  We study the optimal transport problem in the Euclidean space where the cost
function is given by the value function associated with a Linear Quadratic
minimization problem. Under appropriate assumptions, we generalize Brenier's
Theorem proving existence and uniqueness of an optimal transport map. In the
controllable case, we show that the optimal transport map has to be the
gradient of a convex function up to a linear change of coordinates. We give
regularity results and also investigate the non-controllable case.
