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Some elliptic balls which avoid a Nyquist set in ℂn+1

Published online by Cambridge University Press:  14 November 2011

Russell A. Smith
Affiliation:
Department of Mathematics, University of Durham

Synopsis

Conditions are obtained for certain elliptic balls in ℂn+1 to have empty intersection with the Nyquist set of a vector polynomial G(z). Such conditions are shown to yield explicit criteria for the existence of periodic solutions of non-autonomous scalar differential equations of the form A* G(D)y = p.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1978

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References

1Marden, M.Geometry of Polynomials (2nd edn.) (Providence, R. I.: Amer. Math. Soc, 1966).Google Scholar
2Skačkov, B. N.On the stability of the zero solution of a differential equation of the n-th order. Vestnik Leningrad Univ. 17 (19) (1962), 5661.Google Scholar
3Smith, R. A.Absolute stability of certain differential equations. J. London Math. Soc. 7 (1973), 203210.CrossRefGoogle Scholar
4Smith, R. A.On the elliptic ball stability criterion for ordinary differential equations. Proc. Cambridge Philos. Soc. 74 (1973), 497505.CrossRefGoogle Scholar
5Smith, R. A.Inclusion conditions for Hurwitzian and Schur sets in ℂn+1. Math. Proc. Cambridge Philos. Soc. 80 (1976), 113120.CrossRefGoogle Scholar
6Smith, R. A.Forced oscillations of the feedback control equation. Proc. Roy. Soc. Edinburgh Sect. A. 76 (1976), 3142.CrossRefGoogle Scholar