Optimal Transportation for Generalized Lagrangian


Ji LI,Jianlu ZHANG.Optimal Transportation for Generalized Lagrangian[J].Chinese Annals of Mathematics B,2017,38(3):857~868
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Ji LI; Jianlu ZHANG
Abstract: This paper deals with the optimal transportation for generalized Lagrangian $L=L(x, u,t)$, and considers the following cost function: $$c(x, y)=\inf_{\substack{x(0)=x\\x(1)=y\\u\in\mathcal{U}}}\int_0^1L(x(s), u(x(s),s), s)\rmd s,$$ where $\mathcal{U}$ is a control set, and $x$ satisfies the ordinary equation $$\dot{x}(s)=f(x(s),u(x(s),s)).$$ It is proved that under the condition that the initial measure $\mu_0$ 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: \begin{align*} \begin{cases} V_t(t, x)+\sup\limits_{\substack{u\in\mathcal{U}}}\langle V_x(t, x), f(x, u(x(t), t),t)-L(x(t), u(x(t), t),t)\rangle=0, \V(0,x)=\phi_0(x). \end{cases} \end{align*}


Optimal control, Hamilton-Jacobi equation, Characteristic curve, Viscosity solution, Optimal transportation, Kantorovich pair, Initial transport measure


35D40, 35F21, 37J50
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