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
DOI:
https://doi.org/10.1090/S0002-9939-1970-0261118-0
MathSciNet review:
0261118
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Abstract | References | Similar Articles | Additional Information
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.
- Kenneth L. Cooke, Differential-difference equations, Internat. Sympos. Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, New York, 1963, pp. 155–171. MR 0146481
- L. È. Èl′sgol′c, Introduction to the theory of differential equations with deviating arguments, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1966. Translated from the Russian by Robert J. McLaughlin. MR 0192154
- N. N. Krasovskiĭ, Stability of motion. Applications of Lyapunov’s second method to differential systems and equations with delay, Stanford University Press, Stanford, Calif., 1963. Translated by J. L. Brenner. MR 0147744 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.
- M. Namık Oğuztöreli, Time-lag control systems, Mathematics in Science and Engineering, vol. 24, Academic Press, New York-London, 1966. MR 0217394
- F. R. Gantmacher, Matrizenrechnung. II. Spezielle Fragen und Anwendungen, Hochschulbücher für Mathematik, Bd. 37, VEB Deutscher Verlag der Wissenschaften, Berlin, 1959 (German). MR 0107647
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Additional Information
Keywords:
Stability,
linear differential system,
variable coefficients,
variable delays,
quasi-polynomial,
Lyapunov functional
Article copyright:
© Copyright 1970
American Mathematical Society