Linear Theory for Control of Nonlinear Stochastic Systems

We address the role of noise and the issue of efficient computation in stochastic optimal control problems. We consider a class of nonlinear control problems that can be formulated as a path integral and where the noise plays the role of temperature. The path integral displays symmetry breaking and there exists a critical noise value that separates regimes where optimal control yields qualitatively different solutions. The path integral can be computed efficiently by Monte Carlo integration or by a Laplace approximation, and can therefore be used to solve high dimensional stochastic control problems.