Optimal control and feedback design of state-constrained parabolic systems in uncertainty conditions

Abstract

This article 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 a constructive 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 behaviour on the infinite horizon. In this way, solving a number of associated open-loop control and optimization problems, we justify an easily implementable three-positional suboptimal structure of the feedback boundary regulator and compute its optimal parameters, thus ensuring the required state performance and robust stability of the closed-loop, highly nonlinear parabolic control system on the infinite horizon.

Keywords:

• parabolic systems
• Dirichlet boundary controls
• state constraints
• uncertainty perturbations
• feedback control
• suboptimality
• minimax synthesis
• robust stability

AMS Subject Classifications::

• 49K20
• 49K35
• 49N35
• 93B50
• 93D09

Acknowledgements

The author is grateful to Jean-Pierre Raymond and Tom Seidman for helpful discussions and remarks on the material of this article. This research was partly supported by the National Science Foundation under grant DMS-0603846.