Condition-Based Adaptive Scheme for the Stabilization of an Uncertain Reaction–Diffusion Equation With Nonlinear Dynamic Boundary

This article is devoted to the stabilization of an uncertain reaction–diffusion equation with dynamic boundary conditions. The remarkable features of the system under investigation are reflected in the presence of the nonlinear dynamic boundary condition and the serious uncertainties that are involved in not only the dynamic boundary but also the main equation. Consequently, the traditional control schemes on this topic are incapable. Toward the control problem, a novel control scheme via condition-based adaption is proposed in this article. Specifically, an infinite-dimensional backstepping transformation is first introduced to change the original system into a new one, in which certain crucial controller parameter to be tuned later is involved. Then, for the newly obtained system, the controller is designed joint with a crucial condition-based updating mechanism. By the proposed updating mechanism, two pivotal controller parameters are online tuned so as to compensate the unknowns, respectively, involved in both the dynamic boundary and the main equation. Finally, much effort is taken to show that the designed controller guarantees that all the states of the resulting closed-loop system are bounded while converge to zero ultimately. A simulation example is provided to demonstrate the efficiency of the theoretical results.