Asymmetric games for convolution systems with applications to feedback control of constrained parabolic equations

Abstract The paper is devoted to the study of some classes of feedback control problems for linear parabolic equations subject to hard/pointwise constraints on both Dirichlet boundary controls and state dynamic/output functions in the presence of uncertain perturbations within given regions. The underlying problem under consideration, originally motivated by automatic control of the groundwater regime in irrigation networks, is formalized as a minimax problem of optimal control, where the control strategy is sought as a feedback law. Problems of this type are among the most important in control theory and applications — while most challenging and difficult. Based on the Maximum Principle for parabolic equations and on the time convolution structure, we reformulate the problems under consideration as certain asymmetric games, which become the main object of our study in this paper. We establish some simple conditions for the existence of winning and losing strategies for the game players, which then allow us to clarify controllability issues in the feedback control problem for such constrained parabolic systems.

[1]  A. I. Subbotin,et al.  Game-Theoretical Control Problems , 1987 .

[2]  J. M. Ball,et al.  GEOMETRIC THEORY OF SEMILINEAR PARABOLIC EQUATIONS (Lecture Notes in Mathematics, 840) , 1982 .

[3]  Xun Yu Zhou,et al.  Control of Distributed Parameter and Stochastic Systems , 1999, IFIP Advances in Information and Communication Technology.

[4]  Boris S. Mordukhovich,et al.  Robust Suboptimal Control of Constrained Parabolic Systems under Uncertainty Conditions * , 1999 .

[5]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[6]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[7]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[8]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[9]  G. Stampacchia,et al.  Inverse Problem for a Curved Quantum Guide , 2012, Int. J. Math. Math. Sci..

[10]  B. Sh. Mordukhovich.  Optimal control of the groundwater regime on two-way engineering reclamation systems , 1986 .

[11]  M. L. Chambers.  The Mathematical Theory of Optimal Processes , 1965 .

[12]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[13]  R. Triggiani,et al.  Control Theory for Partial Differential Equations: Continuous and Approximation Theories , 2000 .

[14]  Boris S. Mordukhovich.  Minimax Design of Constrained Parabolic Systems , 1998, Control of Distributed Parameter and Stochastic Systems.

[15]  Tamer Başar,et al.  H1-Optimal Control and Related Minimax Design Problems , 1995 .

[16]  Mordukhovich B Sh.  Minimax design of a class of distributed control systems. , 1990 .

[17]  B. Mordukhovich,et al.  Optimization and Feedback Control of Constrained Parabolic Systems Under Uncertain Perturbations , 2003 .