Skip to main content
Log in

Long-term average cost control problems for continuous time Markov processes: A survey

  • Published:
Acta Applicandae Mathematica Aims and scope Submit manuscript

Abstract

This paper addresses the long-term average cost control of continuous time Markov processes. A survey of problems and methods contained in various works is given for continuous control, optimal stopping, and impulse control.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bellman R.: J. Math. Mech. 6 (1957), 679–684.

    Google Scholar 

  2. Bensoussan A.: Stochastic Control by Functional Analysis Methods, North-Holland, Amsterdam, 1982.

    Google Scholar 

  3. Bensoussan A. and Lions J. L.: Applications des inéquations variationnelles en contrôle stochastique, Dunod, Paris, 1978.

    Google Scholar 

  4. Bensoussan A. and Lions J. L.: Contrôle impulsionnel et inéquations quasi-variationnelles, Dunod, Paris, 1982.

    Google Scholar 

  5. Bensoussan A. and Lions J. L.: ‘Nouvelles formulations de problèmes de contrôle impulsionnel et applications’, CRAS, Paris, 1973, pp. 1189–1192.

    Google Scholar 

  6. Bensoussan A. and Lions J. L.: in Theory of Nonlinear Operators, Akademic Verlag, Berlin, 1978.

    Google Scholar 

  7. Bismut J. M. and Skalli B.: Z. Wahz. Verw. G. 39 (1977), 301–313.

    Google Scholar 

  8. Blackwell D. Ann. Math. Stat. 33 (1962), 719–726.

    Google Scholar 

  9. Doshi B. T.: Stoch. Proc. Appl. 4 (1976), 55–77.

    Google Scholar 

  10. Dynkin E. B.: Markov Processes, Vols. I, II, Springer-Verlag, Heidelberg, 1965.

    Google Scholar 

  11. Dynkin E. B.: Dokl. Acad. Nauk USSR, 150 (1963), 238–240.

    Google Scholar 

  12. Dynkin E. B. and Yuskevich A. A.: Controlled Markov Processes, Springer-Verlag, Berlin, 1979.

    Google Scholar 

  13. El Karoui, N.: Mèthodes probabilistes en contrôle stochastique, to be published.

  14. Fayolle G. and Robin M.: in Modeling and Performance Evaluation of Computer Systems, North-Holland, Amsterdam, 1976.

    Google Scholar 

  15. Federgruen A., Hordijk A. and Tijms H. C.: Stoch. Proc. Appl. 9 (1979), 223–235.

    Google Scholar 

  16. Fleming W. and Rishel R.: Optimal Deterministic and Stochastic Control, Springer-Verlag, Berlin, 1975.

    Google Scholar 

  17. Has'minskii, R. Z.: Stochastic Stability of Differential Equations, Sijthoff & Noordhoff, 1980.

  18. Hordijk, A.: ‘Dynamic Programming and Markov Potential Theory’, Mathematical Center Tr. No. 51, Amsterdam, 1974.

  19. Howard R. A.: Dynamic Programming and Markov Processes, MIT Press. Cambridge, Mass, 1960.

    Google Scholar 

  20. Kakumanu P. K.: Siam J. Control 10 (1972), 210–220.

    Google Scholar 

  21. Karatzas, J.: ‘A Class of Singular Stochastic Control Problem’, to appear.

  22. Kogan Y. A.: Theor. Proba. Applic. 14 (1969), 496–502.

    Google Scholar 

  23. Lanery, E.: ‘Etude asymptotique des systèmes markoviens à commande’, Revue inf. Recherche opérationnelle lère annee, No. 5, pp. 3–56.

  24. Lasry, J. M.: ‘Contrôle stochastique ergodique’, Thèse Université de Paris IX, 1974.

  25. Lepeltier, J. P. and Marchal, B.: ‘Théorie générale du contrôle impulsionnel’, Thèse, 1980.

  26. Lippman S. A.: Ann. Math. Stat. 42 (1971), 1717–1726.

    Google Scholar 

  27. Lorenzani, M.: ‘Optimal Stopping for Markov Processes’, Note. Rend. Cont. Acad. Sci., Rome, 1981.

  28. Mandl P.: Analytic Treatment of One-Dimensional Markov Processes, Springer-Verlag, New York, 1968.

    Google Scholar 

  29. Makowski, A.: ‘Local Optimality Conditions for Optimal Stopping’, CSR-81-23/1981, Dept of Elect. Eng. University of Maryland.

