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Mass transportation with LQ cost functions. (English) Zbl 1207.49040
Summary: 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.

MSC:
49N10 Linear-quadratic optimal control problems
49Q20 Variational problems in a geometric measure-theoretic setting
49N60 Regularity of solutions in optimal control
49J27 Existence theories for problems in abstract spaces
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