Pointwise exponential stabilization of a linear parabolic PDE system using non-collocated pointwise observation

This paper deals with the problem of exponential stabilization for a linear distributed parameter system (DPS) using pointwise control and non-collocated pointwise observation, where the system is modeled by a parabolic partial differential equation (PDE). The main objective of this paper is to construct an output feedback controller for pointwise exponential stabilization of the linear parabolic PDE system using the non-collocated pointwise observation. The observer-based output feedback control technique is utilized to overcome the design difficulty caused by the non-collocation pointwise observation. We construct a Luenberger-type PDE state observer to exponentially track the state of the PDE system. A collocated pointwise feedback controller is proposed based on the estimated state. A variation of Poincaré – Wirtinger inequality is derived and the spatial domain is decomposed into multiple sub-domains according to the number of the actuators and sensors. By using Lyapunov's direct method, integration by parts, and the variation of Poincaré – Wirtinger inequality at each sub-domain, sufficient conditions for the existence of the observer-based out feedback controller are developed such that the resulting closed-loop system is exponentially stable, and presented in terms of standard linear matrix inequalities (LMIs). Furthermore, the closed-loop well-posedness analysis is also provided by the C₀-semigroup approach. Extensive numerical simulation results are presented to show the performance of the proposed output feedback controller.