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Stability of Dynamical Systems pp 1–16Cite as

Introduction

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Bibliography

  1. M. A. Aizerman and F. R. Gantmacher, Absolute Stability of Regulator Systems, San Francisco: Holden-Day, 1964.

    Google Scholar 

  2. P. J. Antsaklis and A. N. Michel, Linear Systems, Boston: Birkhäuser, 2006.

    Google Scholar 

  3. K. J. Åström and B. Wittenmark, Adaptive Control, 2nd Edition, New York: Addison-Wesley, 1995.

    MATH  Google Scholar 

  4. F. N. Bailey, “The application of Lyapunov’s Second Method to interconnected systems,” SIAM J. Control, vol. 3, pp. 443–462, 1966.

    Article  MathSciNet  Google Scholar 

  5. D. D. Bainov and P. S. Simeonov, Systems with Impulse Effects: Stability Theory and Applications, New York: Halsted, 1989.

    MATH  Google Scholar 

  6. A. E. Barbashin and N. N. Krasovskii, “On the stability of motion in the large,” Dokl. Akad. Nauk., vol. 86, pp. 453-456, 1952.

    MATH  Google Scholar 

  7. V. Barbu and S. I. Grossman, “Asymptotic behavior of linear integrodifferential systems,” Trans. Amer. Math. Soc., vol. 171, pp. 277–288, 1972.

    Article  MathSciNet  Google Scholar 

  8. R. Bellman, “Vector Lyapunov functions,” SIAM J. Control, vol. 1, pp. 32–34, 1962.

    Article  Google Scholar 

  9. R. Bellman and K. L. Cooke, Differential-Difference Equations, New York: Academic Press, 1963.

    MATH  Google Scholar 

  10. M. S. Branicky, “Multiple Lyapunov functions and other analysis tools for switched and hybrid systems,” IEEE Trans. Autom. Control, vol. 43, pp. 475– 482, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  11. M. G. Crandall, “Semigroups of nonlinear transformations on general Banach spaces,” Contributions to Nonlinear Functional Analysis, E. H. Zarantonello, Ed., New York: Academic Press, 1971.

    Google Scholar 

  12. M. G. Crandall and T. M. Liggett, “Generation of semigroups of nonlinear transformations on general Banach spaces,” Amer. J. Math., vol. 93, pp. 265–298, 1971.

    Article  MATH  MathSciNet  Google Scholar 

  13. J. L. Daleckii and S. G. Krein, Stability of Solutions of Differential Equations in Banach Spaces, Translations of Mathematical Monographs, vol. 43, Providence, RI: American Mathematical Society, 1974.

    Google Scholar 

  14. R. DeCarlo, M. Branicky, S. Pettersson, and B. Lennartson, “Perspectives and results on the stability and stabilizability of hybrid systems,” Proc. IEEE, vol.88, pp. 1069–1082, 2000.

    Article  Google Scholar 

  15. G. F. Franklin and J. D. Powell, Digital Control of Dynamical Systems, Reading,MA: Addison-Wesley, 1980.

    Google Scholar 

  16. A. Friedman, Partial Differential Equations of Parabolic Type, Englewood Cliffs, NJ: Prentice Hall, 1964.

    MATH  Google Scholar 

  17. P. R. Garabedian, Partial Differential Equations, New York: Chelsea, 1986.

    Google Scholar 

  18. L. T. Grujic, A. A. Martynyuk, and M. Ribbens-Pavella, Large Scale Systems Stability Under Structural and Singular Perturbations, Berlin: Springer-Verlag, 1987.

    MATH  Google Scholar 

  19. W. M. Haddad, V. S. Chellaboina, and S. G. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity and Control, Princeton, NJ: Princeton University Press, 2006.

