Discrete-time stochastic optimal control via occupation measures and moment relaxations

We consider discrete-time nonlinear stochastic optimal control problems for which all the data are polynomial. For this class of problems we derive a hierarchy of linear matrix inequality relaxations which is based on occupation measures and moment theory. The dual of the convex problem obtained, which can be interpreted in terms of the Bellman equation, is then used to derive an almost optimal control law. A numerical example illustrates the effectiveness of the approach.