Optimal control of two variational inequalities arising in solid mechanics

Abstract

The optimal control of the static model of infinitesimal elastoplasticity with linear kinematic hardening and the optimal control of Signorini’s problem are considered. We are thus concerned with two optimal control problems governed by an elliptic variational inequality of the first kind. Solution operators associated with variational inequalities are in general not Gâteaux differentiable. The same applies in our case so that the classical optimal control theory cannot be employed. However, under additional regularity assumptions the elastoplasticity operator is shown to be Bouligand differentiable. This enables us to establish second-order sufficient optimality conditions for the optimal control of static elastoplasticity based on a Taylor expansion of a particularly chosen Lagrange function. Moreover, by extending Riesz’ representation theorem to positive functionals f∈H 1 (Ω)' and adapting the ideas of Mignot ’76 we prove that the solution operator of the Signorini problem is directionally differentiable. Subsequently we derive first-order necessary optimality conditions of strong stationary type for the optimal control of Signorini’s problem. The results on the optimal control of static elastoplasticity have been published in large part in the journal ESAIM: Control, Optimisation and Calculus of Variations 21(1), pp. 271-300, 2015.