Skip to main content
SpringerLink
Log in

Approximation of a stochastic ergodic control problem

  • V-Stochastic Systems
  • Conference paper
  • First Online:
New Trends in Nonlinear Control Theory

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 122))

Abstract

We study a degenerate non linear optimal stochastic control problem of ergodic type. We first prove that for each feedback control law, there exists a unique invariant measure which is equivalent to Lebesgue measure. This is proved using an accessibility property of the stochastic differential equation, after the discontinuous part of the drift has been removed via a change of probability measure. We then approximate the problem by ergodic control problems for finite state, continuous time Markov chains. We finally prove that the cost functionals of the approximate problems converge pointwise towards that of the continuous problem.

All the study is done for a particular problem introduced in [1], which is motivated by the optimal control of the shock-absorber of a road vehicle. The numerical results can be found in [1].

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. BELLIZZI, R. BOUC, F. CAMPILLO, and E. PARDOUX. Contrôle optimal semi-actif de suspension de véhicule. In Analysis and Optimization of Systems, A. Bensoussan and J.L. Lions (eds.), INRIA, Antibes, 1988. Lecture Notes in Control and Information Sciences 111, 1988.

    Google Scholar 

  2. A. BENSOUSSAN. Perturbation Methods in Optimal Control. John Wiley & Sons, New-York, 1988.

    Google Scholar 

  3. V.S. BORKAR and M.K. GHOSH. Ergodic control of multidimensional diffusions I: the existence results. SIAM Journal of Control and Optimization, 26(1):112–126, January 1988.

    Google Scholar 

  4. F. DELEBECQUE and J.P. QUADRAT. Contribution of stochastic control singular perturbation averaging and team theories to an example of large-scale systems: management of hydropower production. IEEE Transactions on Automatic Control, AC-23(2):209–221, April 1978.

    Google Scholar 

  5. B.T. DOSHI. Continuous time control of Markov processes on an arbitrary state space: average return criterion. Stochastic Processes and their Applications, 4:55–77, 1976.

    Google Scholar 

  6. N. ETHIER and T.G. KURTZ. Markov Processes — Characterization and Convergence. J. Wiley & Sons, New-York, 1986.

    Google Scholar 

  7. F. LE GLAND. Estimation de Paramètres dans les Processus Stochastiques, en Observation Incomplète — Applications à un Problème de Radio-Astronomie. Thèse de Docteur-Ingénieur, Université de Paris IX — Dauphine, 1981.

    Google Scholar 

  8. P.A. HOWARD. Dynamic Programming and Markov Processes. J. Wiley, New-York, 1960.

    Google Scholar 

  9. H.J. KUSHNER. Optimality conditions for the average cost per unit time problem with a diffusion model. SIAM Journal of Control and Optimization, 16(2):330–346, March 1978.

    Google Scholar 

  10. H.J. KUSHNER. Probability Methods for Approximations in Stochastic Control and for Elliptic Equations. Volume 129 of Mathematics in Science and Engineering, Academic Press, New-York, 1977.

    Google Scholar 

  11. A. LEIZAROWITZ. Controlled diffusion processes on infinite horizon with the overtaking criterion. Applied Mathematics and Optimization, 17:61–78, 1988.

    Google Scholar 

  12. D. MICHEL and E. PARDOUX. An introduction to Malliavin's calculus and some of its applications. (to appear).

    Google Scholar 

  13. J.P. QUADRAT. Sur l'Identification et le Contróle de Systèmes Dynamiques Stochastiques. Thèse, Université de Paris IX — Dauphine, 1981.

    Google Scholar 

  14. R. REBOLLEDO. La méthode des martingales appliquée à l'étude de la convergence en loi de processus. Mémoire 62, Bulletin SMF, 1979.

    Google Scholar 

  15. M. ROBIN. Long-term average cost control problems for continuous time Markov processes: a survey. Acta Applicandae Mathematicae, 1:281–299, 1983.

    Google Scholar 

  16. R.H. STOCKBRIDGE. Time-average control of martingale problems. PhD thesis, University of Wisconsin-Madison, 1987.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. INRIA centre de Sophia Antipolis Route des Lucioles, 06565, Valbonne Cedex, France

    Fabien Campillo & François Le Gland

  2. Inria centre de Sophia Antipolis & Mathématiques, UA 225, Université de Provence, USA

    Etienne Pardoux

Authors
  1. Fabien Campillo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. François Le Gland
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Etienne Pardoux
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

J. Descusse Michel Fliess A. Isidori D. Leborgne

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer-Verlag

About this paper

Cite this paper

Campillo, F., Le Gland, F., Pardoux, E. (1989). Approximation of a stochastic ergodic control problem. In: Descusse, J., Fliess, M., Isidori, A., Leborgne, D. (eds) New Trends in Nonlinear Control Theory. Lecture Notes in Control and Information Sciences, vol 122. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0043045

Download citation

  • DOI: https://doi.org/10.1007/BFb0043045

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51075-8

  • Online ISBN: 978-3-540-46143-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation