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arXiv · 2102.02492

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

Hongyinping Feng·Pei-Hua Lang·Jiankang Liu
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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.

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Hongyinping Feng, Pei-Hua Lang, Jiankang Liu. 2021-02-04. Boundary Stabilization and Observation of an Unstable Heat Equation in a General Multi-dimensional Domain. https://arxiv.org/abs/2102.02492

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