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The Lyapunov functionals method in stability problems for functional differential equations

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Abstract

The paper presents a survey of stability results for retarded functional differential equations with the Lyapunov functional method.

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References

  1. Azbelev, N.V., How It Was (on the Primary Stages of the Development of the Modern Theory of Functional Differential Equations), in Problemy nelineinogo analiza v inzhenernykh sistemakh (Problems of Nonlinear Analysis in Engineering Systems), 2003, vol. 9,issue 1(17), pp. 1–40.

    Google Scholar 

  2. Azbelev, N.V., Maksimov, V.P., and Rakhmatullina, L.F., Introduction to the Theory of Linear Functional Differential Equations, Atlanta: World Federation Publ., 1995.

    MATH  Google Scholar 

  3. Alekseenko, N.V., Solution Stability in Nonlinear Almost Periodic Systems of Functional Differential Equations with Delay, Izv. Vuzov, Mat., 2000, no. 2, pp. 3–6.

  4. Alekseenko, N.V. and Romanovskii, R.K., The Method of Lyapunov Functionals for Linear Difference-Differential Systems with Almost Periodic Coefficients, Diff. Equat., 2001, vol. 37, no. 2, pp. 159–165.

    Article  Google Scholar 

  5. Anan’evskii, I.M. and Kolmanovskii, V.B., Stabilization of Control Systems with Aftereffect, Autom. Remote Control, 1989, no. 9, pp. 1174–1180.

  6. Andreev, A.S., On Asymptotic Stability and Instability of the Zero Solution of a Non-autonomous System, Prikl. Mat. Mekh., 1984, vol. 48, no. 2, pp. 225–232.

    MathSciNet  Google Scholar 

  7. Andreev, A.S., On Asymptotic Stability and Instability with Respect to a Fraction of the Variables, Prikl. Mat. Mekh., 1984, vol. 48, no. 5, pp. 707–713.

    MathSciNet  Google Scholar 

  8. Andreev, A.S., On Studying Partial Asymptotic Stability and Instability Based on Limit Equations, Prikl. Mat. Mekh., 1987, vol. 51, no. 2, pp. 253–260.

    MathSciNet  Google Scholar 

  9. Andreev, A.S., On Studying Partial Asymptotic Stability, Prikl. Mat. Mekh., 1991, vol. 55, no. 4, pp. 539–547.

    Google Scholar 

  10. Andreev, A.S., On Asymptotic Stability and Instability of the Zero Solution of a Non-autonomous Functional Differential Equation, in Problemy analiticheskoi mekhaniki, ustoichivosti i upravleniya dvizheniem (Problems of Analytical Mechanics, Stability and Control of Motions), Novosibirsk: Nauka, 1991, pp. 36–40.

    Google Scholar 

  11. Andreev, A.S., Metody issledovaniya ustoichivosti neavtonomnykh uravnenii (Approaches to Studying Stability of Non-autonomous Equations), Ul’yanovsk: MGU, 1994.

    Google Scholar 

  12. Andreev, A.S., Stability of the Non-autonomous Functional Differential Equation, Dokl. Ross. Akad. Nauk, 1997, vol. 356, no. 7, pp. 151–153.

    Google Scholar 

  13. Andreev, A.S., Problems on the Stability of Functional Differential Equations with Finite and Unbounded Delay, Uch. Zap. Ul’yanovsk. Gos. Univ., Ser. Fund. Probl. Mat. Mekh., Ul’yanovsk: Ul’yanovsk. Gos. Univ., 2003, issue 1(13), pp. 3–11.

    Google Scholar 

  14. Andreev, A.S., On the K.P. Persidsky’s Approach in Instability Problems, Mat. Zh. Almaty, 2004, vol. 4, no. 2(12), pp. 76–82.

    MATH  Google Scholar 

  15. Andreev, A.S., Ustoichivost’ neavtonomnykh funktsional’no-differentsial’nykh uravnenii (Stability of Non-autonomous Functional Differential Equations), Ul’yanovsk: Ul’yanovsk. Gos. Univ., 2005.

    Google Scholar 

  16. Andreev, A.S. and Pavlikov, S.V., On Partial Stability of an Non-autonomous Functional Differential Equation, Prikl. Mat. Mekh., 1999, vol. 63, no. 1, pp. 3–12.

