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The Second Lyapunov Method for Time-Delay Systems

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Functional Differential Equations and Applications (FDEA 2019)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 379))

Abstract

Some classes of systems of delay differential equations are considered. We give a review of methods for the study of the stability of solutions in the case of constant and periodic coefficients in linear terms. Special attention is paid to the development of the second Lyapunov method. A number of authors’ results for linear and nonlinear delay differential equations obtained by using various Lyapunov–Krasovskii functionals are presented. The application of discrete analogs of the constructed functionals to the study of the stability of solutions to delay difference equations is discussed.

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References

  1. Agarwal, R.P., Berezansky, L., Braverman, E., Domoshnitsky, A.: Nonoscillation Theory of Functional Differential Equations with Applications. Springer, New York (2012)

    Book  MATH  Google Scholar 

  2. Aleksenko, N.V., Romanovskii, R.K.: The method of Lyapunov functionals for linear difference-differential systems with almost-periodic coefficients. Differ. Equ. 37, 159–165 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Andronov, A.A., Mayer, A.G.: Simplest linear delay systems (in Russian). Autom. Remote. Control. 7, 95–106 (1946)

    Google Scholar 

  4. Azbelev, N.V., Maksimov, V.P., Rakhmatullina, L.F.: Introduction to the Theory of Functional Differential Equations. Methods and Applications. Hindawi Publishing Corporation, New York (2007)

    Google Scholar 

  5. Azbelev, N.V., Simonov, P.M.: Stability of Solutions to Equations with Ordinary Derivatives. Izd. Perm University, Perm (2001).(in Russian)

    Google Scholar 

  6. Arutyunyan, N.Kh., Kolmanovskii, V.B.: Creep Theory of Inhomogeneous Bodies. Nauka, Moscow (1983).(in Russian)

    Google Scholar 

  7. Baštinec, J., Diblík, J., Khusainov, D.Ya., Ryvolová, A.: Exponential stability and estimation of solutions of linear differential systems of neutral type with constant coefficients. Bound. Value Probl. Art. ID 956121, 1–20 (2010)

    Google Scholar 

  8. Bellman, R.: Stability Theory of Differential Equations. McGraw-Hill Book Company, New York, Toronto, London (1953)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  10. Belotserkovskii, S.M., Kochetkov, Yu.A., Krasovskii, A.A., Novitskii, V.V.: Introduction to Aeroautoelasticity. Nauka, Moscow (1980).(in Russian)

    Google Scholar 

  11. Belykh, L.N.: Analysis of Mathematical Models in Immunology. Nauka, Moscow (1988).(in Russian)

    MATH  Google Scholar 

  12. Berezanskii, L.M.: Development of N.V. Azbelev’s W-method in problems of the stability of solutions of linear functional-differential equations. Differ. Equ. 22, 521–529 (1986)

    Google Scholar 

  13. Chebotarev, N.G., Meyman, N.N.: The Routh-Hurwitz problem for polynomials and entire functions (in Russian). Tr. Mat. Inst. Steklova. 26, 3–331 (1949)

    Google Scholar 

  14. Daleckii, Ju.L., Krein, M.G.: Stability of Solutions to Differential Equations in Banach Space. American Mathematical Society, Providence (1974)

    Google Scholar 

  15. Day, W.A.: The Thermodynamics of Simple Materials with Fading Memory. Springer, New York, Heidelberg (1972)

    Book  MATH  Google Scholar 

  16. Demidenko, G.V.: Stability of solutions to linear differential equations of neutral type. J. Anal. Appl. 7, 119–130 (2009)

    MathSciNet  MATH  Google Scholar 

  17. Demidenko, G.V., Matveeva, I.I.: On stability of solutions to linear systems with periodic coefficients. Siberian Math. J. 42, 282–296 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Demidenko, G.V., Matveeva, I.I.: On asymptotic stability of solutions to nonlinear systems of differential equations with periodic coefficients. Selcuk J. Appl. Math. 3, 37–48 (2002)

    MATH  Google Scholar 

  19. Demidenko, G.V., Matveeva, I.I.: On stability of solutions to quasilinear periodic systems of differential equations. Siberian Math. J. 45, 1041–1052 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Demidenko, G.V., Kolchanov, N.A., Likhoshvai, V.A., Matushkin, Yu.G., Fadeev, S.I.: Mathematical modeling of regulatory circuits of gene networks. Comput. Math. Math. Phys. 44, 2166–2183 (2004)

