Optimal control of a thin rigid inclusion intersecting the boundary of an elastic body

The problem of the optimal control of a long thin rigid inclusion in an elastic body intersecting the external boundary is investigated. It is assumed that the inclusion delaminates, forming a crack between it and the body. Non-linear boundary conditions are specified on the edge of the crack that exclude the mutual penetration of the opposite edges. The solvability of the optimal control problem, in which the performance functional characterizes the displacement of the points of the rigid inclusion and the length of the inclusion located within the elastic body serves as the control function, is proved.