Boundary stabilization and observation of a weak unstable heat equation in a general multi-dimensional domain

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, stabilization of unstable PDE in a general multi-dimensional domain by using the backstepping approach 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.