In this paper we study optimal control problems for systems governed by nonlinear evolution equations. First we develop an
existence theory for systems with a priori feedback using the reduction technique and a convexity-type hypothesis involving
property Q. In doing this we also establish the nonemptiness of the set of admissible state-control pairs, by solving a nonlinear
evolution inclusion. Then we obtain necessary conditions for optimality for a class of problems with terminal cost criterion
and initial condition which is not a priori given but is only required to belong to a given set (systems with insufficient
data in the terminology of Lions).