On stability of linear time-varying second-order differential equations
Authors:
Luu Hoang Duc, Achim Ilchmann, Stefan Siegmund and Peter Taraba
Journal:
Quart. Appl. Math. 64 (2006), 137-151
MSC (2000):
Primary 34A30, 34D20, 35B40
DOI:
https://doi.org/10.1090/S0033-569X-06-00995-X
Published electronically:
January 24, 2006
MathSciNet review:
2211381
Full-text PDF Free Access
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Additional Information
Abstract: We derive sufficient conditions for stability and asymptotic stability of second order, scalar differential equations with differentiable coefficients.
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desoer69 C.E. Desoer, Slowly varying system $\dot x=A(t)x$. IEEE Trans. Automatic Control 14 (1969), 780–781.
dieci-vleck02 L. Dieci and E. Van Vleck, Lyapunov and other spectra: a survey. Collected lectures on the preservation of stability under discretization (Fort Collins, CO, 2001), 197–218, SIAM, Philadelphia, PA, 2002.
hatvani96 L. Hatvani, Integral conditions on the asymptotic stability for the damped linear oscillator with small damping. Proc. Amer. Math. Soc. 124 (1996), 415–422.
hatvani-kristin-totik95 L. Hatvani, T. Krisztin, and V. Totik, A necessary and sufficient condition for the asymptotic stability of the damped oscillator. J. Differential Equations 119 (1995), 209–223.
H66 F.C. Hoppenstaed, Singular perturbations on the infinite interval. Trans. Am. Math. Soc. 123 (1966), 521–535.
hovhannisyan04 G.R. Hovhannisyan, Asymptotic stability for second-order differential equations with complex coefficients. Electron. J. Differential Equations 85 (2004), 20 pp.
ignatyev97 A.O. Ignatyev, Stability of a linear oscillator with variable parameters. Electron. J. Differential Equations 17 (1997), 6 pp.
levin-nohel60 J.J. Levin and J.A. Nohel, Global asymptotic stability for nonlinear systems of differential equations and applications to reactor dynamics. Arch. Rational Mech. Anal. 5 (1960), 194–211.
pucci-serrin93 P. Pucci and J. Serrin, Precise damping conditions for global asymptotic stability for nonlinear second order systems. Acta Math. 170 (1993), no. 2, 275–307.
pucci-serrin94a P. Pucci and J. Serrin, Precise damping conditions for global asymptotic stability for nonlinear second order systems. II. J. Differential Equations 113 (1994), no. 2, 505–534.
pucci-serrin94b P. Pucci and J. Serrin, Asymptotic stability for intermittently controlled nonlinear oscillators. SIAM J. Math. Anal. 25 (1994), no. 3, 815–835.
rosenbrock63 H.H. Rosenbrock, The stability of linear time-dependent control systems. J. Electronics Control 15 (1963), 73–80.
R96 W.J. Rugh (1996), Linear System Theory 2nd ed., Prentice-Hall, Englewood Cliffs, New Jersey.
sacker-sell78 R.J. Sacker and G.R. Sell, A Spectral Theory for Linear Differential Systems. J. Differential Equations 27 (1978), 320–358.
siegmund02 S. Siegmund, Dichotomy spectrum for nonautonomous differential equations. J. Dynam. Differential Equations 14 (2002), no. 1, 243–258.
smith61 R.A. Smith, Asymptotic stability of $x” +a(t)x’ + x = 0$. Quart. J. Math. Oxford Ser. (2) 12 (1961), 123–126.
solo94 V. Solo, On the stability of slowly time-varying linear systems. Math. Control Signals Systems 7 (1994), no. 4, 331–350.
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Additional Information
Luu Hoang Duc
Affiliation:
Fachbereich Mathematik, J.W. Goethe Universität, Frankfurt, Germany
Email:
lhduc@math.uni-frankfurt.de
Achim Ilchmann
Affiliation:
Institut für Mathematik, Technische Universität Ilmenau, Ilmenau, Germany
Email:
achim.ilchmann@tu-ilmenau.de
Stefan Siegmund
Affiliation:
Fachbereich Mathematik, J.W. Goethe Universität, Frankfurt, Germany
MR Author ID:
687393
Email:
siegmund@math.uni-frankfurt.de
Peter Taraba
Affiliation:
Fachbereich Mathematik, J.W. Goethe Universität, Frankfurt, Germany
Email:
taraba@math.uni-frankfurt.de
Received by editor(s):
April 11, 2005
Published electronically:
January 24, 2006
Article copyright:
© Copyright 2006
Brown University