    MATH  MathSciNet  Google Scholar 

  17. Andreev, A.S. and Pavlikov, S.V., A Study of Stability of Functional Differential Equations Based on Sign-definite Lyapunov Functionals, Trudy Srednevolzhsk. Mat. Obshch., 1999, vol. 2(1), pp. 74–75.

    MathSciNet  Google Scholar 

  18. Andreev, A.S. and Pavlikov, S.V., On a Problem of Partial Stability of Functional Differential Equations, Uch. Zapiski Ul’yanovsk. Gis. Univ., Ser. Fund. Probl. Mat. Mekh., Ul’yanovsk: Ul’yanovsk. Gos. Univ., 2003, issue 1(13), pp. 12–21.

    Google Scholar 

  19. Andreev, A.S. and Pavlikov, S.V., Nonfixed Sign Lyapunov Functionals in the Problem of Stability of Functional-differential Equations with Finite Delay, Mekh. Tverdogo Tela, Donetsk: Inst. Prikl. Mat. Mekh., 2004, vyp. 34, pp. 112–120.

    Google Scholar 

  20. Andreev, A.S. and Khusanov, D.H., Limit Equations in the Stability Problem of a Functional Differential Equation, Diff. Equat., 1998, vol. 34, no. 4, pp. 435–440.

    MathSciNet  Google Scholar 

  21. Andreev, A.S. and Khusanov, D.H., On the Lyapunov Functionals Method in the Problem of Asymptotic Stability and Instability, Diff. Equat., 1998, vol. 34, no. 7, pp. 876–885.

    MATH  MathSciNet  Google Scholar 

  22. Bellman, R. and Cook, K.L., Differential-Difference Equations, New York: Academic, 1963.

    MATH  Google Scholar 

  23. Volterra, V., Matematicheskaya teoriya bor’by za sushchestvovanie (The Mathematical Theory of the Struggle for Existence), Moscow: Nauka, 1976.

    Google Scholar 

  24. Volterra, V., Theory of Functionals and of Integral and Integro-differential Equations, New York: Dover, 1959.

    MATH  Google Scholar 

  25. Vorotnikov, V.I., Ustoichivost’ dinamicheskikh sistem po otnosheniyu k chasti peremennykh (Partial Stability of Dynamic Systems), Moscow: Nauka, 1991.

    Google Scholar 

  26. Vorotnikov, V.I. and Rumyantsev, V.V., Ustoichivost’ i upravlenie po chasti koordinat fazovogo vektora dinamicheskikh sistem: teoriya, metody i prilozheniya (Stability and Control in Part of Coordinates of the Phase Vector of Dynamic Systems: Theory, Methods, Applications), Moscow: Nauchnyi Mir, 2001.

    Google Scholar 

  27. Gaishun, I.V., Asymptotic Stability of One System with Delay, Diff. Equat., 1972, vol. 8, no. 5, pp. 906–908.

    Google Scholar 

  28. Gaishun, I.V. and Knyazhishche, L.B., Stability Theorems for Equations with Delay Using Nonmonotonic Lyapunov Functionals, Dokl. Belarus. Nat. Akad. Nauk, 1994, vol. 38, no. 3, pp. 5–8.

    MathSciNet  Google Scholar 

  29. Gaishun, I.V. and Knyazhishche, L.B., Non-monotonic Lyapunov Functionals. Stability Conditions for Equations with Delay, Diff. Equat., 1994, vol. 30, no. 8, pp. 1291–1298.

    MathSciNet  Google Scholar 

  30. Goryachenko, V.D., Metody teorii ustoichivosti v dinamike yadernykh reaktorov (Stability Theory Approaches in the Dynamics of Nuclear Piles), Moscow: Atomizdat, 1971.

    Google Scholar 

  31. Goryachenko, V.D., Metody issledovaniya ustoichivosti yadernykh reaktorov (Studying Stability of Nuclear Piles), Moscow: Atomizdat, 1977.

    Google Scholar 

  32. Drozdov, A.D., Kolmanovskii, V.B., and Tridzhante, D., Stability of Predator-Rey System, Autom. Remote Control, 1992, no. 11, pp. 1697–1704.

  33. Zhikov, V.V. and Levitan, B.M., Pochti periodicheskie funktsii i differentsial’nye uravneniya (Almost Periodic Functions and Differential Equations), Moscow: Mosk. Gos. Univ., 1978.

