perturbations: C. Prieur, J. Winkin and G. Bastin (2008); V. Dos Santos and C. Prieur (2008).

C 1 / H 2 -exponential stability 1. Open problem : Does there exists K such that one has exponential stability for the C 1 -norm but not for the H 2 -norm?

C 1 / H 2 -exponential stability 1. Open problem : Does there exists K such that one has exponential stability for the C 1 -norm but not for the H 2 -norm? 2. Open problem : Does there exists K such that one has exponential stability for the H 2 -norm but not for the C 1 -norm?

Comparison of ρ 2 and ρ 1

Comparison of ρ 2 and ρ 1 Proposition 1.3. For every K ∈ M n , n ( R ) , ρ 1 ( K ) � ρ 2 ( K ) . (3)

Comparison of ρ 2 and ρ 1 Proposition 1.3. For every K ∈ M n , n ( R ) , ρ 1 ( K ) � ρ 2 ( K ) . (3) Example where (3) is strict: for a > 0, let � a � a K a := ∈ M 2 , 2 ( R ) . − a a Then √ ρ 1 ( K a ) = 2 a < 2 a = ρ 2 ( K a ) .

Comparison of ρ 2 and ρ 1 Proposition 1.3. For every K ∈ M n , n ( R ) , ρ 1 ( K ) � ρ 2 ( K ) . (3) Example where (3) is strict: for a > 0, let � a � a K a := ∈ M 2 , 2 ( R ) . − a a Then √ ρ 1 ( K a ) = 2 a < 2 a = ρ 2 ( K a ) . Open problem: Does ρ 1 ( K ) < 1 implies the exponential stability for the C 1 -norm?

Comparison with stability conditions for linear hyperbolic systems For simplicity we assume that Λ i are all positive and consider we consider the special case of linear hyperbolic systems y t + Λ y x = 0 , y ( t , 0) = Ky ( t , 1) , where Λ := diag (Λ 1 , . . . , Λ n ) , with Λ i > 0 , ∀ i ∈ { 1 , . . . , n } . Theorem 1.4. Exponential stability for the C 1 -norm is equivalent to the exponential stability in the H 2 -norm.

A Necessary and sufficient condition for exponential stability Notation: r i = 1 , ∀ i ∈ { 1 , . . . , n } . Λ i Theorem 1.5. ¯ y ≡ is exponentially stable for the system y + Λ y x = 0 , y ( t , 0) = Ky ( t , 1) ˙ if and only if there exists δ > 0 such that � � det ( Id n − ( diag ( e − r 1 z , . . . , e − r n z )) K ) = 0 , z ∈ C ⇒ ( ℜ ( z ) � − δ ) .

An example Let us choose λ 1 := 1, λ 2 := 2 (hence r 1 = 1 and r 2 = 1 / 2 and � a � a , a ∈ R . K a := a a Then ρ 1 ( K ) = 2 | a | . Hence ρ 1 ( K a ) < 1 is equivalent to a ∈ ( − 1 / 2 , 1 / 2).

An example Let us choose λ 1 := 1, λ 2 := 2 (hence r 1 = 1 and r 2 = 1 / 2 and � a � a , a ∈ R . K a := a a Then ρ 1 ( K ) = 2 | a | . Hence ρ 1 ( K a ) < 1 is equivalent to a ∈ ( − 1 / 2 , 1 / 2). However exponential stability is equivalent to a ∈ ( − 1 , 1 / 2).

Robustness issues For a positive integer n , let 4 n 4 n Λ 1 := 4 n + 1 , Λ 2 = 2 n + 1 . Then � y 1 � � sin � 4 n π ( t − ( x / Λ 1 )) � � := � � y 2 sin 4 n π ( t − ( x / Λ 2 )) is a solution of y t + Λ y x , y ( t , 0) = K − 1 / 2 y ( t , 1) which does not tends to 0 as t → + ∞ . Hence one does not have exponential stability. However lim n → + ∞ Λ 1 = 1 and lim n → + ∞ Λ 2 = 2.

Robustness issues For a positive integer n , let 4 n 4 n Λ 1 := 4 n + 1 , Λ 2 = 2 n + 1 . Then � y 1 � � sin � 4 n π ( t − ( x / Λ 1 )) � � := � � y 2 sin 4 n π ( t − ( x / Λ 2 )) is a solution of y t + Λ y x , y ( t , 0) = K − 1 / 2 y ( t , 1) which does not tends to 0 as t → + ∞ . Hence one does not have exponential stability. However lim n → + ∞ Λ 1 = 1 and lim n → + ∞ Λ 2 = 2.The exponential stability is not robust with respect to Λ: small perturbations of Λ can destroy the exponential stability.

Robust exponential stability Notations: ρ 0 ( K ) := max { ρ (diag ( e ιθ 1 , . . . , e ιθ n ) K ); ( θ 1 , . . . , θ n ) tr ∈ R n } , Theorem 1.6 (R. Silkowski (1993)). If the ( r 1 , . . . , r n ) are rationally independent, the linear system y t + Λ y x = 0 , y ( t , 0) = Ky ( t , 1) is exponentially stable if and only if ρ 0 ( K ) < 1 .

Robust exponential stability Notations: ρ 0 ( K ) := max { ρ (diag ( e ιθ 1 , . . . , e ιθ n ) K ); ( θ 1 , . . . , θ n ) tr ∈ R n } , Theorem 1.6 (R. Silkowski (1993)). If the ( r 1 , . . . , r n ) are rationally independent, the linear system y t + Λ y x = 0 , y ( t , 0) = Ky ( t , 1) is exponentially stable if and only if ρ 0 ( K ) < 1 . Note that ρ 0 ( K ) depends continuously on K and that “( r 1 , . . . , r n ) are rationally independent” is a generic condition. Therefore, if one wants to have a natural robustness property with respect to the r i ’s, the condition for exponential stability is ρ 0 ( K ) < 1 .