Skip to main content
SpringerLink
Log in

Necessary and Sufficient Conditions for Optimal Stabilization of Quasi-Linear Stochastic Systems

  • Stochastic Systems
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

A wide class of admissible control strategies that guarantee the mean-square stabilization of a stochastic system is considered. Necessary and sufficient conditions for the opti-mality of a linear time-invariant controller are established. The difference between the stated problem and the optimal control problem on an infinite time interval is demonstrated. The obtained optimality conditions are illustrated by the example of stabilization of an artificial Earth satellite in the neighborhood of a circular orbit.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Panossian, H.V., Review of Linear Stochastic Optimal Control Systems and Applications, J. Vib. Acoust. Stress Reliab. Des., 1989, vol. 111, no. 4, pp. 399–403.

    Article  Google Scholar 

  2. Kwakernaak, H. and Sivan, R., Linear Optimal Control Systems, New York: Wiley-Interscience, 1972. Translated under the title

    MATH  Google Scholar 

  3. Kwakernaak, H. and Sivan, R., Lineinye optimal’nye sistemy upravleniya, Moscow: Mir, 1972.

    Google Scholar 

  4. Chen, S., Li, X., and Zhou, X.-Y., Stochastic Linear Quadratic Regulators with Indefinite Control Weight Costs, SIAM J. Control Optim., 1998, vol. 36, no. 5, pp. 1685–1702.

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, S. and Zhou, X.-Y., Stochastic Linear Quadratic Regulators with Indefinite Control Weight Costs, SIAM J. Control Optim., 2000, vol. 39, no. 4, pp. 1065–1081.

    Article  MathSciNet  MATH  Google Scholar 

  6. Ait Rami, M., Moore, J.B., and Zhou, X.-Y., Indefinite Stochastic Linear Quadratic Control and Generalized Differential Riccati Equation, SIAM J. Control Optim., 2002, vol. 40, no. 4, pp. 1296–1311.

    Article  MathSciNet  MATH  Google Scholar 

  7. Arnold, L., Stochastic Differential Equations: Theory and Applications, New York: Wiley, 1974.

    MATH  Google Scholar 

  8. Wonham, W.M., Optimal Stationary Control of a Linear System with State-Dependent Noise, SIAM J. Control, 1967, vol. 5, no. 3, pp. 486–500.

    Article  MathSciNet  MATH  Google Scholar 

  9. Haussmann, U.G., Optimal Stationary Control with State Control Dependent Noise, SIAM J. Control, 1971, vol. 9, no. 2, pp. 184–198.

    Article  MathSciNet  MATH  Google Scholar 

  10. McLane, P.J., Optimal Stochastic Control of Linear Systems with State- and Control-Dependent Disturbances, IEEE Trans. Automat. Control, 1971, vol. 16, no. 6, pp. 793–798.

    Article  Google Scholar 

  11. El Ghaoui, L., State-Feedback Control of Systems with Multiplicative Noise via Linear Matrix Inequalities, Syst. Control Lett., 1995, vol. 24, no. 3, pp. 223–228.

    Article  MathSciNet  MATH  Google Scholar 

  12. Verriest, E.I. and Florchinger, P., Stability of Stochastic Systems with Uncertain Time Delays, Syst. Control Lett., 1995, vol. 24, no. 1, pp. 41–47.

    Article  MathSciNet  MATH  Google Scholar 

  13. Paraev, Yu.I., Vvedenie v statisticheskuyu dinamiku protsessov upravleniya i fil’tratsii. Biblioteka tekhni-cheskoi kibernetiki (Introduction to Statistical Dynamics of Control Processes and Filtering. Library of Engineering Cybernetics), Moscow: Sovetskoe Radio, 1976.

    Google Scholar 

  14. Rumyantsev, D.S., Khrustalev, M.M., and Tsarkov, K.A., An Algorithm for Synthesis of the Suboptimal Control Law for Quasi-linear Stochastic Dynamical Systems, J. Comput. Syst. Sci. Int., 2014, vol. 53, no. 1, pp. 71–83.

    Article  MathSciNet  MATH  Google Scholar 

  15. Willems, J.L. and Willems, J.C., Feedback Stabilizability for Stochastic Systems with State and Control Dependent Noise, Automatica, 1976, vol. 12, no. 3, pp. 277–283.

    Article  MathSciNet  MATH  Google Scholar 

  16. Kleinman, D.L., Optimal Stationary Control of Linear Systems with Control-Dependent Noise, IEEE Trans. Automat. Control, 1969, vol. 14, no. 6, pp. 673–677.

    Article  MathSciNet  Google Scholar 

  17. Damm, T., Rational Matrix Equations in Stochastic Control, Berlin-Heidelberg: Springer, 2004.

    MATH  Google Scholar 

  18. Khrustalev, M.M. and Onegin, E.E., Analytical Design of Optimal Controllers for Quasi-linear Stochastic Systems on the Infinite Time Interval, Program. Sist.: Teor. Prilozheniya, 2015, vol. 6, no. 2, pp. 29–44.

    Google Scholar 

  19. Øksendal, B., Stochastic Differential Equations, Berlin-Heidelberg: Springer, 2003.

    Book  MATH  Google Scholar 

  20. Khrustalev, M.M., Nash Equilibrium Conditions in Stochastic Differential Games where Players Information about a State Is Incomplete. I, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 1995, vol. 34, no. 6, pp. 194–208.

    Google Scholar 

  21. Khrustalev, M.M., Nash Equilibrium Conditions in Stochastic Differential Games where Players Information about a State Is Incomplete. II. Lagrange Method, J. Comput. Syst. Sci. Int., 1996, vol. 35, no. 1, pp. 67–73.

    MATH  Google Scholar 

  22. Khalina, A.S. and Khrustalev, M.M., System Shape Optimization and Stabilization of Controlled Quasi-linear Stochastic Systems that Operate on an Infinite Time Interval, J. Comput. Syst. Sci. Int., 2017, vol. 56, no. 1, pp. 64–86.

    Article  MathSciNet  MATH  Google Scholar 

  23. Lebedev, A.A., Krasil’shchikov, M.N., and Malyshev, V.V., Optimal’noe upravlenie dvizheniem kosmich-eskikh letatel’nykh apparatov (Optimal Control of Motion of Spacecrafts), Moscow: Mashinostroenie, 1974.

    Google Scholar 

  24. Lebedev, A.A., Bobronnikov, V.T., Krasil’shchikov, M.N., and Malyshev, V.V., Statisticheskaya di-namika i optimizatsiya upravleniya letatel’nykh apparatov (Statistical Dynamics and Optimization of Control of Aircrafts), Moscow: Mashinostroenie, 1985.

    Google Scholar 

Download references

Acknowledgements

This work was carried out within the state task no. 9.7555.2017/BCh.

Author information

Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    M. M. Khrustalev & E. E. Onegin

  2. Moscow Aviation Institute, Moscow, Russia

    M. M. Khrustalev & E. E. Onegin

Authors
  1. M. M. Khrustalev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. E. Onegin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to M. M. Khrustalev or E. E. Onegin.

Additional information

This paper was recommended for publication by A.I. Kibzun, a member of the Editorial Board

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khrustalev, M.M., Onegin, E.E. Necessary and Sufficient Conditions for Optimal Stabilization of Quasi-Linear Stochastic Systems. Autom Remote Control 80, 1252–1264 (2019). https://doi.org/10.1134/S0005117919070038

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117919070038

Keywords

Search

Navigation