Small-Gain-Based Boundary Feedback Design for Global Exponential Stabilization of 1-D Semilinear Parabolic PDEs

TL;DR

This paper introduces a small-gain based approach for designing boundary feedback controllers to achieve global exponential stabilization of 1-D semilinear parabolic PDEs, accommodating nonlinearities with linear growth and nonlocal terms.

Contribution

It develops a novel small-gain methodology for boundary feedback design applicable to a broad class of semilinear parabolic PDEs, including nonlinear and nonlocal terms.

Findings

Successfully designs linear static boundary feedback stabilizers.

Develops nonlinear dynamic boundary feedback controllers.

Demonstrates fundamental limitations on arbitrary gain assignment.

Abstract

This paper presents a novel methodology for the design of boundary feedback stabilizers for 1-D, semilinear, parabolic PDEs. The methodology is based on the use of small-gain arguments and can be applied to parabolic PDEs with nonlinearities that satisfy a linear growth condition. The nonlinearities may contain nonlocal terms. Two different types of boundary feedback stabilizers are constructed: a linear static boundary feedback and a nonlinear dynamic boundary feedback. It is also shown that there are fundamental limitations for feedback design in the parabolic case: arbitrary gain assignment is not possible by means of boundary feedback. An example with a nonlocal nonlinear term illustrates the applicability of the proposed methodology.