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

TL;DR

This paper studies the value function of infinite horizon variational problems in Banach spaces, providing subdifferential estimates and necessary optimality conditions linking Euler-Lagrange and maximum principle.

Contribution

It offers new subdifferential estimates and a necessary optimality condition for nonconvex infinite horizon problems in infinite-dimensional spaces.

Findings

Upper estimate of the Dini-Hadamard subdifferential

Necessary optimality condition via adjoint inclusion

Connection between Euler-Lagrange and maximum principle

Abstract

We investigate the value function of an infinite horizon variational problem in the infinite-dimensional setting. Firstly, 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. Secondly, 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.