Boundary Stabilization and Observation of an Unstable Heat Equation in a General Multi-dimensional Domain

TL;DR

This paper develops a method for stabilizing and observing an unstable heat equation in multi-dimensional domains using spectral truncation and dynamics compensation, extending stabilization techniques beyond one-dimensional cases.

Contribution

It introduces a novel approach combining spectral truncation and dynamics compensation for stabilization and observation of multi-dimensional unstable heat equations, which is a significant extension of existing methods.

Findings

Successfully stabilizes the unstable heat equation in multi-dimensional domains.

Designs a boundary observer based on boundary measurements.

Validates theoretical results with numerical simulations.

Abstract

In this paper, we consider the exponential stabilization and observation of an unstable heat equation in a general multi-dimensional domain by combining the finite-dimensional spectral truncation technique and the recently developed dynamics compensation approach. In contrast to the unstable one-dimensional partial differential equation (PDE), such as the transport equation, wave equation and the heat equation, that can be treated by the well-known PDE backstepping method, stabilization of unstable PDE in a general multi-dimensional domain is still a challenging problem. We treat the stabilization and observation problems separately. A dynamical state feedback law is proposed firstly to stabilize the unstable heat equation exponentially and then a state observer is designed via a boundary measurement. Both the stability of the closed-loop system and the well-posedness of the observer are proved. Some of the theoretical results are validated by the numerical simulations.