{
  "id": "0902.4302",
  "version": "v1",
  "published": "2009-02-25T08:01:02.000Z",
  "updated": "2009-02-25T08:01:02.000Z",
  "title": "Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory",
  "authors": [
    "Guillaume Carlier",
    "Rabah Tahraoui"
  ],
  "journal": "ESAIM COCV 16, 3 (2010) 744-763",
  "categories": [
    "math.OC"
  ],
  "abstract": "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 of viscosity solutions in infinite-dimensions of M. Crandall and P.-L. Lions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-02-25T08:01:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hamilton-jacobi-bellman equation",
      "state equation",
      "optimal control",
      "viscosity solution",
      "initial conditions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0902.4302C"
    }
  }
}