# American Institute of Mathematical Sciences

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

## Strict Lyapunov functions for semilinear parabolic partial differential equations

 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

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.
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). Google Scholar [2] T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations,", Oxford Lecture Series in Mathematics and its Applications, 13 (1998). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] M. Krstic and A. Smyshlyaev, "Boundary Control of PDEs. A Course on Backstepping Designs,", Advances in Design and Control, 16 (2008). Google Scholar [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. Google Scholar [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). Google Scholar [15] M. Malisoff and F. Mazenc, "Constructions of Strict Lyapunov Functions,", Communications and Control Engineering Series, (2009). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] F. Merle and H. Zaag, Stability of the blow-up profile for equations of the type ut$=\delta u+$ |$u$|p-1 $u$,, Duke Math. J., 86 (1997), 143. doi: 10.1215/S0012-7094-97-08605-1. Google Scholar [21] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983). Google Scholar [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. Google Scholar [23] P. Pepe and H. Ito, On saturation, discontinuities and time-delays in iISS and ISS feedback control redesign,, in, (2010), 190. Google Scholar [24] C. Prieur and F. Mazenc, ISS Lyapunov functions for time-varying hyperbolic partial differential equations,, submitted for publication, (2011). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [28] E. D. Sontag, Input to state stability: Basic concepts and results,, Nonlinear and Optimal Control Theory, (2007), 163. Google Scholar [29] M. E. Taylor, "Partial Differential Equations. III. Nonlinear Equations," Applied Mathematical Sciences, 117,, Springer-Verlag, (1997). Google Scholar

show all references

##### References:
 [1] J. Bebernes and D. Eberly, "Mathematical Problems from Combustion Theory,", Applied Mathematical Sciences, 83 (1989). Google Scholar [2] T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equations,", Oxford Lecture Series in Mathematics and its Applications, 13 (1998). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] M. Krstic and A. Smyshlyaev, "Boundary Control of PDEs. A Course on Backstepping Designs,", Advances in Design and Control, 16 (2008). Google Scholar [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. Google Scholar [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). Google Scholar [15] M. Malisoff and F. Mazenc, "Constructions of Strict Lyapunov Functions,", Communications and Control Engineering Series, (2009). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] F. Merle and H. Zaag, Stability of the blow-up profile for equations of the type ut$=\delta u+$ |$u$|p-1 $u$,, Duke Math. J., 86 (1997), 143. doi: 10.1215/S0012-7094-97-08605-1. Google Scholar [21] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983). Google Scholar [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. Google Scholar [23] P. Pepe and H. Ito, On saturation, discontinuities and time-delays in iISS and ISS feedback control redesign,, in, (2010), 190. Google Scholar [24] C. Prieur and F. Mazenc, ISS Lyapunov functions for time-varying hyperbolic partial differential equations,, submitted for publication, (2011). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [28] E. D. Sontag, Input to state stability: Basic concepts and results,, Nonlinear and Optimal Control Theory, (2007), 163. Google Scholar [29] M. E. Taylor, "Partial Differential Equations. III. Nonlinear Equations," Applied Mathematical Sciences, 117,, Springer-Verlag, (1997). Google Scholar
 [1] Minkyu Kwak, Kyong Yu. The asymptotic behavior of solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 483-496. doi: 10.3934/dcds.1996.2.483 [2] Andrei Fursikov. The simplest semilinear parabolic equation of normal type. Mathematical Control & Related Fields, 2012, 2 (2) : 141-170. doi: 10.3934/mcrf.2012.2.141 [3] Shota Sato, Eiji Yanagida. Appearance of anomalous singularities in a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2012, 11 (1) : 387-405. doi: 10.3934/cpaa.2012.11.387 [4] Zhengce Zhang, Bei Hu. Gradient blowup rate for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 767-779. doi: 10.3934/dcds.2010.26.767 [5] Yang Cao, Jingxue Yin. Small perturbation of a semilinear pseudo-parabolic equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 631-642. doi: 10.3934/dcds.2016.36.631 [6] Mourad Choulli, El Maati Ouhabaz, Masahiro Yamamoto. Stable determination of a semilinear term in a parabolic equation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 447-462. doi: 10.3934/cpaa.2006.5.447 [7] Shota Sato, Eiji Yanagida. Asymptotic behavior of singular solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 4027-4043. doi: 10.3934/dcds.2012.32.4027 [8] Franck Boyer, Víctor Hernández-Santamaría, Luz De Teresa. Insensitizing controls for a semilinear parabolic equation: A numerical approach. Mathematical Control & Related Fields, 2019, 9 (1) : 117-158. doi: 10.3934/mcrf.2019007 [9] Martin Gugat, Günter Leugering, Ke Wang. Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function. Mathematical Control & Related Fields, 2017, 7 (3) : 419-448. doi: 10.3934/mcrf.2017015 [10] Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013 [11] Zhengce Zhang, Yanyan Li. Gradient blowup solutions of a semilinear parabolic equation with exponential source. Communications on Pure & Applied Analysis, 2013, 12 (1) : 269-280. doi: 10.3934/cpaa.2013.12.269 [12] Shota Sato, Eiji Yanagida. Singular backward self-similar solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 897-906. doi: 10.3934/dcdss.2011.4.897 [13] Zhengce Zhang, Yan Li. Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 3019-3029. doi: 10.3934/dcdsb.2014.19.3019 [14] Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313 [15] Marek Fila, Michael Winkler, Eiji Yanagida. Convergence to self-similar solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 703-716. doi: 10.3934/dcds.2008.21.703 [16] Karl Kunisch, Lijuan Wang. Bang-bang property of time optimal controls of semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 279-302. doi: 10.3934/dcds.2016.36.279 [17] Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225 [18] Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017 [19] Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223 [20] Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159

2018 Impact Factor: 1.292