Skip to main content
SpringerLink
Log in

Introduction

  • Chapter
Stability of Dynamical Systems

Part of the book series: Systems & Control: Foundations & Applications ((SCFA))

  • 2397 Accesses

Abstract

We summarize the aims and scope of the book and we give an outline of its contents. We also present a brief perspective on the development of stability theory.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 59.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Bibliography

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

    Google Scholar 

  2. P.J. Antsaklis, A.N. Michel, Linear Systems (Birkhäuser, Boston, 2006)

    MATH  Google Scholar 

  3. K.J. Åström, B. Wittenmark, Adaptive Control, 2nd edn. (Addison-Wesley, New York, 1995)

    MATH  Google Scholar 

  4. F.N. Bailey, The application of Lyapunov’s second method to interconnected systems. SIAM J. Control 3, 443–462 (1966)

    Google Scholar 

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

    MATH  Google Scholar 

  6. A.E. Barbashin, N.N. Krasovskii, On the stability of motion in the large. Dokl. Akad. Nauk. 86, 453–456 (1952)

    MATH  Google Scholar 

  7. V. Barbu, S.I. Grossman, Asymptotic behavior of linear integrodifferential systems. Trans. Am. Math. Soc. 171, 277–288 (1972)

    Article  MathSciNet  Google Scholar 

  8. R. Bellman, Vector Lyapunov functions. SIAM J. Control 1, 32–34 (1962)

    MATH  Google Scholar 

  9. R. Bellman, K.L. Cooke, Differential-Difference Equations (Academic, New York, 1963)

    Book  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. M.G. Crandall, Semigroups of nonlinear transformations on general Banach spaces, in Contributions to Nonlinear Functional Analysis, ed. by E.H. Zarantonello (Academic, New York, 1971)

    Google Scholar 

  12. M.G. Crandall, T.M. Liggett, Generation of semigroups of nonlinear transformations on general Banach spaces. Am. J. Math. 93, 265–298 (1971)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  15. G.F. Franklin, J.D. Powell, Digital Control of Dynamical Systems (Addison-Wesley, Reading, 1980)

    Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Book  Google Scholar 

  20. W. Hahn, Theorie und Anwendung der direkten Methode von Ljapunov (Springer, Heidelberg, 1959)

    Book  MATH  Google Scholar 

  21. W. Hahn, Stability of Motion (Springer, Berlin, 1967)

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  24. J.K. Hale, Functional Differential Equations (Springer, Berlin, 1971)

    Book  MATH  Google Scholar 

  25. J.K. Hale, Functional differential equations with infinite delays. J. Math. Anal. Appl. 48, 276–283 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  26. E. Hille, R.S. Phillips, Functional Analysis and Semigroups. American Mathematical Society Colloquium Publications, vol. 33 (American Mathematical Society, Providence, 1957)

    Google Scholar 

  27. L. Hörmander, Linear Partial Differential Equations (Springer, Berlin, 1963)

    Google Scholar 

  28. L. Hörmander, The Analysis of Linear Partial Differential Operators, vols. 1–4 (Springer, Berlin, 1983–1985)

    Google Scholar 

  29. L. Hou, A.N. Michel, Stability analysis of pulse-width-modulated feedback systems. Automatica 37, 1335–1349 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  30. L. Hou, A.N. Michel, Unifying theory for stability of continuous, discontinuous and discrete-time dynamical systems. Nonlinear Anal. Hybrid Syst. 1(2), 154–172 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  31. L. Hou, A.N. Michel, On the continuity of Lyapunov functions in the converse stability theorems for discontinuous dynamical systems. Nonlinear Anal. Hybrid Syst. 2(4), 1021–1029 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  32. L. Hou, A.N. Michel, H. Ye, Some qualitative properties of sampled-data control systems. IEEE Trans. Autom. Control 42, 1721–1725 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  33. P.A. Ioannou, J. Sun, Robust Adaptive Control (Prentice Hall, Upper Saddle River, 1996)

    MATH  Google Scholar 

  34. R.E. Kalman, Lyapunov functions for the problem of Luré in automatic control. Proc. Natl. Acad. Sci. USA 49, 201–205 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  35. E. Kamke, Zur Theorie der Systeme gewöhnlicher Differentialgleichungen, II. Acta Math. 58, 57–85 (1932)

    Article  MathSciNet  Google Scholar 

  36. H.K. Khalil, Nonlinear Systems (Macmillan, New York, 1992)

    MATH  Google Scholar 

  37. N.N. Krasovskii, Stability of Motion (Stanford University Press, Stanford, 1963)

    MATH  Google Scholar 

  38. S.G. Krein, Linear Differential Equations in Banach Spaces. Translation of Mathematical Monographs, vol. 29 (American Mathematical Society, Providence, 1970)

