Optimal location problem for composite bodies with separate and joined rigid inclusions

Nonlinear mathematical models describing an equilibrium state of composite bodies which may come into contact with a fixed non-deformable obstacle are investigated. We suppose that the composite bodies consist of an elastic matrix and one or two built-in volume (bulk) rigid inclusions. These inclusions have a rectangular shape and one of them can vary its location along a straight line. Considering a location parameter as a control parameter, we formulate an optimal control problem with a cost functional specified by an arbitrary continuous functional on the solution space. Assuming that the location parameter varies in a given closed interval, the solvability of the optimal control problem is established. Furthermore, it is shown that the equilibrium problem for the composite body with joined two inclusions can be considered as a limiting problem for the family of equilibrium problems for bodies with two separate inclusions.