Applied Mathematics and Optimization

, Volume 24, Issue 1, pp 197–220

A boundary-value problem for Hamilton-Jacobi equations in hilbert spaces

  • Piermarco Cannarsa
  • Fausto Gozzi
  • Halil Mete Soner
Article

Abstract

We study a Hamilton-Jacobi equation in infinite dimensions arising in optimal control theory for problems involving both exit times and state-space constraints. The corresponding boundary conditions for the Hamilton-Jacobi equation, of mixed nature, have been derived and investigated in [19], [2], [5], and [15] in the finite-dimensional case. We obtain a uniqueness result for viscosity solutions of such a problem and then prove the existence of a solution by showing that the value function is continuous.

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References

  1. 1.
    Aubin JP, Ekeland I (1984) Applied Nonlinear Analysis. Wiley Interscience, New YorkGoogle Scholar
  2. 2.
    Barles G, Perthame B (1988) Exit time problems in optimal control and vanishing viscosity method. SIAM J Control Optim 26:1133–1148Google Scholar
  3. 3.
    Barles G, Perthame B (1987) Discontinuous solutions of deterministic optimal stopping time problems. Math Methods Numer Anal 21(4):557–579Google Scholar
  4. 4.
    Cannarsa P (1989) Regularity properties of solutions to Hamilton-Jacobi equations in infinite dimensions and nonlinear optimal control. Differential Integral Equations 2:479–493Google Scholar
  5. 5.
    Capuzzo-Dolcetta I, Lions PL (to appear) Hamilton-Jacobi equations and state constraints problems. Trans Amer Math SocGoogle Scholar
  6. 6.
    Clarke F (1983) Optimization and Nonsmooth Analysis. Wiley, New YorkGoogle Scholar
  7. 7.
    Crandall MG, Lions PL (1983) Viscosity solutions of Hamilton-Jacobi equations. Trans Amer Math Soc 277:1–42Google Scholar
  8. 8.
    Crandall MG, Lions PL (1985) Hamilton-Jacobi equations in infinite dimensions, I. J Funct Anal 62:379–396Google Scholar
  9. 9.
    Crandall MG, Lions PL (1986) Hamilton-Jacobi equations in infinite dimensions, II. J Funct Anal 65:368–405Google Scholar
  10. 10.
    Crandall MG, Lions PL (1986) Hamilton-Jacobi equations in infinite dimensions, III. J Funct Anal 68:368–405Google Scholar
  11. 11.
    Crandall MG, Lions PL (preprint) Hamilton-Jacobi equations in infinite dimensions, IVGoogle Scholar
  12. 12.
    Crandall MG, Evans LC, Lions PL (1984) Some properties of the viscosity solutions of Hamilton-Jacobi equations. Trans Amer Math Soc 282:487–502Google Scholar
  13. 13.
    Federer H (1959) Curvature measures. Trans Amer Math Soc 93:429–437Google Scholar
  14. 14.
    Giles JR (1982) Convex Analysis with Application in Differentiation of Convex Functions. Pitman, BostonGoogle Scholar
  15. 15.
    Ishii H (to appear) A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations. Ann Scuola Norm Sup Pisa Cl Sci (4)Google Scholar
  16. 16.
    Lions PL (1982) Generalized Solutions of Hamilton-Jacobi Equations. Pitman, BostonGoogle Scholar
  17. 17.
    Lions PL (1985) Optimal control and viscosity solutions. Proc. Conf. Dynamic Programming, Rome, 1983. Springer-Verlag, BerlinGoogle Scholar
  18. 18.
    Loreti P (1987) Some properties of constrained viscosity solutions of Hamilton-Jacobi-Bellman equations. SIAM J Control Optim 25:1244–1252Google Scholar
  19. 19.
    Soner HM (1986) Optimal control with state-space constraint, I. SIAM J Control Optim 24:522–561Google Scholar
  20. 20.
    Soner HM (1988) On the Hamilton-Jacobi-Bellman equations in Banach spaces. J Optim Theory Appl 57:429–437Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1991

Authors and Affiliations

  • Piermarco Cannarsa
    • 1
  • Fausto Gozzi
    • 2
  • Halil Mete Soner
    • 3
  1. 1.Dipartimento di MatematicaPisaItaly
  2. 2.Scuola Normale SuperiorePisaItaly
  3. 3.Department of MathematicsCarnegie-Mellon UniversityPittsburghUSA

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