Kybernetika 56 no. 3, 500-515, 2020

Continuous feedback stabilization for a class of affine stochastic nonlinear systems

Mohamed Oumoun, Lahcen Maniar and Abdelghafour AtlasDOI: 10.14736/kyb-2020-3-0500


We investigate the state feedback stabilization, in the sense of weak solution, of nonlinear stochastic systems when the drift is quadratic in the control and the diffusion term is affine in the control. Based on the generalised stochastic Lyapunov theorem, we derive the necessary conditions and the sufficient conditions, respectively, for the global asymptotic stabilization in probability by a continuous feedback explicitly computed. The interest of this work is that the existing control methods are inapplicable to a lot of systems contained in the class of stochastic systems considered in this paper.


global asymptotic stability in probability, continuous state feedback, control stochastic nonlinear systems


60H10, 93C10, 93D05, 93D15, 93E15


  1. F. Abedi, W. J. Leong and S. S. Chaharborj: On the asymptotical and practical stability of stochastic control systems. Math. Problems Engrg. (2013), 1-10.   DOI:10.1155/2013/560647
  2. Z. Artstein: Stabilization with relaxed control. Nonlinear Anal. Theory Methods Appl. 7 (1983), 1163-1173.   DOI:10.1016/0362-546x(83)90049-4
  3. R. Chabour and M. Oumoun: On a universal formula for the stabilization of control stochastic nonlinear systems. Stochast. Anal. Appl. 17 (1999), 359-368.   DOI:10.1080/07362999908809606
  4. L. Daumail and P. Florchinger: A constructive extension of Artsteins's theorem to the stochastic context. Stochast. Dynamics 2 (2002), 251-263.   DOI:10.1142/s0219493702000418
  5. H. Deng, M. Krstic and R. J. Williams: Stabilization of stochastic nonlinear systems driven by noise of unknown covariance. IEEE Trans. Automat. Control 46 (2001), 1237-1253.   DOI:10.1109/9.940927
  6. P. Florchinger: A universal formula for the stabilization of control stochastic differential equations. Stochast. Anal. Appl. 11 (1993), 155-162.   DOI:10.1080/07362999308809308
  7. P. Florchinger: A universal design of Freeman's formula for the stabilization of stochastic systems. Stochast. Anal. Appl. 34 (2016), 137-146.   DOI:10.1080/07362994.2015.1108203
  8. P. Florchinger: Stabilization of nonlinear stochastic systems without unforced dynamics via time-varying feedback. Kybernetika 54 (2018), 321-335.   DOI:10.14736/kyb-2018-2-0321
  9. J. Fontbona, H. Raminez, V. Riquelme and F. Silva: Stochastic modeling and control of bioreactors. IFACPapersOnLine 50 (2017), 12611-12616.   DOI:10.1016/j.ifacol.2017.08.2203
  10. N. Ikeda and S. Watanabe: Stochastic Differential Equations and Diffusion Processes. Amsterdam, North-Holland 1981.   DOI:10.1002/bimj.4710280425
  11. F. Gao, Y. Wu and X. Yu: Global state feedback stabilization of stochastic high-order nonlinear systems with high-order and low-order nonlinearities. Int. J. Systems Sci. 47 (2016), 16, 3846-3856.   DOI:10.1080/00207721.2015.1129678
  12. H. K. Khalil: Nonlinear Systems. Upper Saddle River, Prentice-Hall, NJ 2002.   CrossRef
  13. R. Z. Khasminskii: Stochastic Stability of Differential Equations. Sijthoff and Noordhoff International Publishers 1980.   DOI:10.1007/978-3-642-23280-0
  14. F. C. Klebaner: Introduction to Stochastic Calculus with Applications. Imperial College Press, London 2005.   DOI:10.1142/p386
