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We consider the problem of steering an initial probability density for the state vector of a linear system to a final one, in finite time, using minimum energy control. In the case where the dynamics correspond to an integrator (ẋ(t) = u(t)) this amounts to a Monge-Kantorovich Optimal Mass Transport (OMT) problem. In general, we show that the problem can again be reduced to solving an OMT problem and that it has a unique solution. In parallel, we study the optimal steering of the state-densityarXiv:1502.01265v1 fatcat:ooay2ofdejeh7gubmnwnsfnule