Suboptimal Feedback Control Design of Constrained Parabolic Systems in Uncertainty Conditions

The talk concerns minimax control problems for linear multidimensional parabolic systems with distributed uncertain perturbations and control functions acting in the Dirichlet boundary conditions. The underlying parabolic control system is functioning under hard/pointwise constraints on control and state variables. The main goal is to design a feedback control regulator that ensures the required state performance and robust stability under any feasible perturbations and minimize an energy-type functional under the worst perturbations from the given area. We develop an efficient approach to the minimax control design of constrained parabolic systems that is based on certain characteristic features of the parabolic dynamics including the transient monotonicity with respect to both controls and perturbations and the turnpike asymptotic behavior on the infinite horizon. In this way, solving a number of associated open-loop control and approximation problems, we justify an easily implemented suboptimal structure of the feedback boundary regulator and compute its optimal parameters ensuring the required state performance and robust stability of the closed-loop, highly nonlinear parabolic control system on the infinite horizon. The primary motivation for this study came from certain environmental models, in particular, those developed within the Dynamical System and Environmental Projects of the International Institute of Applied System Analysis (IIASA), Laxenburg, Austria. Department of Mathematics, Wayne State University, Detroit, Michigan 48202 ([email protected]). This research was partly supported by the USA National Science Foundation under grants DMS-0304989 and DMS-0603846 and by the Australian Research Council under grant DP-0451168.