Singular perturbations in optimal control problem with application to nonlinear structural analysis (Q1363450)

The author considers an optimal control problem for a singularly perturbed elliptic variational inequality. The control appears in coefficients, in the right-hand side, and in the convex set of admissible spaces. The existence of an optimal control and the convergence to a solution of the limit (unperturbed) problem is proven. Some approximations are proposed and studied, as well. The theory is applied to an equilibrium of an elastic membrane perturbed by a small elasto-plastic bending rigidity and resting above a rigid inner obstacle. The role of the control (design variable) is played here by a transversal loading and the obstacle function. Employing standard finite elements, approximate solution is proposed for the above problem of a membrane, but the obstacle is restricted to the boundary of the domain only.