{
  "id": "2408.02856",
  "version": "v1",
  "published": "2024-08-05T22:49:22.000Z",
  "updated": "2024-08-05T22:49:22.000Z",
  "title": "Discrete approximations and optimality conditions for integro-differential inclusions",
  "authors": [
    "Abderrahim Bouach",
    "Tahar Haddad",
    "Boris S. Mordukhovich"
  ],
  "comment": "28 pages",
  "categories": [
    "math.OC"
  ],
  "abstract": "This paper addresses a new class of generalized Bolza problems governed by nonconvex integro-differential inclusions with endpoint constraints on trajectories, where the integral terms are given in the general (with time-dependent integrands in the dynamics) Volterra form. We pursue here a threefold goal. First we construct well-posed approximations of continuous-time integro-differential systems by their discrete-time counterparts with showing that any feasible solution to the original system can be strongly approximated in the $W^{1,2}$-norm topology by piecewise-linear extensions of feasible discrete trajectories. This allows us to verify in turn the strong convergence of discrete optimal solutions to a prescribed local minimizer for the original problem. Facing intrinsic nonsmoothness of original integro-differential problem and its discrete approximations, we employ appropriate tools of generalized differentiation in variational analysis to derive necessary optimality conditions for discrete-time problems (which is our second goal) and finally accomplish our third goal to obtain necessary conditions for the original continuous-time problems by passing to the limit from discrete approximations. In this way we establish, in particular, a novel necessary optimality condition of the Volterra type, which is the crucial result for dynamic optimization of integro-differential inclusions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-05T22:49:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49K24",
      "49K22",
      "49J53",
      "94C99"
    ],
    "keywords": [
      "discrete approximations",
      "novel necessary optimality condition",
      "derive necessary optimality conditions",
      "continuous-time integro-differential systems",
      "employ appropriate tools"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}