Fonctions de Lyapunov pour les EDP: analyse de la stabilit´ e et des perturbations Christophe Prieur Gipsa-lab, CNRS, Grenoble GT- Contrˆ ole et Probl` emes inverses, F´ evrier 2011 1/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011
Introduction Level and flow control in an horizontal reach of an open channel Control = two overflow spillways: $ ! %&' $ " %&' # ! H ( x, t ) ( " %&' ( ! %&' # " Q ( x, t ) ! " ! where H ( x , t ) is the water level and Q ( x , t ) the water flow rate in the reach. 2/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011
Shallow Water Equations Model [Chow, 54] or [Graf, 98]: mass conservation � Q ( x , t ) � ∂ t H ( x , t ) + ∂ x = q ( x ) B momentum conservation � Q 2 ( x , t ) � BH ( x , t ) + gBH 2 ( x , t ) = gBH ( I − J ) + kq Q ∂ t Q ( x , t ) + ∂ x 2 BH where g and B are constant values q the water supply/removal function I is the bottom slope n 2 M Q 2 J ( Q , H ) = S ( H ) 2 R ( H ) 4 / 3 is the slope’s friction 3/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011
Motivations Problem Compute the positions u 0 and u L of the spillways s.t. the control actions depend only on the (measured) H (0 , t ) and H ( L , t ) ∃ a solution of our model (PDE) state → t → + ∞ equilibrium stability properties even in presence of perturbations I , J and q 4/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011
Outline 1.1 Stability analysis of hyperbolic non-homogeneous systems 1.2 Related works 1.3 Applications 2 Sensitivity with respect to large perturbations Notion of ISS Lyapunov functions for PDE Asymp. Stability �⇒ Input-to State Stability 2.1 An ISS Lyapunov function for hyperbolic linear systems 2.2 An ISS Lyapunov function for semilinear parabolic systems Conclusion 5/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011
Outline 1.1 Stability analysis of hyperbolic non-homogeneous systems 1.2 Related works 1.3 Applications 2 Sensitivity with respect to large perturbations Notion of ISS Lyapunov functions for PDE Asymp. Stability �⇒ Input-to State Stability 2.1 An ISS Lyapunov function for hyperbolic linear systems 2.2 An ISS Lyapunov function for semilinear parabolic systems Conclusion 5/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011
Outline 1.1 Stability analysis of hyperbolic non-homogeneous systems 1.2 Related works 1.3 Applications 2 Sensitivity with respect to large perturbations Notion of ISS Lyapunov functions for PDE Asymp. Stability �⇒ Input-to State Stability 2.1 An ISS Lyapunov function for hyperbolic linear systems 2.2 An ISS Lyapunov function for semilinear parabolic systems Conclusion 5/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011
Outline 1.1 Stability analysis of hyperbolic non-homogeneous systems 1.2 Related works 1.3 Applications 2 Sensitivity with respect to large perturbations Notion of ISS Lyapunov functions for PDE Asymp. Stability �⇒ Input-to State Stability 2.1 An ISS Lyapunov function for hyperbolic linear systems 2.2 An ISS Lyapunov function for semilinear parabolic systems Conclusion 5/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011
Outline 1.1 Stability analysis of hyperbolic non-homogeneous systems 1.2 Related works 1.3 Applications 2 Sensitivity with respect to large perturbations Notion of ISS Lyapunov functions for PDE Asymp. Stability �⇒ Input-to State Stability 2.1 An ISS Lyapunov function for hyperbolic linear systems 2.2 An ISS Lyapunov function for semilinear parabolic systems Conclusion 5/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011
Many works in the literature For a survey, see [Malaterre, Rogers, and Schuurmans, 98]. Finite dimensional approach: H ∞ control design is developed in [Litrico, and Georges, 99]. Infinite dimensional approach: Delay-based control [G. Besan¸ con, D. Georges, 09] Lyapunov methods [Dos Santos, Bastin, Coron, and d’Andr´ ea-Novel, 07], [V.T. Pham, G. Besan¸ con, D. Georges, 10] And also new dissipativity condition for quasi-linear hyperbolic systems [Coron, Bastin, and d’Andr´ ea-Novel, 08]. See below. LQ methods [Winkin, Dochain] Backstepping transformations [Smyshlyaev, Cerpa, Krstic, 10] among others 6/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011
