General Distribution Steering: A Sub-Optimal Solution by Convex Optimization

Abstract

General distribution steering is intrinsically an infinite-dimensional problem, when the continuous distributions to steer are arbitrary. We put forward a moment representation of the primal system for control in [42]. However, the system trajectory was a predetermined one without optimization towards a design criterion, which doesn't always ensure a most satisfactory solution. In this paper, we propose an optimization approach to the general distribution steering problem of the first-order discrete-time linear system, i.e., an optimal control law for the corresponding moment system. The domain of all feasible control inputs is non-convex and has a complex topology. We obtain a subset of it by minimizing a weighted sum of squared integral distances alongside the system trajectory. The feasible domain is then proved convex, and the optimal control problem can be treated as a convex optimization or by exhaustive search, based on the type of the cost function. Algorithms of steering for continuous and discrete distributions are then put forward respectively, by adopting a realization scheme of control inputs. We also provide an explicit advantage of our proposed algorithm by truncated power moments to the prevailing Gaussian Mixture Models. Experiments on different types of cost functions are given to validate the performance of our proposed algorithm. Since the moment system is a dimension-reduced counterpart of the primal system, we call this solution a sub-optimal one to the primal general distribution steering problem.