Optimal control of PDEs in a complex space setting; application to the Schr\"odinger equation

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.