Value Function of an Infinite Dimensional Infinite Horizon Problem

We investigate the value function of an infinite horizon problem in the setting of an infinite-dimensional differential inclusion. In particular, we provide an upper estimate of its Gateaux subdifferential in terms of the Clarke subdifferential of the integrand and the Clarke normal cone to the graph of the set-valued dynamics. We also derive a necessary optimality condition in the form of an Euler-Lagrange inclusion, the maximum principle and a sensitivity relation.