Fonctions de Lyapunov pour les EDP : Applications ` a deux syst` emes physiques Christophe Prieur CNRS, Gipsa-lab, Grenoble GT EDP Valence, f´ evrier 2012 1/44 C. Prieur GT-EDP’12
2/44 C. Prieur GT-EDP’12
Lyapunov functional based techniques are central in the study of partial differential equations (PDEs). See [Coron, Bastin, d’Andr´ ea-Novel, 2007], [Krstic, Smyshlyaev, 2008], [Cazenave, Haraux, 1998] ... Two motivations to use strict Lyapunov functions: To demonstrate asymptotic stability by means of a weak 1 Lyapunov functional, the LaSalle invariance principle may be used [Luo, Guo, Morgul, 1999]. It requires a precompactness property for the solutions (may be difficult to prove, or even not satisfied). With strict Lyapunov functions in hands: robustness analysis 2 of the stability with respect to uncertainties and sensitivity of the solutions with respect to external disturbances. 3/44 C. Prieur GT-EDP’12
Lyapunov functional based techniques are central in the study of partial differential equations (PDEs). See [Coron, Bastin, d’Andr´ ea-Novel, 2007], [Krstic, Smyshlyaev, 2008], [Cazenave, Haraux, 1998] ... Two motivations to use strict Lyapunov functions: To demonstrate asymptotic stability by means of a weak 1 Lyapunov functional, the LaSalle invariance principle may be used [Luo, Guo, Morgul, 1999]. It requires a precompactness property for the solutions (may be difficult to prove, or even not satisfied). With strict Lyapunov functions in hands: robustness analysis 2 of the stability with respect to uncertainties and sensitivity of the solutions with respect to external disturbances. 3/44 C. Prieur GT-EDP’12
Lyapunov functional based techniques are central in the study of partial differential equations (PDEs). See [Coron, Bastin, d’Andr´ ea-Novel, 2007], [Krstic, Smyshlyaev, 2008], [Cazenave, Haraux, 1998] ... Two motivations to use strict Lyapunov functions: To demonstrate asymptotic stability by means of a weak 1 Lyapunov functional, the LaSalle invariance principle may be used [Luo, Guo, Morgul, 1999]. It requires a precompactness property for the solutions (may be difficult to prove, or even not satisfied). With strict Lyapunov functions in hands: robustness analysis 2 of the stability with respect to uncertainties and sensitivity of the solutions with respect to external disturbances. 3/44 C. Prieur GT-EDP’12
Lyapunov functional based techniques are central in the study of partial differential equations (PDEs). See [Coron, Bastin, d’Andr´ ea-Novel, 2007], [Krstic, Smyshlyaev, 2008], [Cazenave, Haraux, 1998] ... Two motivations to use strict Lyapunov functions: To demonstrate asymptotic stability by means of a weak 1 Lyapunov functional, the LaSalle invariance principle may be used [Luo, Guo, Morgul, 1999]. It requires a precompactness property for the solutions (may be difficult to prove, or even not satisfied). With strict Lyapunov functions in hands: robustness analysis 2 of the stability with respect to uncertainties and sensitivity of the solutions with respect to external disturbances. 3/44 C. Prieur GT-EDP’12
In this talk: new strict Lyapunov functionals for linear hyperbolic PDEs . We extend the strictification technique developed in [Malisoff, Mazenc, 2009] for ordinary differential equations and the Lyapunov design techniques of [Coron, Bastin, d’Andr´ ea-Novel, 2008] and [Diagne, Bastin, Coron, 2012] to families of PDEs with time-varying parameters. It parallels the semilinear parabolic case [Mazenc, CP, 11] It gives robustness properties of Input-to-State Stability (ISS) type See the surveys [Sontag, 2008] for nonlinear finite-dimensional systems and [Jayawardhana, Logemann, Ryan, 2011] for some classes of nonlinear infinite-dimensional systems. 4/44 C. Prieur GT-EDP’12
In this talk: new strict Lyapunov functionals for linear hyperbolic PDEs . We extend the strictification technique developed in [Malisoff, Mazenc, 2009] for ordinary differential equations and the Lyapunov design techniques of [Coron, Bastin, d’Andr´ ea-Novel, 2008] and [Diagne, Bastin, Coron, 2012] to families of PDEs with time-varying parameters. It parallels the semilinear parabolic case [Mazenc, CP, 11] It gives robustness properties of Input-to-State Stability (ISS) type See the surveys [Sontag, 2008] for nonlinear finite-dimensional systems and [Jayawardhana, Logemann, Ryan, 2011] for some classes of nonlinear infinite-dimensional systems. 4/44 C. Prieur GT-EDP’12
