{"paper":{"title":"Optimal Transportation for Generalized Lagrangian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.DS","authors_text":"Jianlu Zhang, Ji Li","submitted_at":"2013-12-02T06:59:35Z","abstract_excerpt":"In this paper, we study the optimal transportation for generalized Lagrangian $L=L(x, u,t)$, and consider the cost function as following: $$c(x, y)=\\inf_{\\substack{x(0)=x\\\\x(1)=y\\\\u\\in\\mathcal{U}}}\\int_0^1L(x(s), u(x(s),s), s)ds.$$ Where $\\mathcal{U}$ is a control set, and $x$ satisfies the following ordinary equation: $$\\dot{x}(s)=f(x(s),u(x(s),s)).$$ We prove 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 correspond"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.0345","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}