  15. H. J. Kushner: Stochastic Stability and Control. Academic Press, New York 1967.   DOI:10.1002/zamm.19680480428
  16. Q. Lan and S. Li: Global output-feedback stabilization for a class of stochastic nonlinear systems via sampled-data control. Int. J. Robust Nonlinear Control 27 (2017), 3643-3658.   DOI:10.1002/rnc.3758
  17. Q. Lan, H. Niu, Y. Liu and H. Xu: Global output-feedback finite-time stabilization for a class of stochastic nonlinear cascaded systems. Kybernetika 53 (2017), 780-802.   DOI:10.14736/kyb-2017-5-0780
  18. A. L. Lewis: Option Valuation Under Stochastic Volatility II. Finance Press, Newport Beach 2009.   DOI:10.1111/rssa.12262
  19. F. Li and Y. Liu: Global stability and stabilization of more general stochastic nonlinear systems. J. Math. Anal. Appl. 413 (2014), 841-855.   DOI:10.1016/j.jmaa.2013.12.021
  20. Y. Lin and E. D. Sontag: A universal formula for stabilization with bounded controls. Systems Control Lett. 16 (1991), 393-397.   DOI:10.1016/0167-6911(91)90111-q
  21. L. Maniar, M. Oumoun and J. C. Vivalda: On the stabilization of quadratic nonlinear systems. Europ. J Control 35 (2017), 28-33.   DOI:10.1016/j.ejcon.2017.03.001
  22. X. R. Mao: Stochastic Differential Equations and Their Applications. Horwood Publishing, Chichester 1997.   CrossRef
  23. X. Mao, A. Truman and C. Yuan: Euler-Maruyama approximations in mean-reverting stochastic volatility model under regime-switching. J. Appl. Math. Stochast. Anal. (2006), 1-20.   DOI:10.1155/jamsa/2006/80967
  24. M. Ondreját and J. Seidler: A note on weak solutions to stochastic differential equations. Kybernetika 54 (2018), 888-907.   DOI:10.14736/kyb-2018-5-0888
  25. E. D. Sontag: A universal construction of Artstein's theorem on nonlinear stabilization. Systems Control Lett. 13 (1989), 117-123.   DOI:10.1016/0167-6911(89)90028-5
  26. H. Yang, P. E. Kloeden and F. Wu: Weak solution of stochastic differential equations with fractional diffusion coefficient. Stochast. Anal. Appl. 36 (2018), 4, 613-621.   DOI:10.1080/07362994.2018.1434005
  27. W. Zha, J. Zhai and S. Fei: Global adaptive control for a class of uncertain stochastic nonlinear systems with unknown output gain. Int. J. Control Automat. Systems 15 (2017), 3, 1125-1133.   DOI:10.1007/s12555-016-0023-9
  28. B. L. Zhang, Q. L. Han and X. M. Zhang: Recent advances in vibration control of offshore platforms. Nonlinear Dynamics 89 (2017), 755-771.   DOI:10.1007/s11071-017-3503-4
  29. B. L. Zhang, Q. L. Han and X. M. Zhang: Event-triggered $H_\infty$ reliable control for offshore structures in network environments. J. Sound Vibration 368 (2016), 1-21.   DOI:10.1016/j.jsv.2016.01.008
  30. B. L. Zhang, Q. L. Han, X. M. Zhang and X. Yu: Sliding mode control with mixed current and delayed states for offshore steel jacket platforms. IEEE Trans. Control Systems Technol. 22 (2014), 1769-1783.   DOI:10.1109/tcst.2013.2293401
  31. J. Zhang and Y. Liu: Continuous output-feedback stabilization for a class of stochastic high-order nonlinear systems. J. Control Theory Appl. 11 (2013), 343-350.   DOI:10.1007/s11768-013-2166-z
  32. X. Zhang and X. Xie: Global state feedback stabilization of nonlinear systems with high-order and low-order nonlinearities. Int. J. Control 87 (2014), 642-652.   DOI:10.1080/00207179.2013.852252