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<p>The aim of this work is to deal with a discontinuous Hamilton-Jacobi equation in the whole euclidian <i>N</i>-dimensional space, associated to a possibly unbounded optimal control problem. Here, the discontinuities are located on a hyperplane and the typical questions we address concern the existence and uniqueness of solutions, and of course the definition itself of solution. We consider viscosity solutions in the sense of Ishii. The convex Hamiltonians are associated to a control problem with specific cost and dynamics given on each side of the hyperplane. We assume that those are Lipschitz continuous but the main difficulty we deal with is that they are potentially unbounded, as well as the control spaces. This allows to treat superlinear Hamiltonians to which we extend the results in [<CitationRef AdditionalCitationIDS="CR4 CR5" CitationID="CR3">3</CitationRef>–<CitationRef CitationID="CR6">6</CitationRef>]. Moreover, we also build a whole family of value functions, which are still solutions in the sense of Ishii and connect continuously the minimal solution to the maximal one.</p>

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Unbounded Hamilton-Jacobi-Bellman Equations with one co-dimensional discontinuities

  • 期刊论文

  • 发表日期：

    2025年10月30日

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  • Emmanuel Chasseigne,
  • Robson Carlos Reis,
  • Silvia Sastre-Gómez

摘要

The aim of this work is to deal with a discontinuous Hamilton-Jacobi equation in the whole euclidian N-dimensional space, associated to a possibly unbounded optimal control problem. Here, the discontinuities are located on a hyperplane and the typical questions we address concern the existence and uniqueness of solutions, and of course the definition itself of solution. We consider viscosity solutions in the sense of Ishii. The convex Hamiltonians are associated to a control problem with specific cost and dynamics given on each side of the hyperplane. We assume that those are Lipschitz continuous but the main difficulty we deal with is that they are potentially unbounded, as well as the control spaces. This allows to treat superlinear Hamiltonians to which we extend the results in [3–6]. Moreover, we also build a whole family of value functions, which are still solutions in the sense of Ishii and connect continuously the minimal solution to the maximal one.