  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.
