  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.
