Observer-based dynamic local piecewise control of a linear parabolic PDE system with non-collocated pointwise measurements

This paper presents an observer-based dynamic feedback control design for a linear parabolic partial differential equation (PDE) system, where a finite number of actuators and sensors are active over part thereof and at specified points in the spatial domain, respectively. In the proposed design method, a Luenberger-type PDE observer is first constructed by using the non-collocated pointwise measurements to exponentially tract the state of the PDE system. Based on the estimated state, a collocated local piecewise state feedback controller is then proposed. By employing Lyapunov direct method, integration by parts, Wirtinger's inequality and first mean value theorem for integrals, a sufficient condition on exponential stability of the resulting closed-loop system is presented in term of standard linear matrix inequalities (LMIs). Numerical simulation results are presented to show the effectiveness of the proposed design method.