    MathSciNet  MATH  Google Scholar 

  21. Demidenko, G.V., Matveeva, I.I.: Asymptotic properties of solutions to delay differential equations (in Russian). Vestnik Novosib. Gos. Univ. Ser. Mat. Mekh. Inform. 5, 20–28 (2005)

    Google Scholar 

  22. Demidenko, G.V., Matveeva, I.I.: Stability of solutions to delay differential equations with periodic coefficients of linear terms. Siberian Math. J. 48, 824–836 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Demidenko, G.V., Kotova, T.V., Skvortsova, M.A.: Stability of solutions to differential equations of neutral type. J. Math. Sci. 186, 394–406 (2012)

    Article  MathSciNet  Google Scholar 

  24. Demidenko, G.V., Vodop’yanov, E.S., Skvortsova, M.A.: Estimates of solutions to the linear differential equations of neutral type with several delays of the argument. J. Appl. Indust. Math. 7, 472–479 (2013)

    Article  MATH  Google Scholar 

  25. Demidenko, G.V., Matveeva, I.I.: On estimates of solutions to systems of differential equations of neutral type with periodic coefficients. Siberian Math. J. 55, 866–881 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Demidenko, G.V., Matveeva, I.I.: On exponential stability of solutions to one class of systems of differential equations of neutral type. J. Appl. Indust. Math. 8, 510–520 (2014)

    Article  MATH  Google Scholar 

  27. Demidenko, G.V., Matveeva, I.I.: Estimates for solutions to linear systems of neutral type with several delays. J. Anal. Appl. 12, 37–52 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Demidenko, G.V., Matveeva, I.I.: Estimates for solutions to a class of nonlinear time-delay systems of neutral type. Electron. J. Diff. Equ. 2015, 1–14 (2015)

    MathSciNet  MATH  Google Scholar 

  29. Demidenko, G.V., Matveeva, I.I.: Estimates for solutions to a class of time-delay systems of neutral type with periodic coefficients and several delays. Electron. J. Qual. Theory Differ. Equ. 2015, 1–22 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Demidenko, G.V., Matveeva, I.I.: Asymptotic stability of solutions to a class of linear time-delay systems with periodic coefficients and a large parameter. J. Ineq. Appl. 2015, 1–10 (2015)

    MathSciNet  MATH  Google Scholar 

  31. Demidenko, G.V., Matveeva, I.I.: On the robust stability of solutions to linear differential equations of neutral type with periodic coefficients (in Russian). Sib. Zh. Ind. Mat. 18, 18–29 (2015)

    MathSciNet  MATH  Google Scholar 

  32. Demidenko, G.V., Matveeva, I.I.: Exponential stability of solutions to nonlinear time-delay systems of neutral type. Electron. J. Diff. Equ. 2016, 1–20 (2016)

    MathSciNet  MATH  Google Scholar 

  33. Demidenko, G.V., Matveeva, I.I.: Estimates for solutions of one class of nonlinear neutral type systems with several delays. J. Math. Sci. 213, 811–822 (2016)

    Article  MathSciNet  Google Scholar 

  34. Demidenko, G.V., Baldanov, D.S.: Asymptotic stability of solutions to delay difference equations. J. Math. Sci. 221, 815–825 (2017)

    Article  MathSciNet  Google Scholar 

  35. Demidenko, G.V., Matveeva, I.I., Skvortsova, M.A.: Estimates for solutions to neutral differential equations with periodic coefficients of linear terms. Siberian Math. J. 60, 828–841 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  36. Demidenko, G.V., Baldanov, D.S.: Exponential stability of solutions to delay difference equations with periodic coefficients. In: Demidenko, G.V., Romenski, E., Toro, E., Dumbser, M.(eds.) Continuum mechanics, applied mathematics and scientific computing: Godunov’s legacy – A liber amicorum to Professor Godunov. Springer Nature, Cham, Switzerland, pp. 93–100 (2020)

    Google Scholar 

  37. Demidenko, G.V., Matveeva, I.I.: On estimates of solutions to one class of functional difference equations with periodic coefficients. In: Demidenko, G.V., Romenski, E., Toro, E., Dumbser, M.(eds.) Continuum mechanics, applied mathematics and scientific computing: Godunov’s legacy - A liber amicorum to Professor Godunov. Springer Nature, Cham, Switzerland, pp. 101–109 (2020)

    Google Scholar 

  38. Dolgii, Yu.F., Kim, A.V.: On the method of Lyapunov functionals for systems with aftereffect. Differ. Equ. 27, 918–922 (1991)

