Necessary and sufficient conditions in optimal control of mean-field stochastic differential equations with infinite horizon

Abstract

Abstract We consider an infinite horizon optimal control of a system where the dynamics evolve according to a mean-field stochastic differential equation and the cost functional is also of mean-field type. These are systems where the coefficients depend not only on the state variable, but also on its marginal distribution via some linear functional. Under some concavity assumptions on the coefficients as well as on the Hamiltonian, we are able to prove a verification theorem, which gives a sufficient condition for optimality for a given admissible control. In the absence of concavity, we prove a necessary condition for optimality in the form of a weak Pontryagin maximum principle, given in terms of stationarity of the Hamiltonian.