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Discontinuous solutions of Hamilton–Jacobi–Bellman equation under state constraints

  • Hélène FrankowskaEmail author
  • Marco Mazzola
Article

Abstract

This article is devoted to the Hamilton–Jacobi partial differential equation
$$\left\{\begin{array}{lll}\frac{\partial V}{\partial t} = H\left(t, x, - \frac{\partial V}{\partial x}\right) & \hbox{on} & [0, 1]\times {\overline{\Omega}}\\V(1, x) = g(x) & \hbox{on}& {\overline{\Omega}},\end{array}\right.$$
where the Hamiltonian \({{H:[0, 1] \times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}}}\) is convex and positively homogeneous with respect to the last variable, \({{\Omega \subset \mathbb{R}^n}}\) is open and \({{g : \mathbb{R}^n \to \mathbb{R} \cup \{+ \infty\}}}\) is lower semicontinuous. Such Hamiltonians do arise in the optimal control theory. We apply the method of generalized characteristics to show uniqueness of lower semicontinuous solution of this first order PDE. The novelty of our setting lies in the fact that we do not ask regularity of the boundary of Ω and extend the Soner inward pointing condition in a nontraditional way to get uniqueness in the class of lower semicontinuous functions.

Mathematics Subject Classification (2000)

34A60 35D05 35F20 49L25 

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.CNRS and Institut de Mathématiques de JussieuUniversité Pierre et Marie CurieParisFrance
  2. 2.ITN Marie Curie Network SADCO at Institut de Mathématiques de JussieuUniversité Pierre et Marie CurieParisFrance

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