    MATH  Google Scholar 

  34. Kalistratova, T.A., Stability with Respect to a Part of the Variables for a Systems with Delay, Autom. Remote Control, 1986, no. 5, pp. 615–619.

  35. Karapetyan, A.V., Ustoichivost’ statsionarnykh dvizhenii (Stability of Stationary Motion), Moscow: Editorial URSS, 1998.

    Google Scholar 

  36. Kim, A.V., Method of Lyapunov Functionals for Systems with Aftereffect, Autom. Remote Control, 1990, no. 2, pp. 152–157.

  37. Kim, A.V., Pryamoi metod Lyapunova v teorii ustoichivosti sistem s posledeistviem (Lyapunov’s Direct Method in the Stability Theory of Systems with Aftereffect), Ekaterinburg: Ural. Univ., 1992.

    Google Scholar 

  38. Kim, A.V., i-Gladkii analiz i funktsional’no-differentsial’nye uravneniya (i-Smooth Analysis and Functional Differential Equations), Ekaterinburg: Rossk. Akad. Nauk, 1996.

    Google Scholar 

  39. Kim, A.V. and Pimenov, V.G., i-Gladkii analiz i chislennye metody resheniya funktsional’no-differentsialnykh uravnenii (i-Smooth Analysis and Numerical Methods for Functional Differential Equations), Moscow: NIC “Regular and Chaotic Dynamics,” 2004.

    Google Scholar 

  40. Knyazhishche, L.B., Localizing Limit Sets and Asymptotic Stability of Equations with Delay, Dokl. Belarus. Nat. Akad. Nauk, 1997, vol. 41, no. 1, pp. 26–29.

    MATH  MathSciNet  Google Scholar 

  41. Knyazhishche, L.B., Localizing Limit Sets and Asymptotic Stability of Non-autonomous Equations with Delay. I, Diff. Equat., 1998, vol. 34, no. 2, pp. 189–196.

    MathSciNet  Google Scholar 

  42. Knyazhishche, L.B., Localizing Limit Sets and Asymptotic Stability of Non-autonomous Equations with Delay. II, Diff. Equat., 1998, vol. 34, no. 8, pp. 1056–1065.

    MathSciNet  Google Scholar 

  43. Knyazhishche, L.B., Lyapunov Functionals and Conditions for Uniform Asymptotic Stability of Equations with Delay, Dokl. Belarus. Nat. Akad. Nauk, 2000, vol. 44, no. 5, pp. 40–43.

    MathSciNet  Google Scholar 

  44. Knyazhishche, L.B. and Shchavel’, N.A., Nonmonotonic Lyapunov Functionals and Estimates on Solutions of Differential Equations with Delay, Diff. Equat., 1997, vol. 33, no. 2, pp. 205–211.

    MATH  Google Scholar 

  45. Kolmanovskii, V.B., On Stability of Certain Systems with Arbitrary Aftereffect, Dokl. Ross. Akad. Nauk, 1993, vol. 331, no. 4, pp. 421–424.

    Google Scholar 

  46. Kolmanovskii, V.B. and Nosov, V.R., Ustoichivost’ i periodicheskie rezhimy reguliruemykh sistem s posledeistviem (Stability and Periodic Modes of Regulated Systems with Aftereffect), Moscow: Nauka, 1981.

    Google Scholar 

  47. Kolmogorov, A.N. and Fomin, S.V., Elementy teorii funktsii i funktsional’nogo analiza (Elements of Function Theory and Functional Analysis), Moscow: Nauka, 1976.

    Google Scholar 

  48. Corduneanu, C. and Lakshmikantham, V., Equations with Unbounded Delay: A Survey, Nonlin. Anal., 1980, vol. 4, no. 5, pp. 831–877.

    Article  MATH  MathSciNet  Google Scholar 

  49. Krasovskii, N.N., On Applying Lyapunov’s Second Method to Equations with Time Delay, Prikl. Mat. Mekh., 1956, vol. 20, no. 3, pp. 315–327.

    MathSciNet  Google Scholar 

  50. Krasovskii, N.N., Asymptotic Stability of Systems with Aftereffect, Prikl. Mat. Mekh., 1956, vol. 20, no. 4, pp. 513–518.

    MathSciNet  Google Scholar 

  51. Krasovskii, N.N., Nekotorye zadachi teorii ustoichivosti dvizheniya (Certain Problems in Motion Stability Theory), Moscow: Fizmatgiz, 1959.

