Strict Lyapunov functions for semilinear parabolic partial differential equations

Frédéric Mazenc,Christophe Prieur

doi:10.3934/mcrf.2011.1.231

Abstract

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.

Highlights

Lyapunov function based techniques are central in the study of partial differential equations (PDEs)
We show that the function U given in (9) is a strict Lyapunov function for (4) when this system is associated with special families of boundary conditions or when W1 is larger than a positive definite quadratic function
To numerically check the fact that U is a Lyapunov function, let us discretize the semilinear parabolic partial differential equation (53) using an explicit Euler discretization We select the parameters of the numerical scheme so that the CFL condition for the stability holds

Summary

Introduction

Lyapunov function based techniques are central in the study of partial differential equations (PDEs). In [2] a Lyapunov function is used to establish the existence of a global solution to the celebrated heat equation. We show that the function U given in (9) is a strict Lyapunov function for (4) when this system is associated with special families of boundary conditions or when W1 is larger than a positive definite quadratic function.

Results

Conclusion