Observer‐Based Boundary Control for Coupled Parabolic PDEs With Spatially Varying Coefficients

Abstract

This paper addresses the observer‐based output‐feedback boundary control design for a system of ‐coupled parabolic partial differential equations (PDEs) with spatially varying diffusion coefficients without any ordering conditions. First, a boundary observer is proposed to estimate the system states, with its gain determined by considering a stable target system and using a backstepping transformation method. To design an observer‐based boundary controller for ‐coupled parabolic PDE systems, we construct a backstepping transformation that maps the observer dynamics into a stable target system, yielding the associated kernel and boundary control law, which is the main contribution of this work. Furthermore, analytical solutions to two sets of kernel equations are derived to determine the controller and observer gains. By employing Lyapunov stability analysis, we prove that the proposed output‐feedback architecture guarantees exponential stability of the entire closed‐loop system. Finally, numerical simulations are conducted to validate the theoretical results of the proposed method.