    MATH  Google Scholar 

  20. W. Hahn, Theorie und Anwendung der direkten Methode von Ljapunov, Heidelberg: Springer-Verlag, 1959.

    MATH  Google Scholar 

  21. W. Hahn, Stability of Motion, Berlin: Springer-Verlag, 1967.

    MATH  Google Scholar 

  22. A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, New York: Academic Press, 1966.

    MATH  Google Scholar 

  23. J. K. Hale, Ordinary Differential Equations, New York: Wiley-Interscience, 1969.

    MATH  Google Scholar 

  24. J. K. Hale, Functional Differential Equations, Berlin: Springer-Verlag, 1971.

    MATH  Google Scholar 

  25. J. K. Hale, “Functional differential equations with infinite delays,” J. Math. Anal. Appl., vol. 48, pp. 276–283, 1974.

    Article  MATH  MathSciNet  Google Scholar 

  26. E. Hille and R. S. Phillips, Functional Analysis and Semigroups, Amer. Math. Soc. Colloquium Publ. vol. 33, Providence, RI:American Mathematical Society, 1957.

    Google Scholar 

  27. L. Hörmander, Linear Partial Differential Equations, Berlin: Springer-Verlag, 1963

    Google Scholar 

  28. L. Hörmander, The Analysis of Linear Partial Differential Operators, vol. 1, 2,3,4, Berlin: Springer-Verlag, 1983–1985.

    Google Scholar 

  29. L. Hou and A. N. Michel, “Stability analysis of pulse-width-modulated feedback systems,” Automatica, vol. 37, pp. 1335–1349, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  30. L. Hou, A. N. Michel, and H. Ye, “Some qualitative properties of sampled-data control systems,” IEEE Trans. Autom. Control, vol. 42, pp. 1721–1725, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  31. P. A. Ioannou and J. Sun, Robust Adaptive Control, Upper Saddle River, NJ: Prentice Hall, 1996.

    MATH  Google Scholar 

  32. R. E. Kalman, “Lyapunov functions for the problem of Luré in automatic control,” Proc. Nat. Acad. Sci. USA, vol. 49, pp. 201–205, 1963.

    Article  MATH  MathSciNet  Google Scholar 

  33. E. Kamke, “Zur Theorie der Systeme gewöhnlicher Differentialgleichungen, II,” Acta Math., vol. 58, pp. 57–85, 1932.

    Article  MathSciNet  Google Scholar 

  34. H. K. Khalil, Nonlinear Systems, New York: Macmillan, 1992.

    MATH  Google Scholar 

  35. N. N. Krasovskii, Stability of Motion, Stanford, CA: Stanford University Press, 1963.

    MATH  Google Scholar 

  36. S. G. Krein, Linear Differential Equations in Banach Spaces, Translation of Mathematical Monographs, vol. 29, Providence, RI: American Mathematical Society, 1970.

    Google Scholar 

  37. V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, vol. 1 and 2, New York: Academic Press, 1969.

    Google Scholar 

  38. J. P. LaSalle, “The extent of asymptotic stability,” Proc. Nat. Acad. Sci., vol. 48, pp. 363–365, 1960.

    Article  MathSciNet  Google Scholar 

  39. J. P. LaSalle, “Some extensions of Liapunov’s Second Method,” IRE Trans. Circ. Theor., vol. 7, pp. 520–527, 1960.

    MathSciNet  Google Scholar 

  40. J. P. LaSalle, The Stability and Control of Discrete Processes, New York: Springer-Verlag, 1986.

    MATH  Google Scholar 

  41. J. P. LaSalle and S. Lefschetz, Stability of Lyapunov’s Direct Method, NewYork:Academic Press, 1961.

    Google Scholar 

  42. S. Lefschetz, Stability of Nonlinear Control Systems, New York: Academic Press, 1965.

    MATH  Google Scholar 

  43. D. Liberzon and A. S. Morse, “Basic problems in stability and design of switched systems,” IEEE Control Syst. Mag., vol. 19, pp. 59–70, 1999.

    Article  Google Scholar 

  44. A. I. Luré and V. N. Postnikov, “On the theory of stability of control systems,” Prikl. Mat. i. Mehk., vol. 8, pp. 3–13, 1944.

