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Networks and Heterogeneous Media

2007, Volume 2, Issue 2: 313-331. doi: 10.3934/nhm.2007.2.313
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Optimal control problems on stratified domains

  • 1.
    Department of Mathematics, Penn State University, University Park, Pa.16802
  • 2.
    Department of Industrial and Manufacturing Engineering, Penn State University, University Park, PA 16802
  • Received: 01 December 2006 Revised: 01 February 2007

  • Primary: 49L25; Secondary: 49J24.

  • We consider a class of optimal control problems defined on a stratified domain. Namely, we assume that the state space RN admits a stratification as a disjoint union of finitely many embedded submanifolds Mi. The dynamics of the system and the cost function are Lipschitz continuous restricted to each submanifold. We provide conditions which guarantee the existence of an optimal solution, and study sufficient conditions for optimality. These are obtained by proving a uniqueness result for solutions to a corresponding Hamilton-Jacobi equation with discontinuous coefficients, describing the value function. Our results are motivated by various applications, such as minimum time problems with discontinuous dynamics, and optimization problems constrained to a bounded domain, in the presence of an additional overflow cost at the boundary.

    Citation: Alberto Bressan, Yunho Hong. Optimal control problems on stratified domains[J]. Networks and Heterogeneous Media, 2007, 2(2): 313-331. doi: 10.3934/nhm.2007.2.313

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  • We consider a class of optimal control problems defined on a stratified domain. Namely, we assume that the state space RN admits a stratification as a disjoint union of finitely many embedded submanifolds Mi. The dynamics of the system and the cost function are Lipschitz continuous restricted to each submanifold. We provide conditions which guarantee the existence of an optimal solution, and study sufficient conditions for optimality. These are obtained by proving a uniqueness result for solutions to a corresponding Hamilton-Jacobi equation with discontinuous coefficients, describing the value function. Our results are motivated by various applications, such as minimum time problems with discontinuous dynamics, and optimization problems constrained to a bounded domain, in the presence of an additional overflow cost at the boundary.


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