LMI-based H∞ Neumann boundary control for parabolic partial differential equation with actuator saturation

This study proposes a constrained Neumann boundary controller to address the problems of stabilization and disturbance attenuation for the parabolic partial differential equations (PDEs). Initially, by applying the improved Wirtinger’s inequality, the Lyapunov function with spatial integral are applied to derive the sufficient conditions, which are cast into the matrix inequalities, to ensure the exponential stability and H∞ norm of closed-loop PDE system. Then, based on the invariant set theory, Cauchy-Schwarz and improved Poincaré inequality, a constrained H∞ boundary controller is derived to improve the performance of disturbance attenuation while avoiding the actuator saturation. Furthermore, the computational procedure is explicitly given to solve the optimal and feasible problems by applying linear matrix inequality (LMI) tools to obtain the optimal Neumann boundary control gain. Finally, a numerical simulation demonstrates the effectiveness of the proposed methodology.