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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Total stability of sets for nonautonomous differential systems
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by Zhivko S. Athanassov PDF
Trans. Amer. Math. Soc. 295 (1986), 649-663 Request permission

Abstract:

The principal purpose of this paper is to present sufficient conditions for total stability, or stability under constantly acting perturbations, of sets of a sufficiently general kind for nonautonomous ordinary differential equations. To do this, two Liapunov-like functions with specific properties are used. The obtained results include and considerably improve the classical results on total stability of isolated equilibrium points. Applications are presented to study the stability of nonautonomous Lurie-type nonlinear equations.
References
    A. M. Liapounoff [Liapunov], Problème général de la stabilité du mouvement, Ann. of Math. Studies, no. 17, Princeton Univ. Press, Princeton, N. J., 1949.
  • J. P. LaSalle, The stability of dynamical systems, Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. With an appendix: “Limiting equations and stability of nonautonomous ordinary differential equations” by Z. Artstein. MR 0481301
  • V. M. Matrosov, On the stability of motion, J. Appl. Math. Mech. 26 (1963), 1337–1353. MR 0153934, DOI 10.1016/0021-8928(62)90010-2
  • Nicolas Rouche, P. Habets, and M. Laloy, Stability theory by Liapunov’s direct method, Applied Mathematical Sciences, Vol. 22, Springer-Verlag, New York-Heidelberg, 1977. MR 0450715
    • G. N. Duboshin, The problem of the stability of motion under persistently acting perturbations, Trudy Gos. Astr. Inst. Sternberg 14 (1940), no. 1.
  • Zhivko S. Athanassov, Perturbation theorems for nonlinear systems of ordinary differential equations, J. Math. Anal. Appl. 86 (1982), no. 1, 194–207. MR 649865, DOI 10.1016/0022-247X(82)90264-5
  • Zhivko S. Athanassov, On the asymptotic behaviour of nonlinear systems of ordinary differential equations, Glasgow Math. J. 26 (1985), no. 2, 161–170. MR 798744, DOI 10.1017/S0017089500005942
  • Zhivko S. Athanassov, Existence, uniqueness, and stability theorems for bounded solutions of nonlinear differential systems, Ann. Mat. Pura Appl. (4) 140 (1985), 301–330. MR 807642, DOI 10.1007/BF01776854
    • F. Brauer, Some stability and perturbation problems for differential and integral equations, Departamento de Matematica, Universidade de Brasilia, Brasilia, 1974.
  • Wolfgang Hahn, Theory and application of Liapunov’s direct method, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. English edition prepared by Siegfried H. Lehnigk; translation by Hans H. Losenthien and Siegfried H. Lehnigk. MR 0147716
  • V. I. Zubov, Methods of A. M. Lyapunov and their application, P. Noordhoff Ltd., Groningen, 1964. Translation prepared under the auspices of the United States Atomic Energy Commission; edited by Leo F. Boron. MR 0179428
  • N. P. Bhatia and G. P. Szegö, Stability theory of dynamical systems, Die Grundlehren der mathematischen Wissenschaften, Band 161, Springer-Verlag, New York-Berlin, 1970. MR 0289890
  • A. A. Šestakov, Tests for the instability of a set with respect to a nonautonomous differential system, Differencial′nye Uravnenija 13 (1977), no. 5, 958–960, 968 (Russian). MR 0481306
  • A. A. Šestakov, Criteria for the stability of sets with respect to a nonautonomous differential system, Differencial′nye Uravnenija 13 (1977), no. 6, 1079–1090, 1157 (Russian). MR 0481307
    • J. P. LaSalle and G. Szegö, Comments on L. Hwang’s paper "New methods for constructing Liapunov functions for time-invariant control systems", Proc. Second Congr. IFAC, Butterworths, London, 1964, p. 582.
  • J. P. LaSalle, Recent advances in Liapunov stability theory, SIAM Rev. 6 (1964), 1–11. MR 179429, DOI 10.1137/1006001
  • Taro Yoshizawa, Eventual properties and quasi-asymptotic stability of a non-compact set, Funkcial. Ekvac. 8 (1966), 79–90. MR 197893
  • Otomar Hájek, Absolute stability of noncompact sets, J. Differential Equations 9 (1971), 496–508. MR 273154, DOI 10.1016/0022-0396(71)90019-2
  • Otomar Hájek, Ordinary and asymptotic stability of noncompact sets, J. Differential Equations 11 (1972), 49–65. MR 289880, DOI 10.1016/0022-0396(72)90080-0
  • I. Malkin, Stability in the case of constantly acting disturbances, Appl. Math. Mech. [Akad. Nauk SSSR. Prikl. Mat. Mech.] 8 (1944), 241–245 (Russian, with English summary). MR 0014530
    • S. I. Gorshin, On the stability of motion under constantly acting perturbations, Izv. Akad. Nauk Kazakh. SSR Ser. Mat. Mekh. 2 56 (1948), 46-73.
  • Solomon Lefschetz, Stability of nonlinear control systems, Mathematics in Science and Engineering, Vol. 13, Academic Press, New York-London, 1965. MR 0175676
  • M. A. Aizerman and F. R. Gantmacher, Absolute stability of regulator systems, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964. Translated by E. Polak. MR 0183556
  • I. G. Malkin, On the reversibility of Lyapunov’s theorem on asymptotic stability, Akad. Nauk SSSR. Prikl. Mat. Meh. 18 (1954), 129–138 (Russian). MR 0061734
  • José L. Massera, Contributions to stability theory, Ann. of Math. (2) 64 (1956), 182–206. MR 79179, DOI 10.2307/1969955
  • I. G. Malkin, On the construction of Lyapunov functions for systems of linear equations, Akad. Nauk SSSR. Prikl. Mat. Meh. 16 (1952), 239–242 (Russian). MR 0047864
  • Fozi M. Dannan and Saber Elaydi, Lipschitz stability of nonlinear systems of differential equations, J. Math. Anal. Appl. 113 (1986), no. 2, 562–577. MR 826653, DOI 10.1016/0022-247X(86)90325-2
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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 295 (1986), 649-663
  • MSC: Primary 34D10; Secondary 34D20, 93D05
  • DOI: https://doi.org/10.1090/S0002-9947-1986-0833701-9
  • MathSciNet review: 833701