On the value function for nonautonomous optimal control problems with infinite horizon

TL;DR

This paper analyzes the value function in nonautonomous infinite horizon optimal control problems, establishing its Lipschitz continuity, its characterization as a Dini solution, and introducing viscosity solutions, with an existence theorem provided.

Contribution

It proves the value function's Lipschitz property, establishes its equivalence as a Dini and viscosity solution of the HJB equation, and provides an existence theorem for such problems.

Findings

Value function is locally Lipschitz continuous.

Value function is a Dini solution of the HJB equation.

Introduces viscosity solutions and proves their equivalence to Dini solutions.

Abstract

In this paper we consider nonautonomous optimal control problems of infinite horizon type, whose control actions are given by $L^1$-functions. We verify that the value function is locally Lipschitz. The equivalence between dynamic programming inequalities and Hamilton-Jacobi-Bellman (HJB) inequalities for proximal sub (super) gradients is proven. Using this result we show that the value function is a Dini solution of the HJB equation. We obtain a verification result for the class of Dini sub-solutions of the HJB equation and also prove a minimax property of the value function with respect to the sets of Dini semi-solutions of the HJB equation. We introduce the concept of viscosity solutions of the HJB equation in infinite horizon and prove the equivalence between this and the concept of Dini solutions. In the appendix we provide an existence theorem.