  In this paper, we propose an alternative technique to dynamic programming for
solving stochastic control problems. We consider a weak formulation that is
written as an optimization (minimization) problem on the space of
probabilities. We then propose a regularized version of this problem obtained
by splitting the minimization variables and penalizing the entropy between the
two probabilities to be optimized. We show that the regularized problem
provides a good approximation of the original problem when the weight of the
entropy regularization term is large enough. Moreover, the regularized problem
has the advantage of giving rise to optimization problems that are easy to
solve in each of the two optimization variables when the other is fixed. We
take advantage of this property to propose an alternating optimization
algorithm whose convergence to the infimum of the regularized problem is shown.
The relevance of this approach is illustrated by solving a high-dimensional
stochastic control problem aimed at controlling consumption in electrical
systems.