    Google Scholar 

  52. Lakshmikantam, V., Lila, S., and Martynyuk, A.A., Ustoichivost’ dvizheniya: metod sravneniya (Motion Stability: The Comparison Approach), Kiev: Naukova Dumka, 1991.

    Google Scholar 

  53. Lakshmikantam, V. and Martynyuk, A.A., Developing Lyapunov’s Direct Method for Systems with Aftereffect (a Survey), Prikl. Mekh., 1993, vol. 29(39), no. 2, pp. 3–15.

    MathSciNet  Google Scholar 

  54. Lyusternik, L.A. and Sobolev, V.I., Kratkii kurs funktsional’nogo analiza (A Short Course in Functional Analysis), Moscow: Vysshaya Shkola, 1982.

    Google Scholar 

  55. Lyapunov, A.M., Izbrannye trudy. Raboty po teorii ustoichivosti (Selected Works. Works on Stability Theory), Moscow: Nauka, 2007.

    Google Scholar 

  56. Martynyuk, A.A., Kato, D., and Shestakov, A.A., Ustoichivost’ dvizheniya: metod predel’nykh uravnenii (Motion Stability: The Limit Equation Approach), Kiev: Naukova Dumka, 1990.

    Google Scholar 

  57. Matrosov, V.M., Metod vektornykh funktsii Lyapunova: analiz dinamicheskikh svoistv nelineinykh sistem (Lyapunov Vector Functions Approach: Analyzing Dynamic Properties of Nonlinear Systems), Moscow: Fizmatlit, 2000.

    Google Scholar 

  58. Myshkis, A.D., A General Theory of Differential Equations with a Retarded Argument, Uspekhi Mat. Nauk, 1949, vol. 4, no. 5, pp. 99–141.

    Google Scholar 

  59. Myshkis, A.D., Lineinye differentsial’nye uravneniya s zapazdyvayushchim argumentom (Linear Differential Equations with a Retarded Argument), Moscow: Nauka, 1972.

    MATH  Google Scholar 

  60. Myshkis, A.D., On Certain Problems in the Theory of Differential Equations with a Divergent Argument, Uspekhi Mat. Nauk, 1977, vol. 32,issue 2(194), pp. 173–202.

    MATH  Google Scholar 

  61. Myshkis, A.D., The Translation Editor’s Introduction, in Hale, J., Theory of Functional Differential Equations, Moscow: Mir, 1984, pp. 5–6.

    Google Scholar 

  62. Myshkis, A.D. and Elsgolts, L.E., The Current State and the Problems of the Theory of Differential Equations with a Divergent Argument, Uspekhi Mat. Nauk, 1967, vol. 22,issue 2, pp. 21–57.

    Google Scholar 

  63. Pavlikov, S.V., On Stability of the Zero Solution of a Functional Differential Equation of the Second Order, Uch. Zap. Ul’yanovsk. Gos. Univ., Ser. Fundament. Probl. Mat. Mekh., Ul’yanovsk: Ul’yanovsk. Gos. Univ., 1996, issue 2, pp. 32–33.

    Google Scholar 

  64. Pavlikov, S.V., Sign-definite Lyapunov Functional for the Stability Problem of Functional Differential Equations with Finite Delay, Uch. Zap. Ul’yanovsk. Gos. Univ., Ser. Fundament. Probl. Mat. Mekh., Ul’yanovsk: Ul’yanovsk. Gos. Univ., 2002, issue 2(12), pp. 30–39.

    Google Scholar 

  65. Pavlikov, S.V., Limit Equations and Lyapunov Functionals for the Partial Stability Problem, Uch. Zap. Ul’yanovsk. Gos. Univ., Ser. Fundament. Probl. Mat. Mekh., Ul’yanovsk: Ul’yanovsk. Gos. Univ., 2003, issue 1(13), pp. 63–74.

    Google Scholar 

  66. Pavlikov, S.V., On Stabilizing the Motion of a Controlled System with Delay, Mekh. Tverdogo Tela, Donetsk: Inst. Prikl. Mat. Mekh., 2005, issue 35, pp. 212–216.

