Spatial domain decomposition approach to dynamic compensator design for linear space‐varying parabolic MIMO PDEs

This study addresses the problem of dynamic compensator design for exponential stabilisation of linear space‐varying parabolic multiple‐input–multiple‐output (MIMO) partial differential equations (PDEs) subject to periodic boundary conditions. With the aid of the observer‐based feedback control technique, an observer‐based dynamic feedback compensator, whose implementation requires only a few actuators and sensors active over partial areas of the spatial domain, is constructed such that the resulting closed‐loop coupled PDEs is exponentially stable. The spatial domain is divided into multiple subdomains according to the minimum of the actuators' number and the sensors' one. By Lyapunov direct method and two general variants of Poincaré‐Wirtinger inequality at each subdomain, sufficient conditions for the existence of such feedback compensator are developed and presented in terms of algebraic linear matrix inequalities (LMIs) in space. Based on the extreme value theorem, LMI‐based sufficient and necessary conditions are presented for the feasibility of algebraic LMIs in space. Finally, numerical simulation results are presented to support the proposed design method.