Linear Systems

  • Vladimir L. Kharitonov
Part of the Control Engineering book series (CONTRENGIN)


In this chapter we consider the class of neutral type linear systems with one delay. We define the fundamental matrix of such a system and present the Cauchy formula for the solution of an initial value problem. This formula is used to compute a quadratic functional with a given time derivative along the solutions of the time-delay system. It is demonstrated that this functional is defined by a special matrix valued function, which is called a Lyapunov matrix for a time-delay system. A thorough analysis of the basic properties of the matrix is included. Complete type functionals are introduced, and various applications of the functionals are discussed.


Exponential Stability Fundamental Matrix Exponential Estimate Neutral Type Special Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Ahlfors, L.V.: Complex Analysis. McGraw-Hill, New York (1979)zbMATHGoogle Scholar
  2. 2.
    Arnold, V.I.: Differential Equations. MIT Press, Cambridge, MA (1978)Google Scholar
  3. 3.
    Bellman, R., Cooke, K.L.: Differential-Difference Equations. Academic, New York (1963)zbMATHGoogle Scholar
  4. 4.
    Castelan, W.B., Infante, E.F.: On a functional equation arising in the stability theory of difference-differential equations. Q. Appl. Math. 35, 311–319 (1977)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Castelan, W.B., Infante, E.F.: A Liapunov functional for a matrix neutral difference-differential equation with one delay. J. Math. Anal. Appl. 71, 105–130 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Curtain, R.F., Pritchard, A.J.: Infinite-dimensional linear systems theory. In: Lecture Notes in Control and Information Sciences, vol. 8. Springer, Berlin (1978)Google Scholar
  7. 7.
    Datko, R.: An algorithm for computing Liapunov functionals for some differential difference equations. In: Weiss, L. (ed.) Ordinary Differential Equations, pp. 387–398. Academic, New York (1972)Google Scholar
  8. 8.
    Datko, R.: A procedure for determination of the exponential stability of certain diferential-difference equations. Q. Appl. Math. 36, 279–292 (1978)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Datko, R.: Lyapunov functionals for certain linear delay-differential equations in a Hilbert space. J. Math. Anal. Appl. 76, 37–57 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Diekmann, O., von Gils, A.A., Verduyn-Lunel, S.M., Walther, H.-O.: Delay equations, functional-, complex- and nonlinear analysis. Applied Mathematics Sciences Series, vol. 110. Springer, New York (1995)Google Scholar
  11. 11.
    Driver, R.D.: Ordinary and Delay Differential Equations. Springer, New York (1977)zbMATHCrossRefGoogle Scholar
  12. 12.
    Garcia-Lozano, H., Kharitonov, V.L.: Numerical computation of time delay Lyapunov matrices. In: 6th IFAC Workshop on Time Delay Systems, L’Aquila, Italy, 10–12 July 2006Google Scholar
  13. 13.
    Golub, G.H., van Loan, C.F.: Matrix Computations. Johns Hopkins University Press, Baltimore, MD (1983)zbMATHGoogle Scholar
  14. 14.
    Gorecki, H., Fuksa, S., Grabowski, P., Korytowski, A.: Analysis and Synthesis of Time-Delay Systems. Polish Scientific Publishers, Warsaw (1989)zbMATHGoogle Scholar
  15. 15.
    Graham, A.: Kronecker Products and Matrix Calculus with Applications. Ellis Horwood, Chichester, UK (1981)zbMATHGoogle Scholar
  16. 16.
    Gu, K.: Discretized Lyapunov functional for uncertain systems with multiple time-delay. Int. J. Contr. 72, 1436–1445 (1999)zbMATHCrossRefGoogle Scholar
  17. 17.
    Gu, K., Han, Q.-L., Luo, A.C.J., Niculescu, S.-I.: Discretized Lyapunov functional for systems with distributed delay and piecewise constant coefficients. Int. J. Control 74, 737–744 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time Delay Systems. Birkhauser, Boston, MA (2003)zbMATHCrossRefGoogle Scholar
  19. 19.
    Halanay, A.: Differential Equations: Stability, Oscillations, Time Lags. Academic, New York (1966)zbMATHGoogle Scholar
  20. 20.
    Halanay, A., Yorke, J.A.: Some new results and problems in the theory of differential-delay equatons. SIAM Rev. 31, 55–80 (1971)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1971)CrossRefGoogle Scholar
  22. 22.
    Hale, J.K., Infante, E.F., Tsen, F.S.P.: Stability in linear delay equations. J. Math. Anal. Appl. 105, 533–555 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993)Google Scholar
  24. 24.