In this talk: new strict Lyapunov functionals for linear hyperbolic PDEs . We extend the strictification technique developed in [Malisoff, Mazenc, 2009] for ordinary differential equations and the Lyapunov design techniques of [Coron, Bastin, d’Andr´ ea-Novel, 2008] and [Diagne, Bastin, Coron, 2012] to families of PDEs with time-varying parameters. It parallels the semilinear parabolic case [Mazenc, CP, 11] It gives robustness properties of Input-to-State Stability (ISS) type See the surveys [Sontag, 2008] for nonlinear finite-dimensional systems and [Jayawardhana, Logemann, Ryan, 2011] for some classes of nonlinear infinite-dimensional systems. 4/44 C. Prieur GT-EDP’12
In this talk: new strict Lyapunov functionals for linear hyperbolic PDEs . We extend the strictification technique developed in [Malisoff, Mazenc, 2009] for ordinary differential equations and the Lyapunov design techniques of [Coron, Bastin, d’Andr´ ea-Novel, 2008] and [Diagne, Bastin, Coron, 2012] to families of PDEs with time-varying parameters. It parallels the semilinear parabolic case [Mazenc, CP, 11] It gives robustness properties of Input-to-State Stability (ISS) type See the surveys [Sontag, 2008] for nonlinear finite-dimensional systems and [Jayawardhana, Logemann, Ryan, 2011] for some classes of nonlinear infinite-dimensional systems. 4/44 C. Prieur GT-EDP’12
In this talk: new strict Lyapunov functionals for linear hyperbolic PDEs . We extend the strictification technique developed in [Malisoff, Mazenc, 2009] for ordinary differential equations and the Lyapunov design techniques of [Coron, Bastin, d’Andr´ ea-Novel, 2008] and [Diagne, Bastin, Coron, 2012] to families of PDEs with time-varying parameters. It parallels the semilinear parabolic case [Mazenc, CP, 11] It gives robustness properties of Input-to-State Stability (ISS) type See the surveys [Sontag, 2008] for nonlinear finite-dimensional systems and [Jayawardhana, Logemann, Ryan, 2011] for some classes of nonlinear infinite-dimensional systems. 4/44 C. Prieur GT-EDP’12
In this talk: new strict Lyapunov functionals for linear hyperbolic PDEs . We extend the strictification technique developed in [Malisoff, Mazenc, 2009] for ordinary differential equations and the Lyapunov design techniques of [Coron, Bastin, d’Andr´ ea-Novel, 2008] and [Diagne, Bastin, Coron, 2012] to families of PDEs with time-varying parameters. It parallels the semilinear parabolic case [Mazenc, CP, 11] It gives robustness properties of Input-to-State Stability (ISS) type See the surveys [Sontag, 2008] for nonlinear finite-dimensional systems and [Jayawardhana, Logemann, Ryan, 2011] for some classes of nonlinear infinite-dimensional systems. 4/44 C. Prieur GT-EDP’12
In this talk: new strict Lyapunov functionals for linear hyperbolic PDEs . We extend the strictification technique developed in [Malisoff, Mazenc, 2009] for ordinary differential equations and the Lyapunov design techniques of [Coron, Bastin, d’Andr´ ea-Novel, 2008] and [Diagne, Bastin, Coron, 2012] to families of PDEs with time-varying parameters. It parallels the semilinear parabolic case [Mazenc, CP, 11] It gives robustness properties of Input-to-State Stability (ISS) type See the surveys [Sontag, 2008] for nonlinear finite-dimensional systems and [Jayawardhana, Logemann, Ryan, 2011] for some classes of nonlinear infinite-dimensional systems. 4/44 C. Prieur GT-EDP’12
Outline 1 Motivations to design strict Lyapunov functions for non-homogeneous hyperbolic systems 2 Related works: Stability in presence of small perturbations 3 Sensitivity with respect to large perturbations using an ISS-Lyapunov function 4 Application to the Saint-Venant–Exner example 5 Related application: the Tokamak plasma Conclusion Sections 2 and 3 have been presented during the last CDC (sorry to Joseph and Yann). 5/44 C. Prieur GT-EDP’12
Outline 1 Motivations to design strict Lyapunov functions for non-homogeneous hyperbolic systems 2 Related works: Stability in presence of small perturbations 3 Sensitivity with respect to large perturbations using an ISS-Lyapunov function 4 Application to the Saint-Venant–Exner example 5 Related application: the Tokamak plasma Conclusion Sections 2 and 3 have been presented during the last CDC (sorry to Joseph and Yann). 5/44 C. Prieur GT-EDP’12
Outline 1 Motivations to design strict Lyapunov functions for non-homogeneous hyperbolic systems 2 Related works: Stability in presence of small perturbations 3 Sensitivity with respect to large perturbations using an ISS-Lyapunov function 4 Application to the Saint-Venant–Exner example 5 Related application: the Tokamak plasma Conclusion Sections 2 and 3 have been presented during the last CDC (sorry to Joseph and Yann). 5/44 C. Prieur GT-EDP’12
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