    Google Scholar 

  67. Pavlikov, S.V., On the Stability Problem for Functional Differential Equations with Infinite Delay, Uch. Zap. Ul’yanovsk. Gos. Univ., Ser. Fundament. Probl. Mat. Mekh., Ul’yanovsk: Ul’yanovsk. Gos. Univ., 2006, issue 1(13), pp. 28–42.

    Google Scholar 

  68. Pavlikov, S.V., Metod funktsionalov Lyapunova v zadachakh ustoichivosti (The Lyapunov Functionals Method in Stability Problems), Naberezhnye Chelny: Inst. Upravlen., 2006.

    Google Scholar 

  69. Pavlikov, S.V., Stabilization of the Motions of Controlled Mechanical Systems by a Regulator with Delay, Dokl. Math., 2007, vol. 75, no. 1, pp. 174–176.

    Article  Google Scholar 

  70. Pavlikov, S.V., Fixed Sign Lyapunov Functionals in the Problem of Stability of Functional-differential Equation, Prikl. Mat. Mekh., 2007, vol. 71, no. 3, pp. 377–388.

    MATH  MathSciNet  Google Scholar 

  71. Pavlikov, S.V., On Partial Stabilization of a Controlled Mechanical System with Retarded Feedback, Vest. Sam. Gos. Univ., 2007, no. 2(52), pp. 115–123.

  72. Pavlikov, S.V., On Stabilization of the Controlled Mechanical Systems, Autom. Remote Control, 2007, no. 9, pp. 1482–1491.

  73. Pavlikov, S.V., The Stability of Motions of Hereditary Systems with Infinite Delay, Dokl. Math., 2007, vol. 76, no. 2, pp. 678–680.

    Article  MATH  Google Scholar 

  74. Pavlikov, S.V., The Sign-definite Lyapunov Functionals Method in Studying Stability of Functional Differential Equations, Vest. Orenburg. Gos. Univ., 2007, no. 3, pp. 158–162.

  75. Pavlikov, S.V., On the Stability Problem for Functional Differential Equations with Infinite Delay, Izv. Vuzov, Mat., 2008, no. 7, pp. 29–38.

  76. Razumikhin, B.S., On the Stability of Systems with Delay, Prikl. Mat. Mekh., 1956, vol. 20, no. 4, pp. 500–512.

    Google Scholar 

  77. Razumikhin, B.S., Stability of Hereditary Systems, Moscow: Nauka, 1988.

    Google Scholar 

  78. Rumyantsev, V.V., On Partial Motion Stability, Vest. Mosk. Gos. Univ., Ser. Mat., Mekh., Fiz., Astron., Khim., 1957, no. 4, pp. 9–16.

  79. Rumyantsev, V.V., The Lyapunov Functions Method in Motion Stability Theory, in Mekhanika v SSSR za 50 Let, Moscow: Nauka, 1968, vol. 1, pp. 7–66.

    Google Scholar 

  80. Rumyantsev, V.V. and Oziraner, A.S., Ustoichivost’ i stabilizatsiya dvizheniya po otnosheniyu k chasti peremennykh (Partial Stability and Partial Stabilization of Motion), Moscow: Nauka, 1987.

    Google Scholar 

  81. Rouche, N., Habets, P., and LaLoy, M., Stability Theory by Lyapunov’s Direct Method, New York: Springer-Verlag, 1977.

    Google Scholar 

  82. Sergeev, V.S., On Asymptotic Stability of Motion in Certain Classes with Aftereffect, Prikl. Mat. Mekh., 1993, vol. 57, no. 5, pp. 166–174.

    MathSciNet  Google Scholar 

  83. Sergeev, V.S., On Asymptotic Stability and Estimating the Domain of Attraction for Certain Systems with Aftereffect, Prikl. Mat. Mekh., 1996, vol. 60, no. 5, pp. 744–751.

    MathSciNet  Google Scholar 

  84. Tereki, I., Exponential and Polynomial Asymptotic Stability of Functional Differential Equations, in Razvitie i primenenie metoda funktsii Lyapunova (Development and Applications of Lyapunov Functions Method), Novosibirsk: Nauka, 1992, pp. 101–107.

    Google Scholar 

  85. Tikhonov, A.N., On Functional Equations of the Volterra Type and Their Applications to Some Problems in Mathematical Physics, Bull. Mosk. Univ., Sec. A, 1938, vol. 1, no. 8, pp. 1–25.

