Observer-based output feedback compensator design for linear parabolic PDEs with local piecewise control and pointwise observation in space

This study presents an observer-based output feedback compensator design for a linear parabolic partial differential equation (PDE), where a finite number of actuators and sensors are active over partial areas 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 observation to exponentially track the PDE state. Based on the estimated state, a collocated local piecewise state feedback controller is then proposed. By employing a Lyapunov direct method, integration by parts, Wirtinger's inequality and first mean value theorem for definite integrals, sufficient conditions on the exponential stability of the resulting closed-loop coupled PDEs are presented in terms of standard linear matrix inequalities. Furthermore, both open-loop and closed-loop well-posedness analysis results are also established by the C₀-semigroup method. Numerical simulation results are presented to show the effectiveness of the proposed design method.