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An equilibrium problem for an elastic transversely isotropic Timoshenko's plate with a curvilinear crack is considered. On the crack faces, the nonpenetration conditions, which have the form of inequalities (conditions of the Signorini type), are given. By using a sufficiently smooth perturbation determined in the middle plate plane, the variation of plate geometry is specified. The formula of the derivative of the plate energy functional with respect to the perturbation parameter is deduced.

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The mathematical models describing equilibrium of cracked elastic plates with rigid thin stiffeners on the outer boundary are studied. On the crack faces the boundary conditions are specified in the form of inequalities which describe the mutual nonpenetration of the crack faces. We analyze the dependence of solutions on the length of the thin rigid stiffener reinforcing the cracked Kirchhoff-Love plate on the outer edge. The existence is proved of the solution to the optimal control problem. For this problem the cost functional is defined by an arbitrary continuous functional, while the length parameter of the thin rigid stiffener is chosen as a control function.