Moving least-squares approximations for linearly-solvable stochastic optimal control problems

Nonlinear stochastic optimal control problems are fundamental in control theory. A general class of such problems can be reduced to computing the principal eigenfunction of a linear operator. Here, we describe a new method for finding this eigenfunction using a moving least-squares function approximation. We use efficient iterative solvers that do not require matrix factorization, thereby allowing us to handle large numbers of basis functions. The bases are evaluated at collocation states that change over iterations of the algorithm, so as to provide higher resolution at the regions of state space that are visited more often. The shape of the bases is automatically defined given the collocation states, in a way that avoids gaps in the coverage. Numerical results on test problems are provided.