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Stability of solutions of linear systems with retarded arguments

Author: John Abramowich
Journal: Proc. Amer. Math. Soc. 26 (1970), 60-64
MSC: Primary 34.75
MathSciNet review: 0261118
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Abstract: A criterion is given for the stability of the trivial solution of a linear system of differential equations with variable coefficients and variable delays under the assumptions of continuity and boundedness of the coefficients and continuous differentiability of the delays. The criterion is a condition on the roots of a certain ``quasi-polynomial,'' i.e. a polynomial in a variable and exponentials of that variable. The derivation is based on Lyapunov's direct method.

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  • [1] Kenneth L. Cooke, Differential-difference equations, Internat. Sympos. Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, New York, 1963, pp. 155–171. MR 0146481
  • [2] L. È. Èl′sgol′c, Introduction to the theory of differential equations with deviating arguments, Translated from the Russian by Robert J. McLaughlin, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1966. MR 0192154
  • [3] N. N. Krasovskiĭ, Stability of motion. Applications of Lyapunov’s second method to differential systems and equations with delay, Translated by J. L. Brenner, Stanford University Press, Stanford, Calif., 1963. MR 0147744
  • [4] W. Hahn, Theory and application of Lyapunov's direct method, Ergebnisse der Mathematik und ihrer Grenzgebiete, Heft 22, Springer-Verlag, Berlin and New York, 1959; English transl., Prentice-Hall, Englewood Cliffs, N. J., 1963. MR 21 #5048; MR 26 #5230.
  • [5] M. Namık Oğuztöreli, Time-lag control systems, Mathematics in Science and Engineering, vol. 24, Academic Press, New York-London, 1966. MR 0217394
  • [6] F. R. Gantmacher, The theory of matrices. Vols. 1, 2, Translated by K. A. Hirsch, Chelsea Publishing Co., New York, 1959. MR 0107649

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Keywords: Stability, linear differential system, variable coefficients, variable delays, quasi-polynomial, Lyapunov functional
Article copyright: © Copyright 1970 American Mathematical Society