Observer design and boundary output feedback stabilization for semilinear parabolic system over general multidimensional domain

TL;DR

This paper develops a boundary output feedback stabilization method for semilinear parabolic systems over multidimensional domains, utilizing spectral geometry to design observers and controllers that ensure rapid stabilization.

Contribution

It introduces a novel nonlinear observer and a boundary feedback controller that together achieve rapid stabilization for Lipschitz nonlinear parabolic systems over general domains.

Findings

Spectral geometry guides sensor placement effectively.

The observer achieves any prescribed decay rate.

Numerical case confirms stabilization effectiveness.

Abstract

This paper investigates the output feedback stabilization of parabolic equation with Lipschitz nonlinearity over general multidimensional domain using spectral geometry theories. First, a novel nonlinear observer is designed, and the error system is shown to achieve any prescribed decay rate by leveraging the Berezin-Li-Yau inequality from spectral geometry, which also provides effective guidance for sensor placement. Subsequently, a finite-dimensional state feedback controller is proposed, which ensures the quantitative rapid stabilization of the linear part. By integrating this control law with the observer, an efficient boundary output feedback control strategy is developed. The feasibility of the proposed control design is rigorously verified for arbitrary Lipschitz constants, thereby resolving a persistent theoretical challenge. Finally, a numerical case study confirms the effectiveness of the approach.