On the value function for control problems with infinite horizon
G. N. Silva | Baumeister, J. | Leitao , A.
dynamic programming | optimal control | optimality conditions
In this paper we consider optimal control problems of infinite horizon type, whose control actions are given by $L^1$-functions. A characteristic of this type of problems is the utilization of a discount factor in the objective function. We provide an existence theorem, verify that the value function is locally Lipschitz and obtain necessary and sufficient conditions of optimality for the control problem in terms of upper Dini solutions of the Hamilton-Jakobi-Bellman inequality (equation). For a special class of control problems we prove that the value function is a viscosity solution of the Hamilton-Jakobi-Bellman (HJB) equation with certain decay condition. Finally, a ``real life'' example is provided to show the strength of our existence result.