| 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*} |
