Iterative design of suboptimal feedback control for bilinear parabolic PDE systems

Abstract

Optimal control of infinite dimensional systems is one of the central problems in the control of distributed parameter systems. With the development of high performance computers, numerical methods for optimal control design have regained attention and achieved significant progress, mostly in the form of open-loop solutions. We consider in this work an optimal control problem for a bilinear parabolic partial differential equation (PDE) system. Based on the optimality conditions derived from Pontryagin's maximum principle for a reduced-order model, and stated as a two-boundary-value problem, we propose an iterative scheme for suboptimal closed-loop control design. In each iteration step, we take advantage of linear synthesis methods to construct a sequence of controllers. The convergence of the controller sequence is proved in appropriate functional spaces. When compared with previous iterative schemes, the proposed scheme avoids repeated numerical computation of the Riccati equation and therefore reduces significantly the number of ODEs that must be solved at each iteration step. A numerical simulation study shows the effectiveness of this new approach.