    Google Scholar 

  39. V. Lakshmikantham, S. Leela, Differential and Integral Inequalities, vols. 1 and 2 (Academic, New York, 1969)

    Google Scholar 

  40. J.P. LaSalle, The extent of asymptotic stability. Proc. Natl. Acad. Sci. USA 48, 363–365 (1960)

    Article  MathSciNet  Google Scholar 

  41. J.P. LaSalle, Some extensions of Lyapunov’s second method. IRE Trans. Circuit Theory 7, 520–527 (1960)

    Article  MathSciNet  Google Scholar 

  42. J.P. LaSalle, The Stability and Control of Discrete Processes (Springer, New York, 1986)

    MATH  Google Scholar 

  43. J.P. LaSalle, S. Lefschetz, Stability of Lyapunov’s Direct Method (Academic, New York, 1961)

    Google Scholar 

  44. S. Lefschetz, Stability of Nonlinear Control Systems (Academic, New York, 1965)

    MATH  Google Scholar 

  45. D. Liberzon, A.S. Morse, Basic problems in stability and design of switched systems. IEEE Control Syst. Mag. 19, 59–70 (1999)

    Article  Google Scholar 

  46. A.I. Luré, V.N. Postnikov, On the theory of stability of control systems. Prikl. Mat. i. Mehk. 8, 3–13 (1944)

    Google Scholar 

  47. A.M. Liapounoff, Problème générale de la stabilité de mouvement. Ann. Fac. Sci. Univ. Toulouse 9, 203–474 (1907). Translation of a paper published in Communications of the Mathematical Society Kharkow, 1892, reprinted in Annals of Mathematics Studies, vol. 17 (Princeton, Princeton, 1949) The French version was translated into English by A.T. Fuller, and was published in the Int. J. Control 55, 531–773 (1992)

    Google Scholar 

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

    MATH  Google Scholar 

  49. J.L. Massera, On Liapunoff’s conditions of stability. Ann. Mat. 50 705–721 (1949)

    Article  MATH  MathSciNet  Google Scholar 

  50. J.L. Massera, Contributions to stability theory. Ann. Mat. 64, 182–206 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  51. V.M. Matrosov, The method of Lyapunov-vector functions in feedback systems. Autom. Remote Control 33, 1458–1469 (1972)

    MATH  MathSciNet  Google Scholar 

  52. A.N. Michel, Recent trends in the stability analysis of hybrid dynamical systems. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 46, 120–134 (1999)

    Article  MATH  Google Scholar 

  53. A.N. Michel, L. Hou, Stability of dynamical systems with discontinuous motions: beyond classical Lyapunov stability results. SICE J. Control Meas. Syst. Integr. 1(6), 411–422 (2008)

    Article  Google Scholar 

  54. A.N. Michel, L. Hou, Stability results involving time-averaged Lyapunov function derivatives. Nonlinear Anal. Hybrid Syst. 3(1), 51–64 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  55. A.N. Michel, L. Hou, Stability results for finite-dimensional discrete-time dynamical systems involving non-monotonic Lyapunov functions, in Proceedings of the 2010 American Control Conference, Baltimore, June 2010, pp. 2682–2687

    Google Scholar 

  56. A.N. Michel, L. Hou, Relaxation of hypotheses in LaSalle-Krasovskii-type invariance results. SIAM J. Control Optim. 49(4), 1383–1403 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  57. A.N. Michel, L. Hou, Stability theory of continuous-time dynamical systems involving non-monotonic Lyapunov functions. Commun. Appl. Anal. 17, 395–426 (2013)

    MATH  MathSciNet  Google Scholar 

  58. A.N. Michel, L. Hou, Stability theory of discrete-time dynamical systems involving non-monotonic Lyapunov functions. Nonlinear Stud. 21(1), 53–75 (2014)

    MATH  MathSciNet  Google Scholar 

  59. A.N. Michel, B. Hu, Towards a stability theory of general hybrid dynamical systems. Automatica 35, 371–384 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  61. A.N. Michel, R.K. Miller, Qualitative Analysis of Large Scale Dynamical Systems (Academic, New York, 1977)

    MATH  Google Scholar 

  62. A.N. Michel, Y. Sun, Stability of discontinuous Cauchy problems in Banach space. Nonlinear Anal. 65, 1805–1832 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  63. A.N. Michel, Y. Sun, A.P. Molchanov, Stability analysis of discontinuous dynamical systems determined by semigroups. IEEE Trans. Autom. Control 50, 1277–1290 (2005)

    Article  MathSciNet  Google Scholar 

  64. A.N. Michel, K. Wang, Robust stability: perturbed systems with perturbed equilibria. Syst Control Lett. 21, 155–162 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  65. A.N. Michel, K. Wang, B. Hu, Qualitative Theory of Dynamical Systems: The Role of Stability Preserving Mappings, 2nd edn. (Marcel Dekker, New York, 2001)

