Multicriteria Optimal Control and Vectorial Hamilton-Jacobi Equation

  • Nathalie Caroff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4818)


In this paper we investigate a multicriteria optimal control problem associated to a preference relation based on the lexicographic order. We extend different notions of non-smooth analysis and control and show that the vector Value function is the unique vector lower semicontinuous solution to a suitable system of Hamilton-Jacobi equations in the sense of contingent solution or equivalently in the sense of extended viscosity solution.


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  1. 1.
    Aubin, J.P., Cellina, A.: Differential Inclusions. Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
  2. 2.
    Aubin, J.P., Frankowska, H.: Set-valued analysis. Birkhaüser, Berlin (1991)zbMATHGoogle Scholar
  3. 3.
    Barron, E.N., Jensen, R.: Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Partial Differential Equations 15, 1713–1742 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Borwein, J.M., Nieuwenhuis, J.W.: Two kind of Normality in vector optimization. Mathematical Programming 28, 185–191 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Clarke, F.H.: Optimization and non smooth analysis. Wiley-Interscience, New York (1983)Google Scholar
  6. 6.
    Crandall, M.G., Lyons, P.L.: Viscosity solutions of Hamilton-Jacobi equations. Transactions of American Mathematical Society 277, 1–42 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Frankowska, H.: Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 31, 257–272 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Imbert, C., Volle, M.: On vectorial Hamilton-Jacobi equations. Well-posedness in optimization and related topics. Control Cybernet 31, 493–506 (2002)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Volle, M.: Duality Principle for optimization problems Dealing with the Difference of vector-valued Convex Mappings. J. Optim. Theory Appl. 114, 223–241 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Zhu, Q.J.: Hamiltonian necessary conditions for a multiobjective optimal control problem with endpoint constraints. SIAM J. Control Optim. 39, 97–112 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Nathalie Caroff
    • 1
  1. 1.Laboratoire MANOUniversité de PerpignanPerpignan cedexFrance

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