  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.
