Adaptive Control of Uncertain Coupled Reaction–Diffusion Dynamics With Equidiffusivity in the Actuation Path of an ODE System

This article is devoted to the stabilization via adaptive feedback for a class of uncertain coupled reaction-diffusion dynamics in the actuation path of an ordinary differential equation (ODE) system. The system under investigation, a class of coupled parabolic partial differential equation (PDE)-ODE systems, is more representative since the dynamics in actuation path (i.e., the PDE subsystem) are coupled rather than uncoupled parabolic equations and hence includes those in the related literature as special cases. Moreover, serious parametric uncertainties are present in both PDE and ODE subsystems, which bring obstacles to the traditional methods on this issue. To solve the control problem, an infinite-dimensional backstepping transformation with time-varying matrix-valued kernel functions is adopted to change the original system into a new one. Then, an adaptive state-feedback controller is designed for the new system, which guarantees that all the closed-loop system states are bounded while regulating the original system states to zero. Finally, a simulation example is provided to illustrate the effectiveness of the theoretical results.