Applied Mathematics and Optimization

, Volume 10, Issue 1, pp 367–377 | Cite as

On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming

  • I. Capuzzo Dolcetta


An approximation of the Hamilton-Jacobi-Bellman equation connected with the infinite horizon optimal control problem with discount is proposed. The approximate solutions are shown to converge uniformly to the viscosity solution, in the sense of Crandall-Lions, of the original problem. Moreover, the approximate solutions are interpreted as value functions of some discrete time control problem. This allows to construct by dynamic programming a minimizing sequence of piecewise constant controls.


Control Problem Approximate Solution System Theory Mathematical Method Dynamic Programming 
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  1. 1.
    Bertsekas DP, Shreve SE (1978) Stochastic optimal control: The discrete time case. Academic Press, New YorkGoogle Scholar
  2. 2.
    Capuzzo Dolcetta I, Evans LC (to appear) Optimal switching for ordinary differential equations. SIAM J ControlGoogle Scholar
  3. 3.
    Capuzzo Dolcetta I, Matzeu M (1981) On the dynamic programming inequalities associated with the deterministic optimal stopping problem in discrete and continuous time. Num Funct Anal Optim 3:425–450Google Scholar
  4. 4.
    Capuzzo Dolcetta I, Matzeu M, Menaldi JL (to appear) On a system of first order quasi-variational inequalities connected with the optimal switching problem. Systems and Control LettersGoogle Scholar
  5. 5.
    Crandall MG, Evans LC, Lions PL (to appear) Some properties of the viscosity solutions of Hamilton-Jacobi equations. Trans Amer Math SocGoogle Scholar
  6. 6.
    Crandall MG, Lions PL (to appear) Viscosity solutions of Hamilton-Jacobi equations. Trans Amer Math SocGoogle Scholar
  7. 7.
    Evans LC (1980) On solving certain nonlinear partial differential equations by accretive operator methods. Israel J Math 36:365–389Google Scholar
  8. 8.
    Fleming WH, Rishel RW (1975) Deterministic and stochastic optimal control. Springer-Verlag, Berlin-Heidelberg-New YorkGoogle Scholar
  9. 9.
    Gawronski M (1982) Dissertation. Istituto Matematico, Università di Roma, RomeGoogle Scholar
  10. 10.
    Goletti F (1981) Dissertation. Istituto Matematico, Università di Roma, RomeGoogle Scholar
  11. 11.
    Henrici P (1962) Discrete variable methods in ordinary differential equations. J. Wiley, New YorkGoogle Scholar
  12. 12.
    Lions PL (1982) Generalized solutions of Hamilton-Jacobi equations. Pitman, LondonGoogle Scholar
  13. 13.
    Menaldi JL (1982) Le problème de temps d'arret optimal déterministe et l'inéquation variationnelle du premier ordre associée. Appl Math Optim 8:131–158Google Scholar

Copyright information

© Springer-Verlag New York Inc 1983

Authors and Affiliations

  • I. Capuzzo Dolcetta
    • 1
    • 2
  1. 1.Istituto MatematicoUniversità di RomaRomeItaly
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA

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