Optimal control of a quasi-variational obstacle problem

Abstract

We consider an optimal control where the state-control relation is given by a quasi-variational inequality, namely a generalized obstacle problem. We give an existence result for solutions to such a problem. The main tool is a stability result, based on the Mosco-convergence theory, that gives the weak closeness of the control-to-state operator. We end the paper with some examples.

Evolutionary Quasi-Variational Inequalities and the Dynamic Multiclass Network Equilibrium Problem

“…In the case of sequences of set-valued applications, we may formulate Mosco convergence as follows; see [30].…”

“…In fact, is single valued and weakly closed. In addition, following [30], it is possible to find an opportune convex and weakly compact set C such that (C ) ⊂ C . Therefore, we may deduce that has at least one fixed point in C .…”

“…A fundamental role will be played by the concept of Mosco convergence (see [29]), which has been extended to set-valued applications in [30]. Many authors apply set convergence arguments in order to prove convergence or continuity of solutions.…”

“…Our aim is to extend the approach presented in [30] and obtain an existence result under mild and manageable assumptions. A considerable number of results that guarantee the existence of solutions to quasivariational inequalities is based on coercivity conditions on the operator or compactness of the constraint set.…”

Optimal Control of Generalized Quasi-Variational Hemivariational Inequalities and Its Applications

“…3) In recent years the theory of variational and quasi variational inequalities provided us with a convenient mathematical apparatus for studying a wide range of problems arising in numerous applications (see [8,9,11,15]). Optimal control problem for equations, variational and quasi variational inequalities is an expanding and vibrant branch of applied mathematics that has found numerous applications (see [1,[5][6][7][16][17][18]). Although the theory and computational techniques for optimal control for equations and variational inequalities have been studied for quite some time now, it seems that there are still many questions to be answered.…”