Exponentially Stabilizing Observer-Based Feedback Control of a Sampled-Data Linear Parabolic Multiple-Input–Multiple-Output PDE

This article addresses dynamic exponential stabilization problem of a linear scalar parabolic partial differential equation (PDE) with multiple spatially piecewise control inputs and multiple sampled-data measurement outputs in time represented as state averages over spatially noncollocated local piecewise subdomains. A sampled-data-observer-based feedback controller is constructed to guarantee the exponential convergence of the resulting closed-loop PDE. By Lyapunov’s direct method with Poincaré–Wirtinger inequality’s variants, a sufficient condition of the form linear matrix inequalities is derived for such observer-based feedback controller’s existence. This sufficient condition is extended for the case of spatially pointwise control. With the aid of semigroup theory, the closed-loop well-posedness analysis is carried out. The simulation results for a numerical example are provided to demonstrate the effectiveness of the proposed design method and its extension.