Practical Stability

Chapter

Abstract

This chapter investigates stochastic systems with Markovian jump parameters and time-varying delays in terms of their practical stability in probability and in the pth mean, and the practical controllability in probability and in the pth mean, respectively.

References

  1. 1.
    Delfour, M.C., Mitter, S.K.: Hereditary differential systems with constant delays. II.A class of affine systems and the adjoint problem. J. Differ. Equat. 18(75), 18C28 (1975)Google Scholar
  2. 2.
    Eller, D.H., Aggarwal, J.K., Banks, H.T.: Optimal control of linear time-delay systems. IEEE Trans. Autom. Control 14(2), 301–305 (1969)MathSciNetGoogle Scholar
  3. 3.
    Ji, Y., Chizeck, H.J.: Controllability, stability, and continuous-time Markovian jump linear quadratic control. IEEE Trans. Autom. Control 35(7), 777–788 (1990)CrossRefMATHGoogle Scholar
  4. 4.
    Lakshmikantham, V., Leela, S., Martynyuk, A.A.: Practical Stability of Nonlinear Systems. World Scientific, Singapore (1990)CrossRefMATHGoogle Scholar
  5. 5.
    LaSalle, J.P., Lefschetz, S.: Stability by Liapunov’s Direct Method with Applications. Academic Press, New York (1961)MATHGoogle Scholar
  6. 6.
    Luo, J.W., Zou, J.Z., Hou, Z.T.: Comparison principle and stability criteria for stochastic differential delay equations with Markovian switching. Sci. China 46(1), 129–138 (2003)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Mahmoud, S.M., Shi, Y., Nounou, H.N.: Resilient observer-based control of uncertain time-delay systems. Int. J. Innov. Comput. Inf. Control 3(2), 407–418 (2007)Google Scholar
  8. 8.
    Malek-Zavarei, M., Jamshidi, M.: Time-Delay Systems: Analysis Optimization and Applications. Elsevier Science, New York (1987)MATHGoogle Scholar
  9. 9.
    Mao, X.: Stability of stochastic differential equations with Markovian switching. Stoch. Process. Appl. 79(1), 45–67 (1999)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Mao, X.: A note on the Lasalle-type theorems for stochastic differential delay equations. J. Math. Anal. Appl. 268(1), 125–142 (2002)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Mariton, M.: Jump Linear Systems in Automatic Control. Marcel Dekker, New York (1990)Google Scholar
  12. 12.
    Mariton, M., Bertrand, P.: Output feedback for a class of linear systems with stochastic jump parameters. IEEE Trans. Autom. Control 30(9), 898–900 (1985)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Martynyuk, A.: Methods and problems of practical stability of motion theory. Zagadnienia Drgan Nieliniowych 22, 9–46 (1984)Google Scholar
  14. 14.
    Martynyuk, A., Sun, Z.Q.: Practical Stability and Applications (in Chinese). Science Press, Beijing (2004)Google Scholar
  15. 15.
    Ross, D.W., Flugge-Lotz, I.: An optimal control problem for systems with differential-difference equation dynamics. SIAM J. Control 7(4), 609–623 (1969)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Sathananthan, S., Keel, L.H.: Optimal practical stabilization and controllability of systems with Markovian jumps. Nonlinear Anal. 54(6), 1011–1027 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Shi, P., Boukas, E.K.: \(H_{\infty }\) control for Markovian jump linear systems with parametric uncertainty. J. Optim. Theory Appl. 95(1), 75–99 (1997)Google Scholar
  18. 18.
    Shi, P., Boukas, E.K., Agarwal, R.K.: Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay. IEEE Trans. Autom. Control 44(11), 2139–2144 (1999)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Wang, Z.D., Lam, J., Liu, X.H.: Exponential filtering for uncertain Markovian jump time-delay systems with nonlinear disturbances. IEEE Trans. Circuits Syst. II Express Briefs 51(5), 262–268 (2004)CrossRefGoogle Scholar
  20. 20.
    Wang, Z.D., Qiao, H., Burnham, K.J.: On stabilization of bilinear uncertain time-delay stochastic systems with Markovian jumping parameters. IEEE Trans. Autom. Control 47(4), 640–646 (2002)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    You, B.L.: Complemental Tutorial for Ordinary Differential Equations(in Chinese). Higher Education Press, Beijing (1981)Google Scholar
  22. 22.
    Yuan, C., Mao, X.: Asymptotic stability in distribution of stochastic differential equations with Markovian switching. Stoch. Process. Appl. 103(2), 277–291 (2003)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd and Science Press, Beijing 2018

Authors and Affiliations

  1. 1.University of Science and Technology of ChinaHefeiChina
  2. 2.Zhejiang University of TechnologyHangzhouChina
  3. 3.Jinan UniversityJinanChina

Personalised recommendations