  Topology optimization is concerned with the identification of optimal shapes
of deformable bodies with respect to given target functionals. The focus of
this paper is on a topology optimization problem for a time-evolving
elastoplastic medium under kinematic hardening. We adopt a phase-field approach
and argue by subsequent approximations, first by discretizing time and then by
regularizing the flow rule. Existence of optimal shapes is proved both at the
time-discrete and time-continous level, independently of the regularization.
First order optimality conditions are firstly obtained in the regularized
time-discrete setting and then proved to pass to the nonregularized
time-continuous limit. The phase-field approximation is shown to pass to its
sharp-interface limit via an evolutive variational convergence argument.
