Optimal Control of Systems Governed by Mixed Equilibrium Problems Under Monotonicity-Type Conditions with Applications
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In this paper, we introduce and study an optimal control problem governed by mixed equilibrium problems described by the sum of a maximal monotone bifunction and a bifunction which is pseudomonotone in the sense of Brézis/quasimonotone. Our motivation comes from the fact that many control problems, whose state system is a variational inequality problem or a nonlinear evolution equation or a hemivariational inequality problem, can be formulated as a control problem governed by a mixed equilibrium problem. There are different techniques to study optimal control problems governed by nonlinear evolution equations, variational inequalities or hemivariational inequalities in the literature. However, our technique is completely different from existing ones. It is based on the Mosco convergence and recent results in the theory of equilibrium problems. As an application, we study optimal control problems governed by elliptic variational inequalities with additional state constraints.
KeywordsOptimal control problems Equilibrium problems Pseudomontone operators Quasimonotone operators Maximal monotone operators Variational inequalities Mosco convergence
Mathematics Subject Classification49J21 90C33 47H05 90C30 90C29 49J40 47J20
This research was supported by KFUPM funded research project # IN161003 and it was done during the visit of first two authors. Authors are grateful to KFUPM for proving excellent research facilities to carry out this work.
- 20.Gwinner, J.: Nichtlineare Variationsungleichungen mit Anwendungen. PhD Thesis, Universitat Mannheim, 1978Google Scholar
- 36.Mustonen, V.: Quasimonotonicity and the leray-lions condition. In: A.G. Kartsatos (Ed.), Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics, 178:215–222, (1996)Google Scholar
- 40.Showalter, R.E.: Monotone operators in Banach space and nonlinear partial differential equations. Mathematical Surveys and Monographs, vol. 49. American Mathematical Society, Providence (1997)Google Scholar
- 41.Yvon, J.P.: Optimal Control of Systems Governed by Variational Inequalities. In: Conti, R., and Ruberti, R (eds.) Proceedings of the 5th Conference on Optimization Techniques, Part 1. Springer, Berlin (1973)Google Scholar
- 42.Zeidler, E.: Nonlinear Functional Analysis and Its Applications (II A and II B). Springer, New York (2011)Google Scholar
- 43.Zhenhai, L., Migorski, S., Zeng, B.: Existence results and optimal control for a class of quasimixed equilibrium problems involving the \((f,g,h)\)-quasimonotonicity. Appl. Math. Optim (2017). https://doi.org/10.1007/s00245-017-9431-3