Optimal control of an abstract evolution variational inequality with application to homogenized plasticity

Abstract

The paper is concerned with an optimal control problem governed by a state equation in form of a generalized abstract operator differential equation involving a maximal monotone operator. The state equation is uniquely solvable, but the associated solution operator is in general not G\^ateaux-differentiable. In order to derive optimality conditions, we therefore regularize the state equation and its solution operator, respectively, by means of a (smoothed) Yosida approximation. We show convergence of global minimizers for regularization parameter tending to zero and derive necessary and sufficient optimality conditions for the regularized problems. The paper ends with an application of the abstract theory to optimal control of homogenized quasi-static elastoplasticity.

Randomised one-step time integration methods for deterministic operator differential equations

“…Operator differential equations have been applied in peridynamics and elastic materials, e.g. [13,25]. The purpose of this paper is to analyse the error of randomised time integration methods for solving such initial value problems.…”