  In this paper we discuss optimality conditions for abstract optimization
problems over complex spaces. We then apply these results to optimal control
problems with a semigroup structure. As an application we detail the case when
the state equation is the Schr\"{o}dinger one, with pointwise constraints on
the "bilinear" control. We derive first and second order optimality conditions
and address in particular the case that the control enters the state equation
and cost function linearly.
