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

The main goal of this paper is to study the existence of solutions for optimal control problems governed by mixed quasi-equilibrium problems under monotonicity type conditions. More precisely, the state control system takes the general form of a mixed quasi-equilibrium problem described by the sum of a maximal monotone bifunction and a pseudomonotone bifunction in the sense of Brézis. As applications, we study the existence of solutions for optimal control problems governed by quasi-variational inequalities. In particularly, we consider the optimal control of an evolutionary quasi-variational inequality described by a p-Laplacian type operator. The results obtained in this paper are new and can be applied to the study of optimal control of a variety of systems whose formulations can be presented as a mixed equilibrium problem.