    Hinrichsen, D., Pritchard, A.J.: Mathematical Systems Theory 1: Modelling, State Space Analysis, Stability and Robustness. Springer, Heidelberg (2005)Google Scholar
  25. 25.
    Horn, R.A., Johnson, C.A.: Matrix Analysis. Cambridge University Press, Cambridge, UK (1985)zbMATHGoogle Scholar
  26. 26.
    Huang, W.: Generalization of Liapunov’s theorem in a linear delay system. J. Math. Anal. Appl. 142, 83–94 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Infante, E.F.: Some results on the Lyapunov stability of functional equations. In: Hannsgen, K.B., Herdmn, T.L., Stech, H.W., Wheeler, R.L. (eds.) Volterra and Functional Differential Equations. Lecture Notes in Pure and Applied Mathematics, vol. 81, pp. 51–60. Marcel Dekker, New York (1982)Google Scholar
  28. 28.
    Infante, E.F., Castelan, W.V.: A Lyapunov functional for a matrix difference-differential equation. J. Differ. Equat. 29, 439–451 (1978)zbMATHCrossRefGoogle Scholar
  29. 29.
    Jarlebring, E., Vanbiervliet, J., Michiels, W.: Characterizing and computing the 2 norm of time delay systems by solwing the delay Lyapunov equation. In: Proceedings of the 49th IEEE Conference on Decision and Control (2010)Google Scholar
  30. 30.
    Kailath, T.: Linear Systems. Prentice-Hall, Engelewood Cliffs, NJ (1980)zbMATHGoogle Scholar
  31. 31.
    Kharitonov, V.L.: Robust stability analysis of time delay systems: a survey. Annu. Rev. Control 23, 185–196 (1999)Google Scholar
  32. 32.
    Kharitonov, V.L.: Lyapunov functionals and Lyapunov matrices for neutral type time-delay stystems: a single delay case. Int. J. Control 78, 783–800 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Kharitonov, V.L.: Lyapunov matrices for a class of time delay systems. Syst. Control Lett. 55, 610–617 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Kharitonov, V.L.: Lyapunov matrices for a class of neutral type time delay systems. Int. J. Control 81, 883–893 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Kharitonov, V.L.: Lyapunov matrices: Existence and uniqueness issues. Automatica 46, 1725–1729 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Kharitonov, V.L.: Lyapunov functionals and matrices. Ann. Rev. Control 34, 13–20 (2010)CrossRefGoogle Scholar
  37. 37.
    Kharitonov, V.L.: On the uniqueness of Lyapunov matrices for a time-delay system. Syst. Control Lett. 61, 397–402 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Kharitonov, V.L., Hinrichsen, D.: Exponential estimates for time delay systems. Syst. Control Lett. 53, 395–405 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Kharitonov, V.L., Mondie, S., Ochoa, G.: Frequency stability analysis of linear systems with general distributed delays. Lect. Notes Control Inf. Sci. 388, 61–71 (2009)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Kharitonov, V.L., Plischke, E.: Lyapunov matrices for time delay systems. Syst. Control Lett. 55, 697–706 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Kharitonov, V.L., Zhabko, A.P.: Robust stability of time-delay systems. IEEE Trans. Auto.. Control 39, 2388–2397 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Kharitonov, V.L., Zhabko, A.P.: Lyapunov-Krasovskii approach to robust stability analysis of time delay systems. Automatica 39, 15–20 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Kolmanovskii, V., Myshkis, A.: Applied Theory of Functional Differential Equations. Kluwer, Dordrecht, the Netherlands (1992)Google Scholar
  44. 44.
    Kolmanovskii, V.B., Nosov, V.R.: Stability of Functional Differential Equations. Mathematics in Science and Engineering, vol. 180. Academic, New York (1986)Google Scholar
  45. 45.
    Kolmogorov, A., Fomin, S.: Elements of the Theory of Functions and Functional Analysis. Greylock, Rochester, NY (1961)zbMATHGoogle Scholar
  46. 46.
    Krasovskii, N.N.: Stability of Motion. [Russian], Moscow, 1959 [English translation]. Stanford University Press, Stanford, CA (1963)Google Scholar
  47. 47.
    Krasovskii, N.N.: On using the Lyapunov second method for equations with time delay [Russian]. Prikladnaya Matematika i Mekhanika. 20, 315–327 (1956)MathSciNetGoogle Scholar
  48. 48.
    Krasovskii, N.N.: On the asymptotic stability of systems with aftereffect [Russian]. Prikladnaya Matematika i Mekhanika. 20, 513–518 (1956)MathSciNetGoogle Scholar
  49. 49.
