Differential inclusions with unbounded right-hand side and necessary optimality conditions

We study the properties of the trajectories of a differential inclusion with unbounded measurable–pseudo-Lipschitz right-hand side that takes values in a separable Banach space and consider the problem of minimizing a functional over the set of trajectories of such a differential inclusion on an interval. We obtain necessary optimality conditions in the form of Euler–Lagrange differential inclusions for a problem with free right end.