    Book  Google Scholar 

  66. R.K. Miller, Nonlinear Volterra Integral Equations (W.A. Benjamin, Menlo Park, 1971)

    MATH  Google Scholar 

  67. R.K. Miller, A.N. Michel, Asymptotic stability of systems: Results involving the system topology. SIAM J. Control Optim. 18(2), 181–190 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  68. R.K. Miller, A.N. Michel, Ordinary Differential Equations (Dover, Mineola, 2007)

    MATH  Google Scholar 

  69. M. Müller, Über das Fundamentaltheorem in der Theorie der gewöhnlichen Differentialgleichungen. Mat. Zeit. 26, 615–645 (1926)

    Google Scholar 

  70. K.S. Narendra, J.H. Taylor, Frequency Domain Stability for Absolute Stability (Academic, New York, 1973)

    MATH  Google Scholar 

  71. M.A. Pai, Power System Stability (North-Holland, New York, 1981)

    MATH  Google Scholar 

  72. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer, New York, 1983)

    MATH  Google Scholar 

  73. V.M. Popov, Absolute stability of nonlinear systems of automatic control. Autom. Remote Control 22, 857–875 (1961)

    Google Scholar 

  74. G.R. Sell, Lectures on Topological Dynamics and Differential Equations (Van Nostrand, Princeton, 1969)

    Google Scholar 

  75. R. Shorten, F. Wirth, O. Mason, K. Wulff, C. King, Stability criteria for switched and hybrid systems. SIAM Rev. 49(4), 545–592 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  77. Y. Sun, A.N. Michel, G. Zhai, Stability of discontinuous retarded functional differential equations with applications. IEEE Trans. Autom. Control 50, 1090–1105 (2005)

    Article  MathSciNet  Google Scholar 

  78. M. Vidyasagar, Nonlinear Systems Analysis, 2nd edn. (Prentice Hall, Englewood Cliffs, 1993)

    MATH  Google Scholar 

  79. W. Walter, Differential and Integral Inequalities (Springer, Berlin, 1970)

    Book  MATH  Google Scholar 

  80. K. Wang, A.N. Michel, On sufficient conditions for stability of interval matrices. Syst. Control Lett. 20, 345–351 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  81. K. Wang, A.N. Michel, Qualitative analysis of dynamical systems determined by differential inequalities with applications to robust stability. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 41, 377–386 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  82. K. Wang, A.N. Michel, D. Liu, Necessary and sufficient conditions for the Hurwitz and Schur stability of interval matrices. IEEE Trans. Autom. Control 39, 1251–1255 (1996)

    Article  MathSciNet  Google Scholar 

  83. T. Wazewski, Systemés des equations et des inégalités différentielles ordinaires aux deuxiémes membres monotones et leurs applications. Ann. Soc. Poln. Mat. 23, 112–166 (1950)

    MATH  MathSciNet  Google Scholar 

  84. V.A. Yacubovich, Solution of certain matrix inequalities occurring in the theory of automatic control. Dokl. Acad. Nauk. SSSR 144, 1304–1307 (1962)

    Google Scholar 

  85. H. Ye, A.N. Michel, L. Hou, Stability theory for hybrid dynamical systems. IEEE Trans. Autom. Control 43, 461–474 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  86. H. Ye, A.N. Michel, L. Hou, Stability analysis of systems with impulse effects. IEEE Trans. Autom. Control 43, 1719–1923 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  87. H. Ye, A.N. Michel, K. Wang, Robust stability of nonlinear time-delay systems with applications to neural networks. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 43, 532–543 (1996)

    Article  MathSciNet  Google Scholar 

  88. T. Yoshizawa, Stability Theory by Lyapunov’s Second Method (Mathematical Society of Japan, Tokyo, 1966)

    Google Scholar 

  89. G. Zames, Input-output feedback stability and robustness, 1959–85. IEEE Control Syst. Mag. 16, 61–66 (1996)

    Article  Google Scholar 

  90. V.I. Zubov, Methods of A. M. Lyapunov and Their Applications (P. Noordhoff, Groningen, 1964)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN, USA

    Anthony N. Michel

  2. Department of Electrical and Computer Engineering, St. Cloud State University, St. Cloud, MN, USA

    Ling Hou

  3. Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, IL, USA

    Derong Liu

Authors
  1. Anthony N. Michel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ling Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Derong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Michel, A.N., Hou, L., Liu, D. (2015). Introduction. In: Stability of Dynamical Systems. Systems & Control: Foundations & Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-15275-2_1

Download citation

Publish with us

Policies and ethics

Search

Navigation