The unilateral contact problem for a Timoshenko plate and a thin elastic obstacle

The paper deals with the problem of contact between a plate and a beam acting as an obstacle to the plate. The plate is described in the framework of Timoshenko theory of plates. It is assumed that no mutual penetration between the plate and the obstacle can occur, and so an appropriate non-penetration condition is used. We study the existence and uniqueness of a solution for the equilibrium problem as well as passages to the limit with respect to the shear rigidity parameter. The accompanying optimal control problem is investigated in which the rigidity parameter acts as a control parameter, cost functional characterizes the difference between known functions and the displacements obtained by equilibrium problem solving.