|Title||SYSTEMS GOVERNED BY MEAN-FIELD STOCHASTIC EVOLUTION EQUATIONS ON HILBERT SPACES AND THEIR OPTIMAL CONTROL|
|Publication Type||Journal Article|
|Year of Publication||2016|
|Journal||Dynamic Systems and Applications|
|AMS Subject Classification||Existence of Optimal Controls, Hilbert Spaces, McKean-Vlasov mean-field Stochastic Differential Equation, Necessary conditions of optimality., Relaxed Controls|
In this paper we consider a general class of controlled McKean-Vlasov mean-field stochastic evolution equations on Hilbert spaces. We prove existence, uniqueness and regularity properties of mild solutions of these equations. Relaxed controls, covering regular controls, adapted to a current of sub-sigma algebras generated by observable processes and taking values from a Polish space, are used. An appropriate metric topology, based on weak star convergence, is introduced. We prove continuous dependence of solutions on controls with respect to this topology. These results are then used to prove existence of optimal controls for Bolza problem. Then we develop the necessary conditions of optimality using semi-martingale representation theory and show that the adjoint processes arising from the necessary conditions can be constructed from the mild solution of certain backward stochastic mean field evolution equation (BSMEE). The paper is concluded with some applications to mean-field linear quadratic regulator problems.