    MathSciNet  Google Scholar 

  39. Dolgii, Yu.F.: Stability of Periodic Differential-Difference Equations. Izd. Ural University, Ekaterinburg (1996).(in Russian)

    Google Scholar 

  40. El’sgol’ts, L.E.: Qualitative Methods in Mathematical Analysis. American Mathematical Society, Providence (1964)

    Book  MATH  Google Scholar 

  41. El’sgol’ts, L.E.: Introduction to the Theory of Differential Equations with Deviating Argument. Nauka, Moscow (1964).(in Russian)

    MATH  Google Scholar 

  42. El’sgol’ts, L.E., Norkin, S.B.: Introduction to the Theory and Application of Differential Equations with Deviating Arguments. Academic, Academic Press, New York, London (1973)

    MATH  Google Scholar 

  43. Erneux, T.: Applied Delay Differential Equations. Surveys and Tutorials in the Applied Mathematical Sciences, vol. 3. Springer, New York (2009)

    Google Scholar 

  44. Gasilov, G.L.: On the characteristic equation of a system of linear differential equations with periodic coefficients and delays (in Russian). Izv. Vyssh. Uchebn. Zaved. Matematika (4), 60–66 (1972)

    Google Scholar 

  45. Germanovich, O.P.: Linear Periodic Equations of Neutral Type and Their Applications. Izd. Leningrad University, Leningrad (1986).(in Russian)

    Google Scholar 

  46. Gil’, M.I.: Stability of Neutral Functional Differential Equations. Atlantis Studies in Differential Equations, vol. 3. Atlantis Press, Paris (2014)

    Google Scholar 

  47. Godunov, S.K.: Ordinary Differential Equations with Constant Coefficients. Izd. Novosibirsk University, Novosibirsk (1994).(in Russian)

    MATH  Google Scholar 

  48. Godunov, S.K.: Modern Aspects of Linear Algebra. Translations of Mathematical Monographs, vol. 175. American Mathematical Society, Providence (1998)

    Google Scholar 

  49. Gopalsamy K. Stability and Oscillations in Delay Differential Equations of Population Dynamics. Mathematics and its Applications, vol. 74. Kluwer Academic Publishers, Dordrecht (1992)

    Google Scholar 

  50. Goryachenko, V.D.: Methods of Research of Stability of Nuclear Reactors. Atomizdat, Moscow (1977).(in Russian)

    Google Scholar 

  51. Góreckii, H.: Analysis and Synthesis of Control Systems with Delay. Mashinostroenie, Moscow (1974).(in Russian)

    Google Scholar 

  52. Györi, I., Ladas, G.: Oscillation Theory of Delay Differential Equations: With Applications. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1991)

    Google Scholar 

  53. Hahn, W.: On difference differential equations with periodic coefficients. J. Math. Anal. Appl. 3, 70–101 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  54. Hartman, Ph.: Ordinary Differential Equations. Wiley, New York, London, Sydney (1964)

    MATH  Google Scholar 

  55. Hale, J.K.: Theory of Functional Differential Equations. Springer, New York, Heidelberg (1977)

    Book  MATH  Google Scholar 

  56. Halanay, A.: Stability theory of linear periodic systems with delay. Acad. Repub. Popul. Roum., Rev. Math. Pures Appl. 6, 633–653 (1961)

    Google Scholar 

  57. Halanay, A., Wexler, D.: Qualitative Theory of Impulse Systems. Mir, Moscow (1971).(in Russian)

    MATH  Google Scholar 

  58. Kharitonov, V.L., Hinrichsen, D.: Exponential estimates for time delay systems. Syst. Control Lett. 53, 395–405 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  59. Kharitonov, V., Mondié, S., Collado, J.: Exponential estimates for neutral time-delay systems: an LMI approach. IEEE Trans. Automat. Control 50, 666–670 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  60. Kharitonov, V.L.: Time-Delay Systems. Lyapunov Functionals and Matrices. Control Engineering. Birkhäuser/Springer, New York (2013)

    Google Scholar 

  61. Khusainov, D.Ya., Shatyrko, A.V.: The Method of Lyapunov Functions and the Investigation of the Stability of Functional-Differential Systems. Izd. Kiev University, Kiev (1997). (in Russian)

    Google Scholar 

  62. Khusainov, D.Ya., Ivanov, A.F., Kozhametov, A.T.: Convergence estimates for solutions of linear stationary systems of differential-difference equations with constant delay. Differ. Equ. 41, 1196–1200 (2005)

