Optimal transportation for generalized Lagrangian

Abstract

This paper deals with the optimal transportation for generalized Lagrangian L = L(x, u, t), and considers the following cost function: $$c\left( {x,y} \right) = \mathop {\inf }\limits_{\begin{array}{*{20}{c}} {x\left( 0 \right) = x} {x\left( 1 \right) = y} {u \in U} \end{array}} \int_0^1 {L\left( {x\left( s \right),u\left( {x\left( s \right),s} \right),s} \right)ds} ,$$ where U is a control set, and x satisfies the ordinary equation $$\dot x\left( s \right) = f\left( {x\left( s \right),u\left( {x\left( s \right),s} \right)} \right).$$ It is proved that under the condition that the initial measure μ 0 is absolutely continuous w.r.t. the Lebesgue measure, the Monge problem has a solution, and the optimal transport map just walks along the characteristic curves of the corresponding Hamilton-Jacobi equation: $$\left\{ {_{V\left( {0,x} \right) = {\phi _0}\left( x \right).}^{{V_t}\left( {t,x} \right) + \mathop {\sup }\limits_{u \in U} \left\langle {{V_x}\left( {t,x} \right),f\left( {x,u\left( {x\left( t \right),t} \right),t} \right) - L\left( {x\left( t \right),u\left( {x\left( t \right),t} \right),t} \right)} \right\rangle = 0,}} \right.$$