Approximations of Lyapunov functions for ISS analysis of a class of nonlinear parabolic PDEs

TL;DR

This paper develops approximation methods for Lyapunov functions to analyze input-to-state stability of nonlinear parabolic PDEs with boundary disturbances across various function spaces, including Orlicz spaces.

Contribution

It introduces novel Lyapunov function approximations to establish ISS and iISS for PDEs with boundary disturbances in diverse function spaces, extending stability analysis techniques.

Findings

Established ISS and iISS in L^1 and weighted L^1 norms.

Extended ISS analysis to Orlicz spaces and classes.

Provided stability results for PDEs with boundary disturbances in various function spaces.

Abstract

This paper addresses the input-to-state stability (ISS) and integral input-to-state stability (iISS) for a class of nonlinear higher dimensional parabolic partial differential equations (PDEs) with different types of boundary disturbances (Robin or Neumann or Dirichlet) from different spaces by means of approximations of Lyapunov functions. Specifically, by constructing approximations of (coercive and non-coercive) ISS Lyapunov functions we establish: (i) the ISS and iISS in $L^1$-norm (and weighted $L^1$-norm) for PDEs with boundary disturbances from $L^q_{loc}(\mathbb{R}_+;L^1(\partial\Omega))$-space for any $q\in [1,+\infty]$; (ii) the iISS in $L^1$-norm (and weighted $L^1$-norm) for PDEs with boundary disturbances from $L^\Phi_{loc}(\mathbb{R}_+;L^1(\partial\Omega))$-space for certain Young function $\Phi$; and (iii) the ISS and iISS in $L^\Phi$-norm (and weighted $K_\Phi$-class) for PDEs with boundary disturbances from $L^q_{loc}(\mathbb{R}_+;K_\Phi(\partial\Omega))$-class for any $q\in [1,+\infty]$ and certain Young function $\Phi$. The ISS properties stated in (ii) and (iii) are assessed in the framework of Orlicz space or Orlicz class.