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| Volume 4, Number 3, September 2008, pp. 549-570 | ||||||||
| Boris S. Mordukhovich | ||||||||
| Key words: | ||||||||
| dynamic optimization, parabolic systems, boundary controls, state constraints, uncertainty conditions, feedback minimax design, closed-loop stability | ||||||||
| Mathematices Subject Classification: 49K20, 49K35, 49N35, 93B50, 93D09 | ||||||||
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| Abstract: | |||
| The paper is devoted to optimal control and feedback design of state-constrained parabolic systems in uncertainty conditions. Problems of this type are among the most challenging and difficult in dynamic optimization for any kind of dynamical systems. We pay the main attention to considering linear multidimensional parabolic systems with Dirichlet boundary controls and pointwise state constraints, while the methods developed in this study are applicable to other kinds of boundary controls and dynamical systems of the parabolic type. The feedback design problem is formulated in the minimax sense to ensure stabilization of transients within the prescribed diapason and robust stability of the closed-loop control system under all feasible perturbations with minimizing an integral cost functional in the worst perturbation case. Exploiting certain fundamental properties of the parabolic dynamics, we single out the worst perturbations in the minimax control problem and efficiently solve the associated optimal control problems for approximating ODE and the original PDE systems with pointwise state constraints. In this way, using the transient monotonicity and turnpike asymptotic properties of the underlying parabolic dynamics on the infinite horizon, we compute optimal (in the minimax sense) parameters of the easily implemented while rigorously justified three-positional suboptimal structure of the feedback boundary controls that ensure robust stability of the closed-loop and highly nonlinear parabolic control system under consideration. | |||
| A proximity control algorithm to minimize nonsmooth and nonconvex functions | ||