Stochastic Optimal Control via Measure Relaxations

The optimal control problem of stochastic systems is commonly solved via robust [2, 21] or scenario-based [7, 19, 17] optimization methods, which are both challenging to scale to long optimization horizons due to their open-loop nature. Dynamic programming formulations [4], while applicable to stochastic systems, typically involve nonconvex optimization problems and do not support specifying the terminal distribution. Polynomial optimization has been proposed for deterministic nonlinear [11] and hybrid systems [16]. We extend the method to stochastic systems using a weak formulation of the Fokker-Planck equation. As a cost function, we propose to use the Christoffel polynomial, which can be estimated from data.