    Google Scholar 

  45. A. M. Liapounoff, “Problème générale de la stabilité de mouvement,” Annales de la Faculté des Sciences de l’Université de Toulouse, vol. 9, pp. 203–474, 1907. (Translation of a paper published in Comm. Soc. Math., Kharkow, 1892, reprinted in Ann. Math. Studies, vol. 17, Princeton, NJ: Princeton, 1949. The French version was translated into English by A. T. Fuller, and was published in the International Journal of Control, vol. 55, pp. 531–773, 1992.)

    Google Scholar 

  46. I. G. Malkin, “On the question of reversibility of Lyapunov’s theorem on automatic stability,” Prikl. Mat. i Mehk., vol.18, pp. 129–138, 1954 (in Russian).

    MATH  Google Scholar 

  47. J. L. Massera, “On Liapunoff’s conditions of stability,” Ann. Mat., vol. 50, pp. 705–721, 1949.

    Article  MathSciNet  Google Scholar 

  48. J. L. Massera, “Contributions to stability theory,” Ann. Mat., vol. 64, pp. 182–206, 1956.

    Article  MathSciNet  Google Scholar 

  49. V. M. Matrosov, “The method of Lyapunov-vector functions in feedback systems,” Automat. Remote Control, vol. 33, pp. 1458–1469, 1972.

    MATH  MathSciNet  Google Scholar 

  50. A. N. Michel, “Recent trends in the stability analysis of hybrid dynamical systems,” IEEE Trans. Circ. and Syst.–I: Fund. Theor. Appl., vol. 46, pp. 120–134, 1999.

    Article  MATH  Google Scholar 

  51. A. N. Michel and B. Hu, “Towards a stability theory of general hybrid dynamical systems,” Automatica, vol. 35, pp. 371–384, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  52. A. N. Michel and D. Liu, Qualitative Analysis and Synthesis of Recurrent Neural Networks, New York: Marcel Dekker, 2002.

    MATH  Google Scholar 

  53. A. N. Michel and R. K. Miller, Qualitative Analysis of Large Scale Dynamical Systems, New York: Academic Press, 1977.

    MATH  Google Scholar 

  54. A. N. Michel and Y. Sun, “Stability of discontinuous Cauchy problems in Banach space,” Nonlinear Anal., vol. 65, pp. 1805–1832, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  55. A. N. Michel, Y. Sun, and A. P. Molchanov, “Stability analysis of discontinuous dynamical systems determined by semigroups,” IEEE Trans. Autom. Control, vol. 50, pp. 1277–1290, 2005.

    Article  MathSciNet  Google Scholar 

  56. A. N. Michel and K.Wang, “Robust stability: perturbed systems with perturbed equilibria,” Syst. and Control Lett., vol. 21, pp. 155–162, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  57. A. N. Michel, K. Wang, and B. Hu, Qualitative Theory of Dynamical Systems: The Role of Stability Preserving Mappings, 2nd Edition, New York: Marcel Dekker, 2001.

    MATH  Google Scholar 

  58. R. K. Miller, Nonlinear Volterra Integral Equations, Menlo Park, CA: W. A. Benjamin, 1971.

    MATH  Google Scholar 

  59. R. K. Miller and A. N. Michel, Ordinary Differential Equations, New York: Academic Press, 1982.

    MATH  Google Scholar 

  60. M. Müller, “Über das Fundamentaltheorem in der Theorie der gewöhnlichen Differentialgleichungen,” Mat. Zeit., vol. 26, pp. 615–645, 1926.

    Google Scholar 

  61. K. S. Narendra and J. H. Taylor, Frequency Domain Stability for Absolute Stability, New York: Academic Press, 1973.

    MATH  Google Scholar 

  62. M. A. Pai, Power System Stability, New York: North-Holland, 1981.

    MATH  Google Scholar 

  63. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, New York: Springer-Verlag, 1983.