    Google Scholar 

  63. Khusainov, D.Ya., Kozhametov, A.T.: Convergence of solutions of the neutral type nonautonomous systems. Russ. Math. 50, 65–69 (2006)

    Google Scholar 

  64. Kim, A.V.: Direct Lyapunov’s Method in the Stability Theory of Systems with Delays. Izd. Ural University, Ekaterinburg (1992). (in Russian)

    Google Scholar 

  65. Kolmanovskii, V.B., Nosov, V.R.: Stability and Periodic Regimes of Control Systems with Aftereffect. Nauka, Moscow (1981).(in Russian)

    Google Scholar 

  66. Kolmanovskii, V.B., Myshkis, A.D.: Introduction to the Theory and Applications of Functional-Differential Equations. Mathematics and its Applications, vol. 463. Kluwer Academic Publishers, Dordrecht (1999)

    Google Scholar 

  67. Komlenko, Yu.V., Tonkov, E.L.: The Lyapunov-Floquet representation for differential equations with aftereffect. Russ. Math. 39, 38–43 (1995)

    MathSciNet  MATH  Google Scholar 

  68. Korenevskii, D.G.: Stability of Dynamical Systems under Random Perturbations of Parameters. Algebraic Criteria. Naukova Dumka, Kiev (1989). (in Russian)

    Google Scholar 

  69. Korenevskii, D.G.: The Destabilizing Effect of Parametric White Noise in Continuous and Discrete Dynamical Systems. Akademperiodika, Kiev (2008).(in Russian)

    Google Scholar 

  70. Krasovskii, N.N.: On application of second Lyapunov’s method for systems with time delays. Prikl. Mat. Mekh. 20, 315–327 (1956)

    MathSciNet  Google Scholar 

  71. Krasovskii, N.N.: Stability of Motion. Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay. Stanford University Press, Stanford (1963)

    Google Scholar 

  72. Kuang, Y.: Differential Equations with Applications in Population Dynamics. Mathematics in Science and Engineering, vol. 191. Academic Press, Boston (1993)

    Google Scholar 

  73. Lyubich, Yu.I., Tkachenko, V.A.: On a Floquet theory for equations with retarded argument (in Russian). Differ. Uravn. 5, 648–656 (1969)

    Google Scholar 

  74. MacDonald, N.: Biological Delay Systems: Linear Stability Theory. Cambridge Studies in Mathematical Biology, vol. 8. Cambridge University Press, Cambridge (1989)

    Google Scholar 

  75. Malygina, V.V.: On the stability of equations with periodic parameters (in Russian). In: Functional-Differential Equations, Interuniversity Collection Science Works, pp. 41–43. Perm (1987)

    Google Scholar 

  76. Marchuk, G.I.: Mathematical Models in Immunology. Nauka, Moscow (1980).(in Russian)

    MATH  Google Scholar 

  77. Matveeva, I.I.: Estimates of solutions to a class of systems of nonlinear delay differential equations. J. Appl. Indust. Math. 7, 557–566 (2013)

    Article  MathSciNet  Google Scholar 

  78. Matveeva, I.I.: On exponential stability of solutions to periodic neutral-type systems. Siberian Math. J. 58, 264–270 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  79. Matveeva, I.I.: On the exponential stability of solutions of periodic systems of the neutral type with several delays. Differ. Equ. 53, 725–735 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  80. Matveeva, I.I.: On the robust stability of solutions to periodic systems of neutral type. J. Appl. Indust. Math. 12, 684–693 (2018)

    Article  MathSciNet  Google Scholar 

  81. Matveeva, I.I.: Estimates of the exponential decay of solutions to linear systems of neutral type with periodic coefficients. J. Appl. Indust. Math. 13, 511–518 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  82. Matveeva, I.I.: On exponential stability of solutions to linear periodic systems of neutral type with time-varying delay. Siberian Electron. Math. Reports. 16, 748–756 (2019)

    MATH  Google Scholar 

  83. Matveeva, I.I.: Estimates of exponential decay of solutions to one class of nonlinear systems of neutral type with periodic coefficients. Comput. Math. Math. Phys. 60, 601–609 (2020)

    Google Scholar 

  84. Melchor-Aguilar D., Niculescu, S.-I.: Estimates of the attraction region for a class of nonlinear time-delay systems. IMA J. Math. Control Inform. 24, 523–550 (2007)

