Sufficiency conditions for existence of an optimal feedback control in stochastic mechanics

In the framework of stochastic mechanics, the following problem is considered : in a set of admissible feedback controls v, with range in En, find one minimizing the expectation ESX{∝ST L(t,ξ(t), v(t,ξ(t)))dt+WT(ξ(T))} for all (s,x) ∈ [0,T)⊗En, where L(t,x,v) = (1/2)mv2-U(t,x) is the classical action integrand and ξ is a n-dimensional diffusion process (in the weak sense [17]) with drift v and diffusion coefficient D = constant > 0. WT and U are given real functions. Sufficiency conditions for existence of such an optimal feedback control are given.