    MathSciNet  Google Scholar 

  86. Hale, J., Theory of Functional Differential Equations, New York: Springer-Verlag, 1977.

    MATH  Google Scholar 

  87. Khusainov, D.Ya., On Exponential Stability of Linear Systems with Delay, Diff. Equat., 1989, vol. 25, no. 2, pp. 357–359.

    MATH  MathSciNet  Google Scholar 

  88. Khusainov, D.Ya., An Exponential Bound on the Solutions of Linear Systems with Delay with an Arbitrary Deviation of the Argument, Diff. Equat., 1989, vol. 25, no. 9, pp. 1631–1633.

    MATH  MathSciNet  Google Scholar 

  89. Shestakov, A.A., Lyapunov’s Direct Method as a Localization Method by Lyapunov Functions of the Limit Sets of Non-autonomous Dynamical Processes (Based on Limit Equations and Dynamical Systems), in Funktsii Lyapunova i ikh primenenie (Lyapunov Functions and Their Applications), Novosibirsk: Nauka, 1987, pp. 14–48.

    Google Scholar 

  90. Shestakov, A.A., Obobshchennyi pryamoi metod Lyapunova dlya sistem s raspredelennymi parametrami (Generalized Lyapunov’s Direct Method for Systems with Distributed Parameters), Moscow: Nauka, 1990.

    Google Scholar 

  91. Shimanov, S.N., On Nonstability of Motion of Systems with Time Delay, Prikl. Mat. Mekh., 1960, vol. 24, no. 1, pp. 55–63.

    Google Scholar 

  92. Shimanov, S.N., Stability of Systems with Delay, in Proc. of the II All-Union Congress on Theoretical and Applied Mechanics, Moscow, Moscow: Nauka, 1965, issue 1, pp. 170–180.

    Google Scholar 

  93. Scheglov, V.A., Stability of a Linear Differential Equation with Distributed Delay, Diff. Equat., 1996, vol. 32, no. 12, pp. 1665–1669.

    Google Scholar 

  94. Scheglov, V.A., Stability of Solutions of a Second Order Equation with Delay, Diff. Equat., 1998, vol. 34, no. 12, pp. 1710–1713.

    Google Scholar 

  95. El’sgol’ts, L.E., Stability of Solutions of Differential-Difference Equations, Usp. Mat. Nauk, 1954, vol. 9, no. 4, pp. 95–112.

    Google Scholar 

  96. El’sgol’ts, L.E., Vvedenie v teoriyu differentsial’nykh uravnenii s otklonyayushchimsya argumentom (Introduction to the Theory of Differential Equations with Deviating Argument), Moscow: Nauka, 1964.

    Google Scholar 

  97. El’sgol’ts, L.E. and Norkin, S.B., Vvedenie v teoriyu differentsial’nykh uravnenii s otklonyayushchimsya argumentom (Introduction to the Theory of Differential Equations with Deviating Argument), Moscow: Nauka, 1971.

    Google Scholar 

  98. Akinyele, O., On Partial Stability of Differential Equations with Time Delay, Ann. Mat. Pure Appl., 1979, vol. 121, no. 1, pp. 351–372.

    Article  MATH  MathSciNet  Google Scholar 

  99. Andreev, A., On the Stability of Nonautonomous Functional Differential Equations, Nonlin. Anal. TMA, 1997, vol. 30, part 5, pp. 2847–2854.

    Article  MATH  Google Scholar 

  100. Andreev, A. and Khusanov, D., On Asymptotic Stability and Nonstability Functional-differential Equations with Periodic Right Side, Nonlin. Oscillat., 2001, vol. 4, no. 3, pp. 290–298.

    MATH  MathSciNet  Google Scholar 

  101. Andreev, A. and Zappala’, G., On Stability for Perturbed Differential Equations, Le Matematiche, 1996, vol. 51, F. I, pp. 27–41.

    MATH  MathSciNet  Google Scholar 

  102. Artstein, Z., Topological Dinamics of Ordinary Differential Equations, J. Diff. Equat., 1977, vol. 23, pp. 216–223.

    Article  MATH  MathSciNet  Google Scholar 

  103. Artstein, A., Uniform Asymptotic Stability via the Limiting Equations, J. Diff. Equat., 1978, vol. 27, pp. 172–189.

    Article  MATH  MathSciNet  Google Scholar 

  104. Athanassov, Z., Families of Liapunov-Krasovskii Functionals and Stability for Functional Differential Equations, Ann. Mat. Pure Appl. (IV), 1999, vol. 176, pp. 145–165.

