{"paper":{"title":"Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Guillaume Carlier (CEREMADE), Rabah Tahraoui (CEREMADE)","submitted_at":"2009-02-25T08:01:02Z","abstract_excerpt":"This article is devoted to the optimal control of state equations with memory of the form: ?[x(t) = F(x(t),u(t), \\int_0^{+\\infty} A(s) x(t-s) ds), t>0, with initial conditions x(0)=x, x(-s)=z(s), s>0.]Denoting by $y_{x,z,u}$ the solution of the previous Cauchy problem and: \\[v(x,z):=\\inf_{u\\in V} \\{\\int_0^{+\\infty} e^{-\\lambda s} L(y_{x,z,u}(s), u(s))ds \\}\\] where $V$ is a class of admissible controls, we prove that $v$ is the only viscosity solution of an Hamilton-Jacobi-Bellman equation of the form: \\[\\lambda v(x,z)+H(x,z,\\nabla_x v(x,z))+D_z v(x,z), \\dot{z} >=0\\] in the sense of the theory "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0902.4302","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"}