Optimal Transportation for Generalized Lagrangian

2013/12/02 by Li Ji, Li, Ji, Jianlu Zhang +1
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Optimization and Control (math.OC) #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1312.0345

openalex publication_date 2013/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the optimal transportation for generalized Lagrangian L=L(x, u,t), and consider the cost function as following: c(x, y)=inf_\substackx(0)=x
x(1)=y
u\inU∫₀¹L(x(s), u(x(s),s), s)ds. Where U is a control set, and x satisfies the following ordinary equation: x(s)=f(x(s),u(x(s),s)). We prove 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: \begincases V_t(t, x)+sup_\substacku\inU=0.
V(0,x)=ϕ₀(x) \endcases

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