The maximum principle for nonlinear nonconvex systems with set targets

Using Ekeland\´s variational principle ([2]) we obtain a version of Pontryagin\´s maximum principle for the general input-output systems in infinite dimensional systems defined in ??2 below, where trajectories are supposed to hit a "large" set (see the "fat cone condition" in ??5) with optimal value of a general cost functional. Unlike infinite dimensional problems where the target is a point rather than a set ([9]), ([10]), no controllability assumptions on the linearized systems are necessary.