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control of variational inequalities; optimal design; minimization; pseudoplate with obstacles; cost functional; thickness; $G$-convergence; coercive variational inequality; approximate optimization problem; finite element
This paper concerns an obstacle control problem for an elastic (homogeneous) and isotropic) pseudoplate. The state problem is modelled by a coercive variational inequality, where control variable enters the coefficients of the linear operator. Here, the role of control variable is played by the thickness of the pseudoplate which need not belong to the set of continuous functions. Since in general problems of control in coefficients have no optimal solution, a class of the extended optimal control is introduced. Taking into account the results of $G$-convergence theory, we prove the existence of an optimal solution of extended control problem. Moreover, approximate optimization problem is introduced, making use of the finite element method. The solvability of the approximate problem is proved on the basis of a general theorem. When the mesh size tends to zero, a subsequence of any sequence of approximate solutions converges uniformly to a solution of the continuous problem.
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