Optimal Transport of Linear Systems over Equilibrium Measures

We consider the optimal transport problem over convex costs arising from optimal control of linear time-invariant(LTI) systems when the initial and target measures are assumed to be supported on the set of equilibrium points of the LTI system. In this case, the probability measures are singular with respect to the Lebesgue measure, thus not considered in previous results on optimal transport of linear systems. This problem is motivated by applications, such as robotics, where the initial and target configurations of robots, represented by measures, are in equilibrium or stationary. Despite the singular nature of the measures, for many cases of practical interest, we show that the Monge problem has a solution by applying classical optimal transport results. Moreover, the problem is computationally tractable even if the state space of the LTI system is moderately high in dimension, provided the equilibrium set lives in a low dimensional space. In fact, for an important subclass of linear quadratic problems, such as control of the double integrator with linear quadratic cost, the optimal transport map happens to coincide with that of the Euclidean cost. We demonstrate our results by computing the optimal transport map for the minimum energy cost for a two dimensional double integrator, despite the fact that the state space is four dimensional due to position and velocity variables.