Optimal control of the crack angle in the equilibrium problem for a Timoshenko plate with elastic inclusion

Mathematical modeling and research of problems on the deformation of inhomogeneous bodies containing cracks along elastic inclusions involves setting the conjugation conditions at the interface between different materials. Difficulties are associated with the possibility of large stress values appearing near the inclusions. Determining the inhomogeneous bodies with the most optimal parameters is one of the most popular areas of theoretical and experimental research. In this paper, we study the problem of optimal control of the angle of the crack inclination to the median plane in the equilibrium problem for an elastic Timoshenko plate containing an oblique crack at the boundary of an elastic inclusion. The complexity of the problem is due to the fact that the non-penetration condition on the crack faces is given in the form of inequality. The quality functional characterizes the deviation from the specified displacements. We prove solvability of the optimal control problem and establish continuous dependence of the solutions on the value of the crack inclination angle.