Viability Solutions to Hamilton–Jacobi Equations

  • Jean-Pierre Aubin
  • Alexandre M. Bayen
  • Patrick Saint-Pierre


This chapter presents the viability approach to a class of Hamilton-Jacobi equations. We assume not only that the solution depends on time, but on “structured” or “causal” variables. They include age-structured Hamilton-Jacobi-McKendrick equations, useful in population dynamics as well as in transport management (the age variable being replaced by the travel time), as well as Hamilton-Jacobi-Cournot equation, where the “structured” or “causal” variable is the initial state of the underlying control system. Chapters 14, p. 565 and 15, p. 605 apply the results of this chapter to transportation management, finance and economics.


Viscosity Solution Lower Semicontinuous Optimal Evolution Jacobi Equation Viability Solution 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Jean-Pierre Aubin
    • 1
  • Alexandre M. Bayen
    • 2
  • Patrick Saint-Pierre
    • 3
  1. 1.VIMADESParisFrance
  2. 2.Electrical Engineering and Computer Sciences Civil and Environmental EngineeringUniversity of California at BerkeleyBerkeleyUSA
  3. 3.Université Paris Dauphine LEDA-SDFiClamartFrance

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