A New Adaptive Control Scheme for Unstable Heat Equation with Unknown Control Coefficient

In this paper, we propose a novel adaptive control scheme for a one-dimensional unstable heat equation with an unknown control coefficient. A new state observer is designed to estimate the system state, while a new update law is devised to estimate the reciprocal of the control coefficient. In contrast to the conventional state observer, which is generally applicable to all controllers, the newly designed state observer relies on a specific controller factorization. Importantly, neither the unknown control coefficient nor its estimated value appears in the state observer anymore. Consequently, the stabilization of the control plant comes down to the stabilization of the state observer, effectively overcoming the obstacles posed by the unknown control coefficient. All the aforementioned results are rigorously proven mathematically, even though the resulting closed-loop system is nonlinear. Furthermore, some of these results are visually validated through numerical simulations.