Divergent Stability Conditions of Dynamic Systems

1. Afanas’ev, V.N., Kolmanovskii, V.B., and Nosov, V.R., Matematicheskaya teoriya konstruirovaniya sistem upravleniya (Mathematical Theory of Control Systems Design), Moscow: Vysshaya Shkola, 2003.

   Google Scholar 

2. Lyapunov, A.M., Obshchaya zadacha ob ustoichivosti dvizheniya (General Problem on the Stability of Motion), Moscow: GITTL, 1950.

   Google Scholar 

3. Chetaev, N.G., Ustoichivost’ dvizheniya (Stability of Motion), Moscow, 1955.

4. Letov, A.M., Ustoichivost’ nelineinykh reguliruemykh sistem (Stability of Nonlinear Controllable Systems), Moscow: Fizmatgiz, 1962.

   Google Scholar 

5. Malkin, I., Teoriya ustoichivosti dvizheniya (Theory of Motion Stability), Moscow: Nauka, 1966.

   Google Scholar 

6. Zubov, V.I., Ustoichivost’ dvizheniya. Metody Lyapunova i ikh primenenie (Stability of Motion. Lyapunov’s Methods and Their Application), Moscow: Vysshaya Shkola, 1984.

   Google Scholar 

7. Rumyantsev, V.V. and Oziraner, A.S., Ustoichivost’ i stabilizatsiya dvizheniya po otnosheniyu k chasti peremennykh (Stability and Stabilization with Respect to Part of Variables), Moscow: Nauka, 1987.

   MATH  Google Scholar 

8. Yuan, R., Ma, Y.-A., Yuan, B., and Ao, P., Lyapunov Function as Potential Function: A Dynamical Equivalence, Chin. Phys. B, 2014, vol. 23, no. 1, p. 010505.

   Article  Google Scholar 

9. Bikdash, M.U. and Layton, R.A., An Energy-Based Lyapunov Function for Physical Systems, IFAC Proc., 2000, vol. 33, no. 2, pp. 81–86.

   Article  Google Scholar 

10. Willems, J.C., Dissipative Dynamical Systems. Part I: General Theory. Part II: Linear Systems with Quadratic Supply Rates, Arch. Rational Mech. Anal., 1972, vol. 45, no. 5, pp. 321–393.

    Article  MathSciNet  Google Scholar 

11. Zaremba, S.K., Divergence of Vector Fields and Differential Equations, Am. J. Math., 1954, vol. LXXV, pp. 220–234.

    Article  MathSciNet  Google Scholar 

12. Fronteau, J., Le théorèm de Liouville et le problèm général de la stabilité, Genève: CERN, 1965.

13. Brauchli, H.I., Index, Dergenz und Stabilität in Autonomen equations, Zürich: Abhandlung Verlag, 1968.

    MATH  Google Scholar 

14. Shestakov, A.A. and Stepanov, A.N., Index and Divergent Marks of Stability for a Singular Point of an Autonomous System of Differential Equations, Differ. Uravn., 1979, vol. 15, no. 4, pp. 650–661.

    Google Scholar 

15. Masina, O.N. and Druzhinina, O.V., Modelirovanie i analiz ustoichivosti nekotorykh klassov sistem upravleniya (Modeling and Analysis of Some Classes of Control Systems), Moscow: Vychisl. Tsentr Ross. Akad. Nauk, 2011.

    Google Scholar 

16. Druzhinina, O.V., Indeks, divergentsiya i funktsii Lyapunova v kachestvennoi teorii dinamicheskikh sistem (Index, Divergence and Lyapunov Functions in Qualitative Theory of Dynamic Systems), Moscow: URSS, 2013.

    Google Scholar 

17. Rantzer, A. and Parrilo, P.A., On Convexity in Stabilization of Nonlinear Systems, Proc. 39th IEEE Conf. on Decision and Control, Sydney, Australia, 2000, pp. 2942–2946.

18. Rantzer, A., A Dual to Lyapunov’s Stability Theorem, Syst. Control Lett., 2001, vol. 42, pp. 161–168.

    Article  MathSciNet  Google Scholar 

19. Zhukov, V.P., On One Method for Qualitative Study of Nonlinear System Stability, Autom. Remote Control, 1978, vol. 39, no. 6, pp. 785–788.

    MATH  Google Scholar 

20. Zhukov, V.P., On the Method of Sources for Studying the Stability of Nonlinear Systems, Autom. Remote Control, 1979, vol. 40, no. 3, pp. 330–335.

    MATH  Google Scholar 

21. Zhukov, V.P., Necessary and Sufficient Conditions for Instability of Nonlinear Autonomous Dynamic Systems, Autom. Remote Control, 1990, vol. 51, no. 12, pp. 1652–1657.

    MATH  Google Scholar 

22. Krasnosel’skii, M.A., Perov, A.I., Povolotskii, A.I., and Zabreiko, P.P., Vektornye polya na ploskosti (Vector Fields on the Plane), Moscow: Fizmatlit, 1963.

    Google Scholar 

23. Zhukov, V.P., On the Divergence Conditions for the Asymptotic Stability of Second-Order Nonlinear Dynamical Systems, Autom. Remote Control, 1999, vol. 60, no. 7, pp. 934–940.

    MathSciNet  MATH  Google Scholar 

24. Monzon, P., On Necessary Conditions for Almost Global Stability, IEEE Trans. Autom. Control, 2003, vol. 48, no. 4, pp. 631–634.

    Article  MathSciNet  Google Scholar 

25. Loizou, S.G. and Jadbabaie, A., Density Functions for Navigation-Function-Based Systems, IEEE Trans. Autom. Control, 2008, vol. 53, no. 2, pp. 612–617.

    Article  MathSciNet  Google Scholar 

26. Castañeda, Á. and Robledo, G., Differentiability of Palmer’s Linearization Theorem and Converse Result for Density Functions, J. Diff. Equat., 2015, vol. 259, no. 9, pp. 4634–4650.

    Article  MathSciNet  Google Scholar 

27. Karabacak, Ö., Wisniewski, R., and Leth, J., On the Almost Global Stability of Invariant Sets, Proc. 2018 Eur. Control Conf. (ECC 2018), Limassol, Cyprus, 2018, pp. 1648–1653.

28. Furtat, I.B., Stability Analysis of Dynamic Systems Using the Properties of Flow of the Phase Vector through a Closed Convex Surface, Nauchn.-Tekh. Vestn. Inform. Tekhn. Mekh. Opt., 2013, vol. 83, no. 1, pp. 23–27.

    Google Scholar 

29. Khalil, H.K., Nonlinear Systems, Upper Saddle River: Prentice Hall, 2002, 3 ed. Translated under the title Nelineinye sistemy, Moscow-Izhevsk: Inst. Komp. Issled., 2009.

    MATH  Google Scholar 

30. Polyak, B.T. and Topunov, M.V., Suppression of Bounded Exogenous Disturbances: Output Feedback, Autom. Remote Control, 2008, vol. 69, no. 5, pp. 801–818.

    Article  MathSciNet  Google Scholar 

31. Fradkov, A.L., A Scheme of Speed Gradient and Its Application in Problems of Adaptive Control, Autom. Remote Control, 1980, vol. 40, no. 9, pp. 1333–1342.

    MATH  Google Scholar 

Download references