Optimal Control of Systems Governed by Mixed Equilibrium Problems: Existence, Algorithms and Applications

In this project, we introduce and study optimal control governed by mixed equilibrium problems which equilibrium function is the sum of a maximal monotone bifunction and a pseudomonotone (respectively, quasimonotone) function in Brzis sense. Our motivation comes from the fact that many control problems which the state system is a nonlinear evolution equation or a variational inequality problem or a hemivariational inequality problem can be formulated as a control problem governed by a mixed equilibrium problem as described above. We can find in literature different techniques for studying optimal control of problems governed by nonlinear evolution equations, variational inequalities or hemivariational inequalities. Our technique differs completely from those methods; it is based on recent and interesting developments in the theory of equilibrium problems. Equilibrium problems appear frequently in many mathematical models arising from, for instance, physics, engineering, game theory, transportation, economics and network. These mathematical models share an underlying common structure which leads to formulate them in a convenient and unique format. Hence, theoretical results and algorithms obtained for one models can be generally adapted to the others through the common format in a unifying framework. As it has been shown from recent works, equilibrium problems have a great impact and influence in the development of several branches of pure and applied sciences. They include as special cases, optimizations problems, variational inequalities, fixed point, saddle point, Nash equilibria and complementarity problems among others. Therefore, equilibrium problems cover collectively a vast range of applications. Our study for optimal control systems governed by mixed equilibrium problems, for short (OCPMEP), will be devised in three parts: In the first part, we study the existence of solutions for (OCPMEP). Our study will be based on some recent interesting results established for mixed equilibrium problems described by a bifunction which is the sum of a maximal monotone bifunction and a pseudomonotone (quasimonotone) bifunction in Brzis sense. This new approach for studying equilibrium problems has shown in recent works to be of great interests since it improve and unify most of the studies on nonlinear evolutions equations and open new directions of investigations for many applied problems like variational inequalities and hemivariational inequalities among others. In different topics of this first part, we shall study by this new approach the optimal control of problems governed by nonlinear evolution equations, respectively variational and quasi-variational inequalities, a particular interest will be reserved to the control of obstacle problems. We will show that our approach is interesting in the sense that it can improve and unify many studies on those problems. A particular interesting topic in this first part is to study, by this new approach, the optimal control of systems governed by hemivariational inequalities. It is well known that the study of optimal control of hemivariational inequalities is more complicated than those of variational inequalities due to the lack of convexity of energy functionals, and therefore techniques for solving variational inequalities can not be applied to study hemivariational inequalities. Equilibrium problems seams to be a very powerful tool to study hemivariational inequalities and their control, in the sense that the techniques for solving variational inequalities could be used to study hemivariational inequalities trough an equilibrium problem formulation. In a second part of the project, we investigate the sensitivity of optimal solutions for control problems governed by mixed equilibrium problems, i.e. we are interested in the behavior of optimal solutions under perturbations of systems governed by mixed equilibrium problems as well as of perturbations of cost functionals. The third part of the project is consecrated to approximate the optimal control problem governed by mixed equilibrium problems under monotonicity type conditions by a method based on the Lagrangian and penalty function.