Strict Lyapunov functions for semilinear parabolic partial differential equations

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2011, 1(2): 231-250. doi: 10.3934/mcrf.2011.1.231

Strict Lyapunov functions for semilinear parabolic partial differential equations

Frédéric Mazenc ^1, and Christophe Prieur ^2,

1.	Team INRIA DISCO, CNRS-Supelec, 3 rue Joliot Curie, 91192 Gif-sur-Yvette, France
2.	Department of Automatic Control, Gipsa-lab, 961 rue de la Houille Blanche, BP 46, 38402 Grenoble Cedex

Received November 2010 Revised March 2011 Published June 2011

Abstract
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For families of partial differential equations (PDEs) with particular boundary conditions, strict Lyapunov functions are constructed. The PDEs under consideration are parabolic and, in addition to the diffusion term, may contain a nonlinear source term plus a convection term. The boundary conditions may be either the classical Dirichlet conditions, or the Neumann boundary conditions or a periodic one. The constructions rely on the knowledge of weak Lyapunov functions for the nonlinear source term. The strict Lyapunov functions are used to prove asymptotic stability in the framework of an appropriate topology. Moreover, when an uncertainty is considered, our construction of a strict Lyapunov function makes it possible to establish some robustness properties of Input-to-State Stability (ISS) type.

Keywords: semilinear parabolic equation., Strictification, Lyapunov function.

Mathematics Subject Classification: Primary: 93D05, 35K55; Secondary: 93C1.

Citation: Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control & Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231

References:

