Strict Lyapunov functionals for nonlinear parabolic partial differential equations

Frédéric Mazenc (1) , Christophe Prieur (2)

1 DISCO - Dynamical Interconnected Systems in COmplex Environments
2 GIPSA-SYSCO - GIPSA - Systèmes non linéaires et complexité

For families of partial differential equations (PDE) with particular boundary conditions, strict Lyapunov functionals are constructed. The PDE 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 functionals for the nonlinear source term. The strict Lyapunov functionals 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 functional makes it possible to establish some robustness properties of Input-to-State Stability (ISS) type.

Domaines: Automatique / Robotique

Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functionals for nonlinear parabolic partial differential equations. IFAC WC 2011 - 18th IFAC World Congress, Aug 2011, Milan, Italy. ⟨hal-00573988⟩