    MATH  Google Scholar 

  64. V. M. Popov, “Absolute stability of nonlinear systems of automatic control,” Automation Remote Control, vol. 22, pp. 857–875, 1961.

    MATH  Google Scholar 

  65. G. R. Sell, Lectures on Topological Dynamics and Differential Equations, Princeton, NJ: Van Nostrand, 1969.

    Google Scholar 

  66. D. D. Siljak, Large-Scale Dynamical Systems: Stability and Structure, New York: North Holland, 1978.

    Google Scholar 

  67. Y. Sun, A. N. Michel, and G. Zhai, “Stability of discontinuous retarded functional differential equations with applications,” IEEE Trans. Autom. Control, vol. 50, pp. 1090–1105, 2005.

    Article  MathSciNet  Google Scholar 

  68. M.Vidyasagar, Nonlinear Systems Analysis, 2nd Edition, Englewood Cliffs, NJ: Prentice Hall, 1993.

    MATH  Google Scholar 

  69. W.Walter, Differential and Integral Inequalities, Berlin: Springer-Verlag, 1970.

    MATH  Google Scholar 

  70. K. Wang and A. N. Michel, “On sufficient conditions for stability of interval matrices,” Syst. Control Lett., vol. 20, pp. 345–351, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  71. K. Wang and A. N. Michel, “Qualitative analysis of dynamical systems determined by differential inequalities with applications to robust stability,” IEEE Trans. Circ. Syst.–I: Fund. Theor. Appl., vol. 41, pp. 377–386, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  72. K.Wang, A. N. Michel, and D. Liu, “Necessary and sufficient conditions for the Hurwitz and Schur stability of interval matrices,” IEEE Trans. Autom. Control, vol. 39, pp. 1251–1255, 1996.

    Article  MathSciNet  Google Scholar 

  73. T.Wazewski, “Systemés des equations et des inégalités différentielles ordinaires aux deuxiémes membres monotones et leurs applications,” Ann. Soc. Poln. Mat., vol. 23, pp. 112–166, 1950.

    MATH  MathSciNet  Google Scholar 

  74. V.A.Yacubovich, “Solution of certain matrix inequalities occurring in the theory of automatic control,” Dokl. Acad. Nauk. SSSR, pp. 1304–1307, 1962.

    Google Scholar 

  75. H. Ye, A. N. Michel, and L. Hou, “Stability theory for hybrid dynamical systems,” IEEE Trans. Autom. Control, vol. 43, pp. 461–474, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  76. H. Ye, A. N. Michel, and L. Hou, “Stability analysis of systems with impulse effects,” IEEE Trans. Autom. Control, vol. 43, pp. 1719–1923, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  77. H. Ye, A. N. Michel, and K. Wang, “Robust stability of nonlinear time-delay systems with applications to neural networks,” IEEE Trans. Circ. Syst.–I: Fund. Theor. Appl., vol. 43, pp. 532–543, 1996.

    Article  MathSciNet  Google Scholar 

  78. T. Yoshizawa, Stability Theory by Lyapunov’s Second Method, Tokyo: Math. Soc. Japan, 1966.

    Google Scholar 

  79. G. Zames, “Input-output feedback stability and robustness, 1959–85,” IEEE Control Syst. Mag., vol. 16, pp. 61–66, 1996.

    Article  Google Scholar 

  80. V. I. Zubov, Methods of A. M. Lyapunov and Their Applications, Groningen, The Netherlands: P. Noordhoff, 1964.

    Google Scholar 

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Authors and Affiliations

  1. Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, U.S.A

    Anthony N. Michel

  2. Department of Electrical and Computer Engineering, St. Cloud State University, St. Cloud, MN 56301, U.S.A

    Ling Hou

  3. Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, IL 60607, U.S.A

    Derong Liu

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  1. Anthony N. Michel
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  3. Derong Liu
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Michel, A.N., Hou, L., Liu, D. (2008). Introduction. In: Stability of Dynamical Systems. Systems&Control: Foundations&Applications. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4649-3_1

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