  We propose an open loop methodology based on sample statistics to solve
chance constrained stochastic optimal control problems with probabilistic
safety guarantees for linear systems where the additive Gaussian noise has
unknown mean and covariance. We consider a joint chance constraint for
time-varying polytopic target sets under assumptions that the disturbance has
been sufficiently sampled. We derive two theorems that allow us to bound the
probability of the state being more than some number of sample standard
deviations away from the sample mean. We use these theorems to reformulate the
chance constraint into a series of convex and linear constraints. Here,
solutions guarantee chance constraint satisfaction. We demonstrate our method
on a satellite rendezvous maneuver and provide comparisons with the scenario
approach.
