Adaptive Output Feedback Stabilization for an Unstable Heat Equation with Unknown Parameter

This paper addresses the adaptive output feedback stabilization problem for a one-dimensional unstable heat equation with an unknown parameter. A state observer is designed to estimate the system state, while a parameter update law is introduced to estimate the unknown parameter. Notably, the convergence of the state observer is independent of the convergence of the unknown parameter. Moreover, under a simple persistent excitation condition, the unknown parameter can be effectively estimated. To overcome the challenges in control design posed by the unknown parameter, we stabilize the state observer instead of the original control system. This approach facilitates the design of an output feedback controller. Numerical simulations are provided to demonstrate the effectiveness of the proposed method.