  In this paper we discuss the optimal control of a quasilinear parabolic state
equation. Its form is leaned on the kind of problems arising for example when
controlling the anisotropic Allen-Cahn equation as a model for crystal growth.
Motivated by this application we consider the state equation as a result of a
gradient flow of an energy functional. The quasilinear term is strongly
monotone and obeys a certain growth condition and the lower order term is
non-monotone. The state equation is discretized implicitly in time with
piecewise constant functions. The existence of the control-to-state operator
and its Lipschitz-continuity is shown for the time discretized as well as for
the time continuous problem. Latter is based on the convergence proof of the
discretized solutions. Finally we present for both the existence of global
minimizers. Also convergence of a subsequence of time discrete optimal controls
to a global minimizer of the time continuous problem can be shown. Our results
hold in arbitrary space dimensions.