    Google Scholar 

  85. Michiels, W., Niculescu, S.-I.: Stability, Control, and Computation for Time-Delay Systems. An Eigenvalue-Based Approach. Advances in Design and Control, vol. 27. Society for Industrial and Applied Mathematics, Philadelphia (2014)

    Google Scholar 

  86. Mikhalevich, V.S., Kozorez, V.V., Rashkovan, V.M., Khusainov, D.Ya., Cheborin, O.G.: “Magnetic Potential Well” — Stabilization Effect of Superconducting Dynamical Systems. Naukova Dumka, Kiev (1991). (in Russian)

    Google Scholar 

  87. Mitropol’skij, Yu.A., Martynyuk, D.I.: Periodic and Quasi-Periodic Oscillations of Systems with Delay. Vishcha Shkola, Kiev (1979). (in Russian)

    Google Scholar 

  88. Mondié, S., Kharitonov, V.L.: Exponential estimates for retarded time-delay systems: an LMI approach. IEEE Trans. Automat. Control. 50, 268–273 (2005)

    Google Scholar 

  89. Murray, J.D.: Lectures on Nonlinear Differential-Equation Models in Biology. Clarendon Press, Oxford (1977)

    MATH  Google Scholar 

  90. Myshkis, A.D.: Linear Differential Equations with Retarded Argument. Gostekhizdat, Moscow, Leningrad (1951).(in Russian)

    MATH  Google Scholar 

  91. Myshkis, A.D.: Linear Differential Equations with Retarded Argument. Nauka, Moscow (1972).(in Russian)

    MATH  Google Scholar 

  92. Pertsev, N.V.: Application of the monotone method and of \(M\)-matrices to the analysis of the behavior of solutions of some models of biological processes (in Russian). Sib. Zh. Ind. Mat. 5, 110–122 (2002)

    MathSciNet  MATH  Google Scholar 

  93. Pertsev, N.V.: Global solvability and estimates of solutions to the Cauchy problem for the retarded functional differential equations that are used to model living systems. Siberian Math. J. 59, 113–125 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  94. Pinney, E.: Ordinary Difference-Differential Equations. University of California Press, Berkeley, Los Angeles (1958)

    MATH  Google Scholar 

  95. Pontryagin, L.S.: On zeros of some transcendental functions (in Russian). Izv. Akad. Nauk SSSR, Ser. Mat. 6 115–134 (1942)

    Google Scholar 

  96. Razumikhin, B.S.: On stability of systems with time lag (in Russian). Prikl. Mat. Mekh. 20, 500–512 (1956)

    MathSciNet  Google Scholar 

  97. Razumikhin, B.S.: Direct Method of Investigation of Stability of Systems with Aftereffect (in Russian). Preprint. VNIISI, Moscow, 75 p. (1984)

    Google Scholar 

  98. Richard, J.P.: Time-delay systems: an overview of some recent advances and open problems. Automatica. 39, 1667–1694 (2003)

    Google Scholar 

  99. Romanovskii, R.K., Trotsenko, G.A.: The method of Lyapunov functionals for neutral type linear difference-differential systems with almost periodic coefficients. Siberian Math. J. 44, 355–362 (2003)

    Article  MathSciNet  Google Scholar 

  100. Romanovskii, R.K., Bel’gart, L.V., Dobrovol’skii, S.M., Rogozin, A.V., Trotsenko, G.A.: Method of Lyapunov Functions for Almost Periodic Systems. Publishing House SB RAS, Novosibirsk (2015). (in Russian)

    Google Scholar 

  101. Romanyukha, A.A., Rudnev, S.G.: A variational principle for modelling infection immunity by the example of pneumonia (in Russian). Mat. Model. 13, 65–84 (2001)

    Google Scholar 

  102. Rubanik, V.P.: Oscillations of Quasilinear Systems with Delay. Nauka, Moscow (1969).(in Russian)

    MATH  Google Scholar 

  103. Shil’man, S.V.: The Method of Generating Functions in the Theory of Dynamical Systems. Nauka, Moscow (1978).(in Russian)

    Google Scholar 

  104. Shimanov, S.N.: Stability of Linear Systems with Periodic Coefficients and Delay. Izd. Ural University, Sverdlovsk (1983). (in Russian)

    Google Scholar 

  105. Skvortsova, M.A.: Estimates of solutions to a system describing birds’ migration. J. Anal. Appl. 13, 15–27 (2015)

    MathSciNet  MATH  Google Scholar 

  106. Skvortsova, M.A.: Asymptotic properties of solutions to a system describing the spread of avian influenza. Siberian Electron. Math. Reports. 13, 782–798 (2016)