[1]	J. Bebernes and D. Eberly, "Mathematical Problems from Combustion Theory,", Applied Mathematical Sciences, 83 (1989).
[2]	T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations,", Oxford Lecture Series in Mathematics and its Applications, 13 (1998).
[3]	X.-Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations,, Journal of Differential Equations, 78 (1989), 160.
[4]	J.-M. Coron, G. Bastin and B. d'Andréa-Novel, Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems,, SIAM Journal on Control and Optimization, 47 (2008), 1460. doi: 10.1137/070706847.
[5]	J.-M. Coron and B. d'Andréa-Novel, Stabilization of a rotating body beam without damping,, IEEE Transactions on Automatic Control, 43 (1998), 608. doi: 10.1109/9.668828.
[6]	J.-M. Coron, B. d'Andréa-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws,, IEEE Transactions on Automatic Control, 52 (2007), 2. doi: 10.1109/TAC.2006.887903.
[7]	J.-M. Coron and E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equations,, SIAM Journal on Control and Optimization, 43 (2004), 549. doi: 10.1137/S036301290342471X.
[8]	J.-M. Coron and E. Trélat, Global steady-state stabilization and controllability of 1D semilinear wave equations,, Commun. Contemp. Math., 8 (2006), 535. doi: 10.1142/S0219199706002209.
[9]	A. K. Dramé, D. Dochain and J. J. Winkin, Asymptotic behavior and stability for solutions of a biochemical reactor distributed parameter model,, IEEE Transactions on Automatic Control, 53 (2008), 412. doi: 10.1109/TAC.2007.914948.
[10]	O. V. Iftime and M. A. Demetriou, Optimal control of switched distributed parameter systems with spatially scheduled actuators,, Automatica J. IFAC, 45 (2009), 312. doi: 10.1016/j.automatica.2008.07.012.
[11]	I. Karafyllis, P. Pepe and Z.-P. Jiang, Input-to-output stability for systems described by retarded functional differential equations,, European Journal of Control, 14 (2008), 539. doi: 10.3166/ejc.14.539-555.
[12]	M. Krstic and A. Smyshlyaev, "Boundary Control of PDEs. A Course on Backstepping Designs,", Advances in Design and Control, 16 (2008).
[13]	M. Krstic and A. Smyshlyaev, Adaptive boundary control for unstable parabolic PDEs. I. Lyapunov design,, IEEE Transactions on Automatic Control, 53 (2008), 1575. doi: 10.1109/TAC.2008.927798.
[14]	Z.-H. Luo, B.-Z. Guo and O. Morgul, "Stability and Stabilization of Infinite Dimensional Systems with Applications,", Communications and Control Engineering Series, (1999).
[15]	M. Malisoff and F. Mazenc, "Constructions of Strict Lyapunov Functions,", Communications and Control Engineering Series, (2009).
[16]	D. Matignon and C. Prieur, Asymptotic stability of linear conservative systems when coupled with diffusive systems,, ESAIM Control Optim. Cal. Var., 11 (2005), 487. doi: 10.1051/cocv:2005016.
[17]	F. Mazenc, M. Malisoff and O. Bernard, A simplified design for strict Lyapunov functions under Matrosov conditions,, IEEE Transactions on Automatic Control, 54 (2009), 177. doi: 10.1109/TAC.2008.2008353.
[18]	F. Mazenc, M. Malisoff and Z. Lin, Further results on input-to-state stability for nonlinear systems with delayed feedbacks,, Automatica J. IFAC, 44 (2008), 2415. doi: 10.1016/j.automatica.2008.01.024.
[19]	F. Mazenc and D. Nesic, Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem,, Mathematics of Control, 19 (2007), 151. doi: 10.1007/s00498-007-0015-7.
[20]	F. Merle and H. Zaag, Stability of the blow-up profile for equations of the type u_t$=\delta u+$ \|$u$\|^p-1 $u$,, Duke Math. J., 86 (1997), 143. doi: 10.1215/S0012-7094-97-08605-1.
[21]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).
[22]	P. Pepe, Input-to-state stabilization of stabilizable, time-delay, control-affine, nonlinear systems,, IEEE Transactions on Automatic Control, 54 (2009), 1688. doi: 10.1109/TAC.2009.2020642.
[23]	P. Pepe and H. Ito, On saturation, discontinuities and time-delays in iISS and ISS feedback control redesign,, in, (2010), 190.
[24]	C. Prieur and F. Mazenc, ISS Lyapunov functions for time-varying hyperbolic partial differential equations,, submitted for publication, (2011).
[25]	M. Slemrod, A note on complete controllability and stabilizability for linear control systems in Hilbert space,, SIAM Journal on Control, 12 (1974), 500. doi: 10.1137/0312038.
[26]	A. Smyshlyaev and M. Krstic, Adaptive boundary control for unstable parabolic PDEs. II. Estimation-based designs,, Automatica J. IFAC, 43 (2007), 1543. doi: 10.1016/j.automatica.2007.02.014.
[27]	A. Smyshlyaev and M. Krstic, Adaptive boundary control for unstable parabolic PDEs. III. Output feedback examples with swapping identifiers,, Automatica J. IFAC, 43 (2007), 1557. doi: 10.1016/j.automatica.2007.02.015.
[28]	E. D. Sontag, Input to state stability: Basic concepts and results,, Nonlinear and Optimal Control Theory, (2007), 163.
[29]	M. E. Taylor, "Partial Differential Equations. III. Nonlinear Equations," Applied Mathematical Sciences, 117,, Springer-Verlag, (1997).

show all references

References:

