Optimal control of loads for an equilibrium problem describing a point contact of an elastic body with a sharp-shaped stiffener

We consider a non-classical 2D mathematical equilibrium model describing a possible mechanical contact of a composite structure having a sharp-shaped edge. Nonlinearity of the model is caused by conditions of inequality type for a corresponding variational problem. The main feature of this basic model consists in its geometrical configuration, which determines non-convexity of the problem under consideration. Namely, the composite in its reference state touches a wedge-shaped rigid obstacle at a single contact point. On the basis of this model, we consider an induced family of problems depending on different functions of external loads. For a given set of functions, describing admissible external loads, we formulate an optimal control problem, where functions of external loads serve as a control. A cost functional is given with the help of an arbitrary weakly upper semicontinuous functional defined on the Sobolev space of feasible solutions. The solvability of the optimal control problem is proved. Furthermore, for a sequence of solutions corresponding to a maximizing sequence, a strong convergence in the corresponding Sobolev space is proven.