Hamilton–Jacobi–Bellman Equations on Multi-domains

  • Zhiping Rao
  • Hasnaa Zidani
Part of the International Series of Numerical Mathematics book series (ISNM, volume 164)


A system of Hamilton–Jacobi (HJ) equations on a partition of \(\mathbb {R}^{d}\) is considered, and a uniqueness and existence result of viscosity solution is analyzed. While the notion of viscosity solution is by now well known, the question of uniqueness of solution, when the Hamiltonian is discontinuous, remains an important issue. A uniqueness result has been derived for a class of problems, where the behavior of the solution, in the region of discontinuity of the Hamiltonian, is assumed to be irrelevant and can be ignored (see (Camilli, Siconolfi in Adv. Differ. Equ. 8(6):733–768, 2003)). Here, we provide a new uniqueness result for a more general class of Hamilton–Jacobi equations.


Hamilton–Jacobi equations Discontinuous Hamiltonian Viscosity solution Optimal control 

Mathematics Subject Classification (2010)

35F21 49L25 49L20 



The authors are grateful to Peter Wolenski and Antonio Siconolfi for many helpful discussions.


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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Commands (ENSTA ParisTech, INRIA Saclay)Palaiseau CedexFrance

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