  30. Menaldi, J. L.: ‘On the Degenerate variational Inequality with Neumann Boundary Conditions, Siam J. Control Opt., to be published.

  31. Menaldi J. L.: Siam J. Control Opt. 18 (1980), 697–721.

    Google Scholar 

  32. Miller B.: J. Appl. Prob. 18 (1981), 361–377.

    Google Scholar 

  33. Prabhu N. and Stidham S.: in Mathematical Methods in Queueing Theory, (Lecture Notes in Economics and Mathematical Systems, 98) Springer-Verlag, Heidelberg, 1974.

    Google Scholar 

  34. Puterman, M.: ‘On the Optimal Control of Diffusion Processes’, TR No. 24, Stanford University, 1972.

  35. Robin, M.: ‘Contrôle impulsionnel des processus de Markov’, Thesis, University of Paris IX, 1978.

  36. Robin M.: Siam J. Control Opt. 19 (1981), 333–358.

    Google Scholar 

  37. Robin M.: in System Modeling and Optimization. (Lecture Notes in Control and Information Sciences, 38), Springer-Verlag, Heidelberg, 1982.

    Google Scholar 

  38. Robin, M.: ‘Asymptotics in Quasi-Variational Inequalities and Ergodic Control Problems’, to be published, Systems and Control Letters 1983.

  39. Rosberg Z., Varayia P. and Walrand J.: IEEE Tr. Auto-Control, AC 27 (1982), 600–610.

    Google Scholar 

  40. Ross S. M.: Ann. Math. Stat. 42 (1971), 1767–1768.

    Google Scholar 

  41. Ross S. M.: J. Appl. Prob. 7 (1970), 649–656.

    Google Scholar 

  42. Ross S. M.: Applied Probability Models with Optimization Applications, Holden Day, San Francisco, 1970.

    Google Scholar 

  43. Schweitzer P.: ‘A Non-Expansive Mapping Approach to Undiscounted Semi-Markovian Decision Processes’, WP. 8109, 1981, Graduate School of Management, University of Rochester, U.S.A.

    Google Scholar 

  44. Stroock D. and Varadhan S.: Comm. Pure Appl. Math. 24 (1971), 147–225.

    Google Scholar 

  45. Sulem, A.: to be published.

  46. Stettner, L.: ‘On Impulsive Control with Long-Run Average Cost Criterion’, to appear in Studia Mathematica.

  47. Shiryaev A. N.: Optimal Stopping Rules, Springer-Verlag, New York, 1978.

    Google Scholar 

  48. Tarres, R.: ‘Comportement asymptotique d'un problème de contrôle stochastique’, Cahier de math. de la Décision No. 8215, Université de Paris IX, 1982.

  49. Vanderduyn-Shouten F. A. and Hordijk A.: ‘On the Existence of Average Optimal Policies in Markov Decision Drift Processes with General State and Action Space’, Vrije Universiteit, Amsterdam, TR No. 73, 1981.

    Google Scholar 

  50. Wijngaard J.: Math. Oper. Res. 2 (1977), 91–102.

    Google Scholar 

  51. Yu Chung Liao- ‘On Optimal Control of a Brownian Motion and Approximation of Queueing Processes’, PhD Thesis, Brown University, R.I. 1982.

  52. Yuskevich A. A.: Theor. Proba. Applic. 26 (1981); 796–802.

    Google Scholar 

  53. Zabczyk J.: in Stochastic Control Theory and Stochastic Differential Systems (Lecture Notes in Control and Information Sciences 16), Springer-Verlag, Heidelberg, 1979.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Robin, M. Long-term average cost control problems for continuous time Markov processes: A survey. Acta Appl Math 1, 281–299 (1983). https://doi.org/10.1007/BF00046603

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00046603

AMS (Mos) subject classifications (1980)

Key words

Navigation