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Modelling and control in pseudoplate problem with discontinuous thickness. (English) Zbl 1212.49009
Summary: 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 \(\Gamma\)-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.
MSC:
49J40 Variational inequalities
49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
70Q05 Control of mechanical systems
49M15 Newton-type methods
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