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On the elliptic ball stability criterion for ordinary differential equations

Published online by Cambridge University Press:  24 October 2008

Russell A. Smith
Russell A. Smith
Affiliation:
University of Durham
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Abstract

It is proved that the elliptic ball criterion is a necessary condition on the matrix M(t) for the linear feedback control equation f(D)x + BM(t)g(D)x = 0 to have a special kind of quadratic Lyapunov function. That it is also a sufficient condition has already been proved elsewhere. These two facts lead to a comparison theorem which enables the existence of a quadratic Lyapunov function for one equation to be deduced from that for another equation.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 74 , Issue 3 , November 1973 , pp. 497 - 505
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

REFERENCES

(1)Brockett, R. W. and Lee, H. B.Frequency domain instability criteria for time-varying and nonlinear systems. Proc. IEEE 55 (1967), 604–619.Google Scholar
(2)Lim, Y. S. and Kazda, L. F.A Study of second order nonlinear systems. J. Math. Anal. Appl. 8 (1964), 423–444.Google Scholar
(3)Harden, M.Geometry of polynomials (Providence R.I., American Mathematical Society, 1966).Google Scholar
(4)Newman, M. H. A.Elements of the topology of plane sets of points, 2nd edn. (Cambridge University Press, 1951).Google Scholar
(5)Smith, R. A.On absolute stability of certain differential equations. J. London Math. Soc. (to appear).Google Scholar
(6)Starzinskii, V. M.Sufficient conditions for stability of a mechanical system with one degree of freedom [Russian]. Prikl. Mat. Meh. 16 (1952), 369–374.Google Scholar