Optimal control of differential quasivariational-hemivariational inequalities with applications

In this paper, we study a new class of differential quasivariational-hemivariational inequalities of the elliptic type. The problem consists of a system coupling the Cauchy problem for an ordinary differential equation with the variational-hemivariational inequalities, unilateral constraints, and history-dependent operators. First, based on the Minty formulation and the continuity of the solution map of a parametrized quasivariational-hemivariational inequality, and a fixed point theorem for a history-dependent operator, we prove a result on the well-posedness. Next, we examine optimal control problems for differential quasivariational-hemivariational inequalities, including a time-optimal control problem and a maximum stay control problem, for which we show the existence of solutions. In all the optimal control problems, the system is controlled through a distributed and boundary control, a control in initial conditions, and a control that appears in history-dependent operators. Finally, we illustrate the results by considering a nonlinear controlled system for a time-dependent elliptic equation with unilateral constraints.