A Lyapunov-based design of dynamic feedback compensator for linear parabolic MIMO PDEs

Abstract

Abstract This paper discusses dynamic feedback compensator design for a linear parabolic partial differential equation (PDE) with multiple inputs and multiple outputs. Actuating control inputs are provided by actuators distributed over partial areas (or active at specified positions) of the spatial domain, and observation outputs are taken from the non-collocated sensors distributed over partial areas of the spatial domain. An observer-based dynamic feedback compensator is constructed via the observer-based feedback control technique to exponentially stabilize the multi-input–multi-output PDE in the spatial $\mathscr{L}^2$ norm. By constructing an appropriate Lyapunov function candidate and using two variants of Poincaré–Wirtinger inequality, sufficient conditions on the existence of such observer-based dynamic feedback compensator are developed and presented in terms of linear matrix inequalities. The well posedness of the closed-loop coupled PDEs is also analyzed within the framework of $C_0$ semigroup theory. Finally, numerical simulation results are given to show the effectiveness of the proposed method.