Output feedback finite-time boundary control for an unstable heat PDE with spatially varying coefficients

This article studies the output feedback finite-time boundary control for unstable heat systems with the spatially varying coefficient. First, a finite-time observer with switched gains under a state-dependent switching law is designed in order to estimate the states of the system in a finite-time only exerting one displacement boundary measurement. Next, an observer-based linear finite-time control is planned. Namely, a linear switched control under a state-dependent switching law is proposed to vanish every solution of an unstable heat partial differential equation with spatially varying coefficients in a finite time. We also present explicit forms for the proposed observer gains and output feedback finite-time controller. Finally, some numerical simulations are provided to confirm the theoretical results.