    MathSciNet  MATH  Google Scholar 

  107. Skvortsova, M.: Asymptotic properties of solutions in Marchuk’s basic model of disease. Func. Diff. Equ. 24, 127–135 (2017)

    MathSciNet  Google Scholar 

  108. Skvortsova, M.A.: Asymptotic properties of solutions in a model of antibacterial immune response (in Russian). Siberian Electron. Math. Reports. 15, 1198–1215 (2018)

    MATH  Google Scholar 

  109. Skvortsova, M.A.: On estimates of solutions in a predator-prey model with two delays (in Russian). Siberian Electron. Math. Reports. 15, 1697–1718 (2018)

    MATH  Google Scholar 

  110. Solodov, A.V., Solodova, E.A.: Systems with Variable Delay. Nauka, Moscow (1980).(in Russian)

    Google Scholar 

  111. Stokes, A.P.: A Floquet theory for functional differential equations. Proc. Nat. Acad. Sci. USA 48, 1330–1334 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  112. Svirezhev, Yu.M., Pasekov, V.P.: Fundamentals of Mathematical Genetics. Nauka, Moscow (1982).(in Russian)

    Google Scholar 

  113. Tsypkin, Ya.Z.: Theory of Linear Impulse Systems. Gos. Izd. Fiz.-Mat. Lit., Moscow (1963). (in Russian)

    Google Scholar 

  114. Vielle, B., Chauvet, G.: Delay equation analysis of human respiratory stability. Math. Biosci. 152, 105–122 (1998)

    Google Scholar 

  115. Vlasov, V.V., Medvedev, D.A.: Functional-differential equations in Sobolev spaces and related problems of spectral theory. J. Math. Sci. 164, 659–841 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  116. Volterra, V.: Lessons on the Mathematical Theory of Struggle for Life. Nauka, Moscow (1976).(in Russian)

    Google Scholar 

  117. Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Clarendon Press, Oxford (1965)

    MATH  Google Scholar 

  118. Wolkowicz, G.S.K., Xia, H.: Global asymptotic behavior of a chemostat model with discrete delays. SIAM. J. Appl. Math. 57, 1019–1043 (1997)

    MathSciNet  MATH  Google Scholar 

  119. Yskak, T.: Stability of solutions to systems of differential equations with distributed delay. Func. Diff. Equ. 25, 97–108 (2018)

    MathSciNet  MATH  Google Scholar 

  120. Yskak, T.: On the stability of systems of linear differential equations of neutral type with distributed delay. J. Appl. Indust. Math. 13, 575–583 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  121. Zubov, V.I.: Analytical Dynamics of a System of Solids. Izd. Leningrad University, Leningrad (1983). (in Russian)

    Google Scholar 

  122. Zverkin, A.M.: On the theory of linear delay differential equations with periodic coefficients (in Russian). Dokl. Akad. Nauk SSSR 128, 882–885 (1959)

    MathSciNet  MATH  Google Scholar 

  123. Zverkin, A.M.: Differential difference equations with periodic coefficients (in Russian). In: Bellman, R.E., Cooke, K.L. Differential-Difference Equations. pp. 498–535. Supplement to the Russian translation of the book. Mir, Moscow (1967)

    Google Scholar 

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Acknowledgements

This work was supported by the Russian Foundation for Basic Research (project no. 19-01-00754).

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Acad. Koptyug avenue 4, 630090, Novosibirsk, Russia

    G. V. Demidenko & I. I. Matveeva

  2. Novosibirsk State University, Pirogov street 1, 630090, Novosibirsk, Russia

    G. V. Demidenko & I. I. Matveeva

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  1. G. V. Demidenko
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  1. Department of Mathematics, Ariel University, Ariel, Israel

    Alexander Domoshnitsky

  2. Department of Mathematics, Ariel University, Ariel, Israel

    Alexander Rasin

  3. Department of Mathematics, Birla Institute of Technology, Mesra, Ranchi, Jharkhand, India

    Seshadev Padhi

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Demidenko, G.V., Matveeva, I.I. (2021). The Second Lyapunov Method for Time-Delay Systems. In: Domoshnitsky, A., Rasin, A., Padhi, S. (eds) Functional Differential Equations and Applications. FDEA 2019. Springer Proceedings in Mathematics & Statistics, vol 379. Springer, Singapore. https://doi.org/10.1007/978-981-16-6297-3_11

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