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

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 $L_{loc}^q({\mathbb{R}}_{+};L^1(\partial \Omega ))$-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 $L_{loc}^{\Phi }({\mathbb{R}}_{+};L^1(\partial \Omega ))$-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 $L_{loc}^q({\mathbb{R}}_{+};K_{\Phi }(\partial \Omega ))$-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.

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