Many works in the literature For a survey, see [Malaterre, Rogers, and Schuurmans, 98]. Finite dimensional approach: H ∞ control design is developed in [Litrico, and Georges, 99]. Infinite dimensional approach: Delay-based control [G. Besan¸ con, D. Georges, 09] Lyapunov methods [Dos Santos, Bastin, Coron, and d’Andr´ ea-Novel, 07], [V.T. Pham, G. Besan¸ con, D. Georges, 10] And also new dissipativity condition for quasi-linear hyperbolic systems [Coron, Bastin, and d’Andr´ ea-Novel, 08]. See below. LQ methods [Winkin, Dochain] Backstepping transformations [Smyshlyaev, Cerpa, Krstic, 10] among others 6/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011
Contribution Here: Perturbations are taken into account asymp. stability when perturbations are vanishing bounded state with bounded perturbations Methods that are used Riemann invariants [Li Ta-tsien, 94] Lyapunov method 7/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011
Contribution Here: Perturbations are taken into account asymp. stability when perturbations are vanishing bounded state with bounded perturbations Methods that are used Riemann invariants [Li Ta-tsien, 94] Lyapunov method 7/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011
Contribution Here: Perturbations are taken into account asymp. stability when perturbations are vanishing bounded state with bounded perturbations Methods that are used Riemann invariants [Li Ta-tsien, 94] Lyapunov method 7/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011
Related issue: Leak localization Instead of a control problem, we may also consider an observation problem. Leak detection for quasi-linear system Instead of controlling the state, we may regulate the error using a similar approach, let us cite In Australia: E. Weyer, I. Mareels In France: X. Litrico, N. Bedjaoui, G. Besan¸ con 8/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011
Related issue: Leak localization Instead of a control problem, we may also consider an observation problem. Leak detection for quasi-linear system Instead of controlling the state, we may regulate the error using a similar approach, let us cite In Australia: E. Weyer, I. Mareels In France: X. Litrico, N. Bedjaoui, G. Besan¸ con 8/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011
General context: non-homogeneous systems in R 2 When fixing an equilibrium and using the Riemann invariant coordinates (as in [Li, 94]), we may rewrite the previous eq. as a non-homogeneous quasi-linear hyperbolic system : Let us consider ξ : [0 , L ] × [0 , + ∞ ) → R 2 such that ∂ t ξ + Λ( ξ ) ∂ x ξ = h ( ξ ) (1) where Λ: ε 0 B → R 2 × 2 is a C 1 function satisfying Λ = diag( λ 1 , λ 2 ) , and λ 1 (0) < 0 < λ 2 (0) , and h : ε 0 B → R 2 is C 1 s.t. h (0) = 0 . The boundary conditions are � ξ 1 ( L , t ) � � ξ 1 (0 , t ) � = g (2) , ξ 2 (0 , t ) ξ 2 ( L , t ) where g : ε 0 B → R 2 is C 1 s.t. g (0) = 0. In [de Halleux, CP, Coron, d’Andr´ ea-Novel, Bastin, 03] and [Li, 94]: h ≡ 0 9/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011
General context: non-homogeneous systems in R 2 When fixing an equilibrium and using the Riemann invariant coordinates (as in [Li, 94]), we may rewrite the previous eq. as a non-homogeneous quasi-linear hyperbolic system : Let us consider ξ : [0 , L ] × [0 , + ∞ ) → R 2 such that ∂ t ξ + Λ( ξ ) ∂ x ξ = h ( ξ ) (1) where Λ: ε 0 B → R 2 × 2 is a C 1 function satisfying Λ = diag( λ 1 , λ 2 ) , and λ 1 (0) < 0 < λ 2 (0) , and h : ε 0 B → R 2 is C 1 s.t. h (0) = 0 . The boundary conditions are � ξ 1 ( L , t ) � � ξ 1 (0 , t ) � = g (2) , ξ 2 (0 , t ) ξ 2 ( L , t ) where g : ε 0 B → R 2 is C 1 s.t. g (0) = 0. In [de Halleux, CP, Coron, d’Andr´ ea-Novel, Bastin, 03] and [Li, 94]: h ≡ 0 9/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011
General context: non-homogeneous systems in R 2 When fixing an equilibrium and using the Riemann invariant coordinates (as in [Li, 94]), we may rewrite the previous eq. as a non-homogeneous quasi-linear hyperbolic system : Let us consider ξ : [0 , L ] × [0 , + ∞ ) → R 2 such that ∂ t ξ + Λ( ξ ) ∂ x ξ = h ( ξ ) (1) where Λ: ε 0 B → R 2 × 2 is a C 1 function satisfying Λ = diag( λ 1 , λ 2 ) , and λ 1 (0) < 0 < λ 2 (0) , and h : ε 0 B → R 2 is C 1 s.t. h (0) = 0 . The boundary conditions are � ξ 1 ( L , t ) � � ξ 1 (0 , t ) � = g (2) , ξ 2 (0 , t ) ξ 2 ( L , t ) where g : ε 0 B → R 2 is C 1 s.t. g (0) = 0. In [de Halleux, CP, Coron, d’Andr´ ea-Novel, Bastin, 03] and [Li, 94]: h ≡ 0 9/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011
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