Adaptive stabilization for a reaction–diffusion equation with uncertain nonlinear actuator dynamics

Abstract

This paper is devoted to the stabilization of a reaction–diffusion equation with uncertain nonlinear actuator dynamics described by a cascaded parabolic PDE–ODE system. Essentially different from the systems in the related literature, control input appears in the ODE subsystem rather than in the PDE one, and moreover, the actuator dynamics are nonlinear and possess uncertainties rather than linear without uncertainty. Note that the existing control schemes so far are ineffective to handle such systems, and hence the stabilization problem for these systems is still unsolved. For this, backstepping approaches in both infinite and finite-dimension are smartly combined with the adaptive dynamic compensation technique in order to solve the control problem. Specifically, by using an infinite-dimensional backstepping transformation and its inverse, the original PDE subsystem is first changed to a new one from which control design is much convenient. Then, based on the new system together with the actuator dynamics, an adaptive state-feedback controller is explicitly constructed by using a couple of finite-dimensional backstepping transformations, which guarantees that all the states of the resulting closed-loop systems are bounded. Moreover, both the control input and the states of the original system converge to zero ultimately. Finally, a simulation example is provided to validate the effectiveness of the established theoretical results.