Density Steering of Gaussian Mixture Models for Discrete-Time Linear Systems

In this paper, we study the finite-horizon optimal density steering problem for discrete-time stochastic linear dynamical systems for the case when the state distribution can be represented by Gaussian mixture models. First, we revisit the covariance steering problem for Gaussian distributions and derive its optimal control policy in closed-form. Subsequently, we leverage the latter (deterministic) control policy to define a randomized control policy which ensures that the state distribution will remain a Gaussian mixture over the whole time horizon. By leveraging these results, we reduce the Gaussian mixture steering problem to a linear program. We also discuss the problem of steering general distributions using Gaussian mixture approximations. Finally, we present the results of nontrivial numerical experiments which demonstrate that our approach can be applied to general distribution steering problems.