Optimal control of plasticity systems

Abstract

The thesis is concerned with an optimal control problem with an evolution variational inequality (EVI), involving a maximal monotone operator, as a constraint. This abstract setting can be applied to various cases of elasto plasticity. It is shown that elasto and homogenized plasticity, elasto plasticity with an inertia term and also perfect plasticity can be transformed into a certain EVI. Such an EVI is analyzed in the context of optimal control. Then optimal control problems for each of the mentioned plasticity systems are considered, where the findings in the abstract case are either directly applied (elasto and homogenized plasticity and partly elasto plasticity with an inertia term) or at least partially used (perfect plasticity). In each case, the existence of a global solution to the corresponding optimal control problem is shown. The state equations, and thus the control problems, are then regularized and results regarding approximation of global minimizers by global minimizers of the regularized problems are proved. For the optimal control problem, constrained by the abstract EVI, first and second order optimality conditions are derived, whereas only first order conditions are investigated for optimal control problems governed by plasticity systems. A certain difficulty arises in the case of perfect plasticity due to the non-uniqueness of the displacement and the fact that it is only of bounded deformation. This is the main reason for restricting the optimal control problem to the stress as the only state variable when it comes to optimality conditions. Moreover, for this case numerical experiments are presented. The finite element toolbox FEniCS was used to solve the involved partial differential equations.