Value Functions and Transversality Conditions for Infinite Horizon Optimal Control Problems

Abstract

This paper investigates infinite horizon optimal control problems with fixed left endpoints with nonconvex, nonsmooth data. We derive the nonsmooth maximum principle and the adjoint inclusion for the value function as necessary conditions for optimality that indicate the relationship between the maximum principle and dynamic programming. The necessary conditions under consideration are extensions of those of [8] to an infinite horizon setting. We then present new sufficiency conditions consistent with the necessary conditions, which are motivated by the useful result by [26] whose sufficiency theorem is valid for nonconvex, nondifferentiable Hamiltonians. The sufficiency theorem presented in this paper employs the strengthened adjoint inclusion of the value function as well as the strengthened maximum principle. If we restrict our result to convex models, it is possible to characterize minimizing processes and provide necessary and sufficient conditions for optimality. In particular, the role of the transversality conditions at infinity is clarified.