Value function, relaxation, and transversality conditions in infinite horizon optimal control

We investigate the value function V:R+×Rn→R+∪{+∞} of the infinite horizon problem in optimal control for a general-not necessarily discounted-running cost and provide sufficient conditions for its lower semicontinuity, continuity, and local Lipschitz regularity. Then we use the continuity of V(t,⋅) to prove a relaxation theorem and to write the first order necessary optimality conditions in the form of a, possibly abnormal, maximum principle whose transversality condition uses limiting/horizontal supergradients of V(0,⋅) at the initial point. When V(0,⋅) is merely lower semicontinuous, then for a dense subset of initial conditions we obtain a normal maximum principle augmented by sensitivity relations involving the Fréchet subdifferentials of V(t,⋅). Finally, when V is locally Lipschitz, we prove a normal maximum principle together with sensitivity relations involving generalized gradients of V for arbitrary initial conditions. Such relations simplify drastically the investigation of the limiting behavior at infinity of the adjoint state.