  The purpose of this work is the development of space-time discretization
schemes for phase-field optimal control problems. First, a time discretization
of the forward problem is derived using a discontinuous Galerkin formulation.
Here, a challenge is to include regularization termsand the crack
irreversibility constraint. The optimal control setting is formulated by means
of the Lagrangian approach from which the primal part, adjoint, tangent and
adjoint Hessian are derived. Herein the overall Newton algorithm is based on a
reduced approach by eliminating the state constraint. From the low-order
discontinuous Galerkin discretization, adjoint time-stepping schemes are
finally obtained. Our algorithmic developments are substantiated and
illustrated with some numerical experiments.
