<p>A new mathematical model describing a contact of a Timoshenko plate with an inclined obstacle is studied. We assume that in the initial state, a given part of the boundary of the plate bottom surface touches a non-deformable obstacle. Furthermore, we assume that the obstacle shape is composed of rectilinear segments, and each segment (generatrix) forms an angle with the plane of the plate corresponding to its front surface. A boundary condition of Signorini’s type is imposed in the form of an inequality, which depends on a slope coefficient or the obstacle inclination angle. A corresponding variational problem is formulated as a minimization of an energy functional over a convex set subject to a nonlinear non-penetration condition of inequality type. It is shown that the problem has a unique solution. Assuming that angles of inclination can vary with respect to a positive parameter, we consider a family of equilibrium problems. The parameter varies in a given closed interval and determines the variation of the angles between generatrices and the plate front surface. Taking the mentioned parameter as a control, we formulate an optimal control problem for a cost functional that characterizes the deviation of the solution from the specified displacements. We prove the existence of a solution for the optimal control problem. The continuous dependence of solutions with respect to the parameter is also established.</p>