The intuitive dynamic programming approach to optimal stochastic navigation

  • Jürgen Franke


This paper supplies an intuitive probabilistic interpretation of the necessary conditions on locally optimal solutions of the optimization problems treated in „Optimal Navigation with Random Terminal Time in the Presence of Phase Constraints” by means of a functional analytic approach. The dynamic programming approach yields rigorous results only under restrictive and hard-to-verify assumptions on the cost function, but it follows an interpretation of an optimal control as direction of fastest fecrease of the conditional expected residual costs.


Cost Function Stochastic Process Analytic Approach Probability Theory Dynamic Programming 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bellman, R.: Introduction to the Mathematical Theory of Control Processes Vol. II. New York-London: Academic Press 1971Google Scholar
  2. 2.
    Berkovitz, L.D.: Optimal Control Theory. Berlin-Heidelberg-New York: Springer 1974Google Scholar
  3. 3.
    Dreyfus, S.E.: Dynamic Programming and the Calculus of Variations. New York-London: Academic Press 1965Google Scholar
  4. 4.
    Dreyfus, S.E., Law, A.M.: The Art and Theory of Dynamic Programming. New York-London: Academic Press 1977Google Scholar
  5. 5.
    Franke, J.: Optimal Navigation with Random Terminal Time in the Presence of Phase Constraints. [To appear in this journal].Google Scholar
  6. 6.
    Girsanov, I.V.: Lectures on Mathematical Theory of Extremum Problems. Lecture Notes in Economics and Mathematical Systems 67. Berlin-Heidelberg-New York: Springer 1972Google Scholar
  7. 7.
    Lukka, M.: On the Optimal Search Tracks for a Moving Target. SIAM J. Appl. Math. 32, 126–132 (1977)Google Scholar
  8. 8.
    Osborn, H.: On the Foundations of Dynamic Programming. J. Math. Mech. 8, 867–872 (1959)Google Scholar
  9. 9.
    Pontryagin, L.S. et al.: Mathematische Theorie optimaler Prozesse. München-Wien: Oldenbourg 1964Google Scholar
  10. 10.
    Pursiheimo, U.: On the Optimal Search for a Target whose Motion is Conditionally Deterministic with Stochastic Initial Conditions on Location and Parameters. SIAM J. Appl. Math. 32, 105–114 (1977)Google Scholar
  11. 11.
    Pursiheimo, U.: On the Optimal Search for a Moving Target in Discrete Space. In: Optimization Techniques, pp. 177–185. Lecture Notes in Control and Information Sciences 6. Berlin-Heidelberg-New York: Springer 1978Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Jürgen Franke
    • 1
  1. 1.Fachbereich MathematikUniversität FrankfurtFrankfurt a.M.Federal Republic of Germany

Personalised recommendations