  We propose a novel Galerkin discretization scheme for stochastic optimal
control problems on an indefinite time horizon. The control problems are
linear-quadratic in the controls, but possibly nonlinear in the state
variables, and the discretization is based on the fact that problems of this
kind can be transformed into linear boundary value problems by a logarithmic
transformation. We show that the discretized linear problem is dual to a Markov
decision problem, the precise form of which depends on the chosen Galerkin
basis. We prove a strong error bound in $L^{2}$ for the general scheme and
discuss two special cases: a variant of the known Markov chain approximation
obtained from a basis of characteristic functions of a box discretization, and
a sparse approximation that uses the basis of committor functions of metastable
sets of the dynamics; the latter is particularly suited for high-dimensional
systems, e.g., control problems in molecular dynamics. We illustrate the method
with several numerical examples, one being the optimal control of Alanine
dipeptide to its helical conformation.
