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Several optimal control problems with the same state problem—a variational inequality with a monotone operator—are considered. The inequality represents bending of an elastic, nonhomogeneous, anisotropic Kirchhoff plate resting on some unilateral elasto-rigid foundation and point supports. Both the thickness of the plate and the coefficient of the unilateral elastic foundation play the role of design variables. Cost functionals include the work of external forces (compliance)), total reaction forces of the foundation or the weight of the plate. The solvability of all the problems is proved. Moreover, approximate methods for the optimal control and weight minimization problems are proposed, making use of finite elements. The design variables are approximated by piecewise affine functions. The solvability of the approximate problems is proved and some convergence analysis is presented.