ISS estimates in the spatial sup-norm for nonlinear 1-D parabolic PDEs

¹ Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece.
² Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA.

^* Corresponding author: iasonkar@central.ntua.gr

Received: 28 August 2020
Accepted: 16 May 2021

Abstract

This paper provides novel Input-to-State Stability (ISS)-style maximum principle estimates for classical solutions of nonlinear 1-D parabolic Partial Differential Equations (PDEs). The derivation of the ISS-style maximum principle estimates is performed in two ways: by using an ISS Lyapunov Functional for the sup norm and by exploiting well-known maximum principles. 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.