    Lakshmikantam, V., Leela, S.: Differential and Integral Inequalities. Academic, New York (1969)Google Scholar
  50. 50.
    Levinson, N., Redheffer, R.M.: Complex Variables. Holden-Day, Baltimore, MD (1970)zbMATHGoogle Scholar
  51. 51.
    Louisel, J.: Growth estimates and asymptotic stability for a class of differential-delay equation having time-varying delay. J. Math. Anal. Appl. 164, 453–479 (1992)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Louisell, J.: Numerics of the stability exponent and eigenvalue abscissas of a matrix delay system. In: Dugard, L., Verriest, E.I. (eds.) Stability and Control of Time-delay Systems. Lecture Notes in Control and Information Sciences, vol. 228, pp. 140–157. Springer, New York (1997)Google Scholar
  53. 53.
    Louisell, J.: A matrix method for determining the imaginary axis eigenvalues of a delay system. IEEE Trans. Autom. Control 46, 2008–2012 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Malek-Zavarei, M., Jamshidi, M.: Time delay systems: analysis, optimization and applications. North-Holland Systems and Control Series, vol. 9. North-Holland, Amsterdam (1987)Google Scholar
  55. 55.
    Marshall, J.E., Gorecki, H., Korytowski, A., Walton, K.: Time-Delay Systems: Stability and Performance Criteria with Applications. Ellis Horwood, New York (1992)zbMATHGoogle Scholar
  56. 56.
    Mondie, S.: Assesing the exact stability region of the single delay scalar equation via its Lyapunov function. IMA J. Math. Control Inf. (2012). doi: ID:DNS004Google Scholar
  57. 57.
    Myshkis, A.D.: General theory of differential equations with delay [Russian]. Uspekhi Matematicheskikh Nauk. 4, 99–141 (1949)zbMATHGoogle Scholar
  58. 58.
    Niculescu, S.-I.: Delay Effects on Stability: A Robust Control Approach. Springer, Heidelberg (2001)zbMATHGoogle Scholar
  59. 59.
    Ochoa, G., Mondie, S., Kharitonov, V.L.: Time delay systems with distributed delays: critical values. In: Proceedings of the 8th IFAC Workshop on Time Delay Systems, Sinaia, Romania, 1–3 Sept 2009Google Scholar
  60. 60.
    Plishke, E.: Transient effects of linear dynamical systems. Ph.D. thesis, University of Bremen, Bremen, Germany (2005)Google Scholar
  61. 61.
    Razumikhin, B.S.: On the stability of systems with a delay [Russian]. Prikladnaya Matematika i Mekhanika. 20, 500–512 (1956)Google Scholar
  62. 62.
    Razumikhin, B.S.: Application of Liapunov’s method to problems in the stability of systems with a delay [Russian]. Automatika i Telemekhanika. 21, 740–749 (1960)Google Scholar
  63. 63.
    Repin, Yu.M.: Quadratic Lyapunov functionals for systems with delay [Russian]. Prikladnaya Matematika i Mekhanika. 29, 564–566 (1965)MathSciNetGoogle Scholar
  64. 64.
    Richard, J.-P.: Time-delay systems: an overview of some recent advances and open problems. Automatica 39, 1667–1694 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Rudin W.: Real and Complex Analysis. McGraw-Hill, New York (1973)Google Scholar
  66. 66.
    Rudin, W.: Functional Analysis. McGraw-Hill, New York (1987)Google Scholar
  67. 67.
    Stépán, G.: Retarded Dynamical Systems: Stability and Characteristic Function. Wiley, New York (1989)Google Scholar
  68. 68.
    Velazquez-Velazquez, J., Kharitonov, V.L.: Lyapunov-Krasovskii functionals for scalar neutral type time delay equation. Syst. Control Lett. 58, 17–25 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Volterra, V.: Sulle equazioni integrodifferenciali della teorie dell’elasticita. Atti. Accad. Lincei. 18, 295 (1909)zbMATHGoogle Scholar
  70. 70.
    Volterra, V.: Theorie mathematique de la lutte pour la vie [French]. Gauthier-Villars, Paris (1931)Google Scholar
  71. 71.
    Zhou, K., Doyle, J.C., Glover, K.: Robust and Optimal Control. Prentice-Hall, Upper Saddle River, NJ (1996)zbMATHGoogle Scholar
  72. 72.
    Zubov, V.I.: The Methods of A.M. Lyapunov and Their Applications. Noordhoff, Groningen, the Netherlands (1964)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  • Vladimir L. Kharitonov
    • 1
  1. 1.Faculty of Applied Mathematics and Processes of ControlSaint Petersburg State UniversitySaint PetersburgRussia

Personalised recommendations