Dynamic Compensator Design of Linear Parabolic MIMO PDEs in $N$-Dimensional Spatial Domain

Abstract

This article employs the observer-based output feedback control technique to deal with dynamic compensator design for a linear $N$ -D parabolic partial differential equation (PDE) with multiple local piecewise control inputs and multiple noncollocated local piecewise observation outputs. These control inputs and observation outputs are provided by only few actuators and noncollocated sensors active over partial areas (or entire) of the spatial domain. An observer-based dynamic feedback compensator is constructed for exponential stabilization of the linear PDE. Poincare–Wirtinger inequality and its variant in $N$ -D spatial domain are presented for the closed-loop stability analysis. By the Lyapunov direct method and Poincare–Wirtinger inequality and its variant, sufficient conditions on the existence of such feedback compensator of the linear PDE are developed, and presented in term of standard linear matrix inequalities. Well-posedness is also analyzed for both open-loop PDE and resulting closed-loop coupled PDEs within the $C_0$ -semigroup framework. Finally, numerical simulation results are presented to support the proposed design method.