    Article  MATH  MathSciNet  Google Scholar 

  105. Becker, L.C., Burton, T.A., and Zhang, S., Functional Differential Equations and Jensen’s Inequality, J. Math. Anal. Appl., 1989, vol. 138, no. 1, pp. 135–156.

    Article  MathSciNet  Google Scholar 

  106. Bernfeld, S.R., Corduneanu, C., and Ignatyev, A.O., On the Stability of Invariant Sets of Functional Differential Equations, Nonlin. Anal., 2003, vol. 55, pp. 641–656.

    Article  MATH  MathSciNet  Google Scholar 

  107. Burton, T.A., Uniform Asymptotic Stability in Functional Differential Equations, Proc. Am. Math. Soc., 1978, vol. 68, pp. 195–199.

    Article  MATH  MathSciNet  Google Scholar 

  108. Burton, T.A., Stability Theory for Delay Equations, Funkcial. Ekvac., 1979, vol. 22, no. 1, pp. 67–76.

    MATH  MathSciNet  Google Scholar 

  109. Burton, T.A., Casal, A., and Somolinos, A., Upper and Lower Bounds for Liapunov Functionals, Funkcial. Ekvac., 1989, vol. 32, no. 1, pp. 23–55.

    MATH  MathSciNet  Google Scholar 

  110. Burton, T.A. and Hatvani, L., Stability Theorems for Nonautonomous Functional Differential Equations by Liapunov Functionals, Tohoku Math. J., 1989, vol. 41, pp. 65–104.

    Article  MATH  MathSciNet  Google Scholar 

  111. Burton, T.A. and Hatvani, L., On Nonuniform Asymptotic Stability for Nonautonomous Functional Differential Equations, Diff. Integral Equat., 1990, vol. 3, pp. 285–293.

    MATH  MathSciNet  Google Scholar 

  112. Burton, T.A. and Makay, G., Marachkov Stability Results for Functional Differential Equations, EJQTDE, 1998, no. 1, pp. 1–17.

  113. Conley, C.C. and Miller, R.K., Asymptotic Stability without Iniform Stability. Almost Periodic Coefficients, J. Diff. Equat., 1965, vol. 1, no. 1, pp. 333–336.

    Article  MATH  MathSciNet  Google Scholar 

  114. Corduneanu, C., On Partial Stability for Delay Systems, Ann. Polon. Math., 1975, vol. 29, pp. 357–362.

    MATH  MathSciNet  Google Scholar 

  115. Corduneanu, C. and Ignatyev, O.A., Stability of Invariant Sets of Functional Differential Equations with delay, Nonlinear Func. Anal. Appl., 2005, no. 1, pp. 11–24.

  116. Haddock, J., The “Evolution” of Invariance Principles a la Liapunov’s Direct Method, Adv. Nonl. Dynamics, 1997, vol. 5, pp. 261–272.

    MathSciNet  Google Scholar 

  117. Haddock, J. and Terjéki, J., On the Location of Positive Limit Sets for Autonomous Functional Differential Equations with Infinite Delay, J. Diff. Equat., 1990, vol. 86, pp. 1–32.

    Article  MATH  Google Scholar 

  118. Hale, J., A Stability Theorem for Functional-differential Equations, Proc. NAS, 1963, vol. 50, pp. 942–946.

    Article  MATH  MathSciNet  Google Scholar 

  119. Hale, J.K. and Kato, J., Phase Space for Retarded Equations with Infinite Delay, Funkcial. Ekvac., 1978, vol. 21, pp. 11–41.

    MATH  MathSciNet  Google Scholar 

  120. Hatvani, L., On the Asymptotic Stability of the Solutions of Functional Differential Equations, in Qualitative Theory Diff. Equat. Colloq. Math. Soc. J. Bolyai., North Holland, Amsterdam, 1990, vol. 53, pp. 227–238.

    MathSciNet  Google Scholar 

  121. Hatvani, L., On the Asymptotic Stability in Differential Systems by Liapunov Direct Method, Proc. 1st World Congr. Nonlinear Anal., Tampa, 1992, pp. 1341–1348.

