Optimal Transportation for Generalized Lagrangian


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


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Springer Journals Database

Total: 16 articles

  • [1] Luca Granieri, On action minimizing measures for the Monge-Kantorovich problem, Nonlinear Differential Equations and Applications NoDEA, 2007 (1)
  • [2] Aldo Pratelli, Equivalence between some definitions for the optimal mass transport problem and for the transport density on manifolds, Annali di Matematica Pura ed Applicata, 2005 (2)
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