Convex Stochastic Optimal Control for Nonlinear Systems via Gaussian Mixture Steering

In this paper, we propose a convex optimization-based algorithm for chance-constrained stochastic optimal control of nonlinear stochastic systems. Due to the presence of disturbances and system nonlinearity, the initial state distribution - although assumed to be gaussian - undergoes deformation as the system evolves. To accurately capture this nonlinear propagation of state uncertainty, we employ the Gaussian Mixture Model (GMM) method. The GMM represents the non-gaussian evolution of state uncertainty using the mean and covariance dynamics of a set of sub-gaussian components, thereby approximating the behavior of the original nonlinear stochastic system. We extend the conventional linear covariance analysis (LinCov)-based convex covariance steering approach to the GMM setting, enabling efficient and effective steering of the state distribution toward a desired target distribution while satisfying chance constraints. Finally, the proposed algorithm is applied to solve the low-thrust trajectory optimization problem for Earth-to-Mars transfer under uncertainty. Monte Carlo simulations are conducted to validate the effectiveness and performance of the resulting control policy.