Ji LI;Jianlu ZHANG;Department of Mathematics, Nanjing University;
Department of Mathematics, Nanjing University;
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 x(0)=x x(1)=y u∈U∫_0~1 L(x(s), u(x(s), s), s)ds, where U is a control set, and x satisfies the ordinary equation x(s) = f(x(s), u(x(s), s)).It is proved that under the condition that the initial measure μ0 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:V_t(t, x) + sup u∈U<V_x(t, x), f(x, u(x(t), t), t)-L(x(t), u(x(t), t), t)> = 0,V(0, x) = Φ0(x).
Optimal control;;Hamilton-Jacobi equation;;Characteristic curve;;Viscosity solution;;Optimal transportation;;Kantorovich pair;;Initial transport measure
To explore the background and basis of the node document
Documents that have the similar content to the node document