Optimal Control of Systems Governed by Mixed Equilibrium Problems Under Monotonicity-Type Conditions with Applications

  • O. Chadli
  • Q. H. Ansari
  • S. Al-Homidan
  • M. AlshahraniEmail author


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.


Optimal control problems Equilibrium problems Pseudomontone operators Quasimonotone operators Maximal monotone operators Variational inequalities Mosco convergence 

Mathematics Subject Classification

49J21 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.


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Copyright information

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Authors and Affiliations

  • O. Chadli
    • 1
  • Q. H. Ansari
    • 2
    • 3
  • S. Al-Homidan
    • 3
  • M. Alshahrani
    • 3
    Email author
  1. 1.Faculty of Economics and Social Sciences, Department of EconomicsIbn Zohr UniversityAgadirMorocco
  2. 2.Department of MathematicsAligarh Muslim UniversityAligarhIndia
  3. 3.Department of Mathematics & StatisticsKing Fahd University of Petroleum & MineralsDhahranSaudi Arabia

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