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Control in obstacle-pseudoplate problems with friction on the boundary. Approximate optimal design and worst scenario problems. (English) Zbl 1053.74032
The authors present an approximation of the worst scenarion problem formulated in their previous paper [Appl. Math. 28, No. 4, 407–426 (2001; Zbl 1042.49036)]. Approximate optimal design problem is solved in the finite element space of piecewise linear functions. The approximate state problem has the form of finite-dimensional variational inequality $$a(H_h;u_h(e_h),v_h-u_h(e_h))+b_h(Z_h;u_h(e_h),v_h-u_h(e_h))$$ $$+ \Phi_h(e_h)(v_h)-\Phi_h(e_h)(u_h)(e_h) \geq [p,v_h-u_h(e_h)]_h-2\omega\langle H_h,v_h-u_h(e_h)\rangle_0$$ $$\text{for all}\;v_h\in K_h(H_h)$$, with the approximate thickness $$H_h$$ of the pseudoplate and the convex set $$K_h(H_h)$$. The approximate optimal design problems have the form $$\;e_J^{*h}=\text{argmin}_{e_h\in U_{ad}^h} \mathcal L_J^h(e_h,u_h(e_h)),$$ where $$\mathcal T_h$$ is a fixed triangulation. The existence theorems for the approximate state and optimal design problems are proved. When the mesh size tends to zero, a subsequence of any sequence of approximate solutions converges uniformly to the solution of the continuous problem.

##### MSC:
 74M05 Control, switches and devices (“smart materials”) in solid mechanics 74S05 Finite element methods applied to problems in solid mechanics 74K20 Plates 74P05 Compliance or weight optimization in solid mechanics
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