Value Functions and Optimality Conditions for Nonconvex Variational Problems with an Infinite Horizon in Banach Spaces

We investigate the value function of an infinite horizon variational problem in the infinite-dimensional setting. First, we provide an upper estimate of its Dini-Hadamard subdifferential in terms of the Clarke subdifferential of the Lipschitz continuous integrand and the Clarke normal cone to the graph of the set-valued mapping describing dynamics. Second, we derive a necessary condition for optimality in the form of an adjoint inclusion that grasps a connection between the Euler-Lagrange condition and the maximum principle. The main results are applied to the derivation of the necessary optimality condition of the spatial Ramsey growth model.