[1]	J. Bebernes and D. Eberly, "Mathematical Problems from Combustion Theory,", Applied Mathematical Sciences, 83 (1989).
[2]	T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations,", Oxford Lecture Series in Mathematics and its Applications, 13 (1998).
[3]	X.-Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations,, Journal of Differential Equations, 78 (1989), 160.
[4]	J.-M. Coron, G. Bastin and B. d'Andréa-Novel, Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems,, SIAM Journal on Control and Optimization, 47 (2008), 1460. doi: 10.1137/070706847.
[5]	J.-M. Coron and B. d'Andréa-Novel, Stabilization of a rotating body beam without damping,, IEEE Transactions on Automatic Control, 43 (1998), 608. doi: 10.1109/9.668828.
[6]	J.-M. Coron, B. d'Andréa-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws,, IEEE Transactions on Automatic Control, 52 (2007), 2. doi: 10.1109/TAC.2006.887903.
[7]	J.-M. Coron and E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equations,, SIAM Journal on Control and Optimization, 43 (2004), 549. doi: 10.1137/S036301290342471X.
[8]	J.-M. Coron and E. Trélat, Global steady-state stabilization and controllability of 1D semilinear wave equations,, Commun. Contemp. Math., 8 (2006), 535. doi: 10.1142/S0219199706002209.
[9]	A. K. Dramé, D. Dochain and J. J. Winkin, Asymptotic behavior and stability for solutions of a biochemical reactor distributed parameter model,, IEEE Transactions on Automatic Control, 53 (2008), 412. doi: 10.1109/TAC.2007.914948.
[10]	O. V. Iftime and M. A. Demetriou, Optimal control of switched distributed parameter systems with spatially scheduled actuators,, Automatica J. IFAC, 45 (2009), 312. doi: 10.1016/j.automatica.2008.07.012.
[11]	I. Karafyllis, P. Pepe and Z.-P. Jiang, Input-to-output stability for systems described by retarded functional differential equations,, European Journal of Control, 14 (2008), 539. doi: 10.3166/ejc.14.539-555.
[12]	M. Krstic and A. Smyshlyaev, "Boundary Control of PDEs. A Course on Backstepping Designs,", Advances in Design and Control, 16 (2008).
[13]	M. Krstic and A. Smyshlyaev, Adaptive boundary control for unstable parabolic PDEs. I. Lyapunov design,, IEEE Transactions on Automatic Control, 53 (2008), 1575. doi: 10.1109/TAC.2008.927798.
[14]	Z.-H. Luo, B.-Z. Guo and O. Morgul, "Stability and Stabilization of Infinite Dimensional Systems with Applications,", Communications and Control Engineering Series, (1999).
[15]	M. Malisoff and F. Mazenc, "Constructions of Strict Lyapunov Functions,", Communications and Control Engineering Series, (2009).
[16]	D. Matignon and C. Prieur, Asymptotic stability of linear conservative systems when coupled with diffusive systems,, ESAIM Control Optim. Cal. Var., 11 (2005), 487. doi: 10.1051/cocv:2005016.
[17]	F. Mazenc, M. Malisoff and O. Bernard, A simplified design for strict Lyapunov functions under Matrosov conditions,, IEEE Transactions on Automatic Control, 54 (2009), 177. doi: 10.1109/TAC.2008.2008353.
[18]	F. Mazenc, M. Malisoff and Z. Lin, Further results on input-to-state stability for nonlinear systems with delayed feedbacks,, Automatica J. IFAC, 44 (2008), 2415. doi: 10.1016/j.automatica.2008.01.024.
[19]	F. Mazenc and D. Nesic, Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem,, Mathematics of Control, 19 (2007), 151. doi: 10.1007/s00498-007-0015-7.
[20]	F. Merle and H. Zaag, Stability of the blow-up profile for equations of the type u_t$=\delta u+$ \|$u$\|^p-1 $u$,, Duke Math. J., 86 (1997), 143. doi: 10.1215/S0012-7094-97-08605-1.
[21]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).
[22]	P. Pepe, Input-to-state stabilization of stabilizable, time-delay, control-affine, nonlinear systems,, IEEE Transactions on Automatic Control, 54 (2009), 1688. doi: 10.1109/TAC.2009.2020642.
[23]	P. Pepe and H. Ito, On saturation, discontinuities and time-delays in iISS and ISS feedback control redesign,, in, (2010), 190.
[24]	C. Prieur and F. Mazenc, ISS Lyapunov functions for time-varying hyperbolic partial differential equations,, submitted for publication, (2011).
[25]	M. Slemrod, A note on complete controllability and stabilizability for linear control systems in Hilbert space,, SIAM Journal on Control, 12 (1974), 500. doi: 10.1137/0312038.
[26]	A. Smyshlyaev and M. Krstic, Adaptive boundary control for unstable parabolic PDEs. II. Estimation-based designs,, Automatica J. IFAC, 43 (2007), 1543. doi: 10.1016/j.automatica.2007.02.014.
[27]	A. Smyshlyaev and M. Krstic, Adaptive boundary control for unstable parabolic PDEs. III. Output feedback examples with swapping identifiers,, Automatica J. IFAC, 43 (2007), 1557. doi: 10.1016/j.automatica.2007.02.015.
[28]	E. D. Sontag, Input to state stability: Basic concepts and results,, Nonlinear and Optimal Control Theory, (2007), 163.
[29]	M. E. Taylor, "Partial Differential Equations. III. Nonlinear Equations," Applied Mathematical Sciences, 117,, Springer-Verlag, (1997).

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