ISS Estimates in the Spatial Sup-Norm for Nonlinear 1-D Parabolic PDEs

TL;DR

This paper develops new ISS-style maximum principle estimates for classical solutions of highly nonlinear 1-D parabolic PDEs, using an ISS Lyapunov functional to handle nonlinear and non-local boundary conditions.

Contribution

It introduces a novel methodology for deriving ISS estimates in the sup norm for nonlinear 1-D parabolic PDEs, including boundary and in-domain non-local terms.

Findings

Provides fading memory ISS estimates in the sup norm

Handles nonlinear and non-local boundary conditions

Demonstrates effectiveness through three examples

Abstract

This paper provides novel Input-to-State Stability (ISS)-style maximum principle estimates for classical solutions of highly nonlinear 1-D parabolic Partial Differential Equations (PDEs). The derivation of the ISS-style maximum principle estimates is performed by using an ISS Lyapunov Functional for the sup norm. The estimates provide fading memory ISS estimates in the sup norm of the state with respect to distributed and boundary inputs. The obtained results can handle parabolic PDEs with nonlinear and non-local in-domain terms/boundary conditions. Three illustrative examples show the efficiency of the proposed methodology for the derivation of ISS estimates in the sup norm of the state.