The problem of the contact between two plates, one of which has a vertical crack which reaches the outer edge, is considered.
It is assumed that, in the natural state, the plates are a specified distance from one another. The displacements of points on the plates satisfy two constraints of the inequality type. One of these describes the condition of non-penetration between the plates and is specified at internal points of the region, while the other describes the mutual non-penetration of the edges of the crack and is specified on the boundary of the region. The presence of a crack means that, first, the solution of the problem is sought in a region with a non-smooth boundary, and, second, the boundary conditions on the boundary of the region are given in the form of inequalities. It is proved that the equilibrium problem is solvable. Additional smoothness of the solution up to internal points of the crack is established. It is shown that the problem of controlling external loads with an objective functional, characterizing the opening of the crack, is solvable. For cracks of zero opening it is shown that the solution belongs to class C" in the region of the bolmdary for smooth external data. The convergence of the solutions of optimal-control problems as the thickness of the plates approaches zero is analysed.