Approximations of Lyapunov functionals for ISS analysis of a class of higher dimensional nonlinear parabolic PDEs

Abstract This paper introduces a novel method, namely the approximations of Lyapunov functionals, for input-to-state stability (ISS) analysis of a class of higher dimensional nonlinear parabolic partial differential equations (PDEs) with variable coefficients. Specifically, for any q ∈ [ 1 , + ∞ ] and the considered nonlinear parabolic PDEs with different types of boundary disturbances in L l o c q ( R + ; L 1 ( ∂ Ω ) ) and initial data in L 1 ( Ω ) , we show that ISS-like estimates in L 1 -norm (or weighted L 1 -norm) can be established by constructing approximations of (coercive and non-coercive) Lyapunov functionals. Some examples are provided to illustrate the application of the proposed method.