Constructive boundary observer-based control of high-dimensional semilinear heat equations

TL;DR

This paper introduces a boundary observer-based control method for high-dimensional semilinear heat equations, enabling effective output-feedback stabilization despite challenges posed by slow eigenvalue growth and boundary-only measurements.

Contribution

It develops a novel modal-decomposition-based design that enlarges Lipschitz constants, allows boundary-only shape functions, and provides LMI-based conditions for stability and robustness in high dimensions.

Findings

Successfully stabilizes 2D and 3D heat equations.

Enlarges class of shape functions for boundary control.

Provides LMI conditions for robustness against noise.

Abstract

This paper presents a constructive finite-dimensional output-feedback design for semilinear $M$-dimensional ($M\geq 2$) heat equations with boundary actuation and sensing. A key challenge in high dimensions is the slower growth rate of the Laplacian eigenvalues. The novel features of our modal-decomposition-based design, which allows to enlarge Lipschitz constants, include a larger class of shape functions that may be distributed over a part of the boundary only, the corresponding lifting transformation and the full-order controller gain found from the design LMIs. We further analyze the robustness of the closed-loop system with respect to either multiplicative noise (vanishing at the origin) or additive noise (persistent). Effective LMI conditions are provided for specifying the minimal observer dimension and maximal Lipschitz constants that preserve the stability (mean-square exponential stability for multiplicative noise and noise-to-state stability for additive noise). Numerical examples for 2D and 3D cases demonstrate the efficacy and advantages of our method.