  122. Hatvani, L., On the Asymptotic Stability by Lyapunov Functionals with Semidefinite Derivatives, Nonlinear Anal., TMA, 1997, vol. 30, no. 8, pp. 4713–4721.

    Article  MATH  MathSciNet  Google Scholar 

  123. Hatvani, L., Annulus Arguments in the Stability Thery for Functional Differential Equations, Diff. Integral Equat., 1997, vol. 10, no. 5, pp. 975–1002.

    MATH  MathSciNet  Google Scholar 

  124. Hatvani, L., On Lyapunov’s Direct Method for Nonautonomous Fde’s, Functional Diff. Equat., 1998, vol. 5, no. 3–4, pp. 315–323.

    MATH  MathSciNet  Google Scholar 

  125. Hatvani, L., On the Asymptotic Stability for Functional Differential Equations by Lyapunov Functionals, Nonlinear Anal., 2001, vol. 47, no. 7, pp. 4333–4343.

    Article  MATH  MathSciNet  Google Scholar 

  126. Hino, X., Stability Properties for Functional Differential Equations with Infinite Delay, Tohoku Math. J., 1983, vol. 35, pp. 597–605.

    Article  MATH  MathSciNet  Google Scholar 

  127. Hino, Y., Murakami, S., and Naito, T., Functional Differential Equations with Infinite Delay, Berlin: Springer-Verlag, 1991.

    MATH  Google Scholar 

  128. Ignatyev, A.O., On the Asymptotic Stability in Functional Differential Equations, Proc. Am. Math. Soc., 1999, vol. 127, no. 6, pp. 1753–1760.

    Article  MATH  MathSciNet  Google Scholar 

  129. Ignatyev, A.O., On the Partial Equiasymptotic Stability in Functional Differential Equations, J. Math. Anal. Appl., 2002, vol. 268, pp. 615–628.

    Article  MATH  MathSciNet  Google Scholar 

  130. Kato, J., Stability Problem in Functional Differential Equations with Infinite Delay, Funcial. Eqvac., 1978, vol. 21, pp. 63–80.

    MATH  Google Scholar 

  131. Kato, J., Uniform Asymptotic Stability and Total Stability, Tohoku Math. J., 1970, vol. 22, pp. 254–269.

    Article  MATH  MathSciNet  Google Scholar 

  132. Makay, G., An Example on the Asymptotic Stability for Functional Differential Equations, Nonl. Anal., TMA, 1994, vol. 23, pp. 365–368.

    Article  MATH  MathSciNet  Google Scholar 

  133. Murakami, S., Perturbation Theorems for Functional Differential Equations with Infinite Delay via Limiting Equations, J. Diff. Equat., 1985, vol. 59, pp. 314–335.

    Article  MATH  MathSciNet  Google Scholar 

  134. Saperstone, S.N., Semidynamical System in the Infinite Dimentional Space, New York: Springer-Verlag, 1981.

    Google Scholar 

  135. Sell, G.R., Nonautonomous Differential Equations and Topological Dynamics. I, II, Trans. Am. Math. Soc., 1967, vol. 22, pp. 241–283.

    Article  MathSciNet  Google Scholar 

  136. Wang, T., Weakening the Condition W 1(| φ(θ) |) ≤ V (t, φ) ≤ W 2(‖φ‖) for Uniform Asymptotic Stability, Nonlin. Anal. TMA, 1994, vol. 23, no. 2, pp. 251–264.

    Article  MATH  Google Scholar 

  137. Yoshizawa, T., Stability Theory and the Existence of Periodic Solutions and Almost-Periodic Solutions. Applied Math. Sciences, New York: Springer-Verlag, 1975, vol. 14.

    Google Scholar 

  138. Zhang, B., Asymptotic Stability in Functional Differential Equations by Lyapunov Functionals, Trans. Am. Math. Soc., 1995, vol. 347, no. 4, pp. 1375–1382.

    Article  MATH  Google Scholar 

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Original Russian Text © A.S. Andreev, 2009, published in Avtomatika i Telemekhanika, 2009, No. 9, pp. 4–55.

This work was supported by the Russian Foundation for Basic Research, project no. 08-01-00741 and the program “Developing the scientific potential of universities,” project no. 2.1.1/6194.

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Andreev, A.S. The Lyapunov functionals method in stability problems for functional differential equations. Autom Remote Control 70, 1438–1486 (2009). https://doi.org/10.1134/S0005117909090021

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