Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Integral conditions on the asymptotic stability
for the damped linear oscillator
with small damping

Author: L. Hatvani
Journal: Proc. Amer. Math. Soc. 124 (1996), 415-422
MSC (1991): Primary 34D20, 34A30
MathSciNet review: 1317039
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The equation $x''+h(t)x'+k^2x=0$ is considered under the assumption $0\le h(t)\le \overline{h}<\infty$ $(t\ge 0)$. It is proved that $\limsup _{t \to \infty}\left(t^{-2/3}\int_0 ^t h\right)>0$ is sufficient for the asymptotic stability of $x=x'=0$, and $2/3$ is best possible here. This will be a consequence of a general result on the intermittent damping, which means that $h$ is controlled only on a sequence of non-overlapping intervals.

References [Enhancements On Off] (What's this?)

  • 1 Z. Artstein and E. F. Infante, On the asymptotic stability of oscillators with unbounded damping, Quart. Appl. Mech. 34 (1976), 195-199. MR 57:6665
  • 2 R. J. Ballieu and K. Peiffer, Attractivity of the origin for the equation $\ddot x+f(t,x,\dot x)|\dot x|^\alpha \dot x% +g(x) % =0$, J. Math. Anal. Appl. 34 (1978), 321-332. MR 80a:34057
  • 3 T. A. Burton and J. W. Hooker, On solutions of differential equations tending to zero, J. Reine Angew. Math. 267 (1974), 151-165. MR 50:702
  • 4 G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1952. MR 13:727e
  • 5 L. Hatvani, T. Krisztin, and V. Totik, A necessary and sufficient condition for the asymptotic stability of the damped oscillator, J. Differential Equations (to appear).
  • 6 L. Hatvani and V. Totik, Asymptotic stability of the equilibrium of the damped oscillator, Differential Integral Equations 6 (1993), 835-848. MR 94c:34080
  • 7 J. Karsai, On the global asymptotic stability of the zero solution of the equation $\ddot x + % g(t,x,\dot x)\dot x +f(x)=0$, Studia Sci. Math. Hungar. 19 (1984), 385-393. MR 87m:34064
  • 8 ------, On the asymptotic stability of the zero solution of certain nonlinear second order differential equations, Differential Equations: Qualitative Theory (Szeged, 1984), Colloq. Math. Soc. János Bolyai, vol. 47, North-Holland, Amsterdam and New York, 1987, pp. 495-503. MR 88e:34087
  • 9 V. Kertész, Stability investigation of the differential equation of damped oscillation, Alkalmaz. Mat. Lapok 8 (1982), 323-339. (Hungarian, English summary) MR 85h:34058b
  • 10 J. J. Levin and J. A. Nohel, Global asymptotic stability of nonlinear systems of differential equations and applications to reactor dynamics, Arch. Rational Mech. Anal. 5 (1960), 194-211. MR 22:10285
  • 11 P. Pucci and J. Serrin, Precise damping conditions for global asymptotic stability for nonlinear second order systems, Acta Math. 170 (1993), 275-307. MR 94i:34103
  • 12 ------, Precise damping conditions for global asymptotic stability for non-linear second order systems, II, J. Differential Equations 113 (1994), 505-534. CMP 95:02
  • 13 ------, Asymptotic stability for intermittently controlled nonlinear oscillators, SIAM J. Math. Anal. 25 (1994), 815-835. MR 95c:34092
  • 14 R. A. Smith, Asymptotic stability of $x'' +a(t)x' +x=0$, Quart. J. Math. Oxford Ser. (2) 12 (1961), 123-126. MR 23:A1894
  • 15 A. G. Surkov, On asymptotic stability of some two dimensional linear systems, Differentsial'nye Uravneniya 20 (1984), 1452-1454. (Russian) MR 85j:34098

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 34D20, 34A30

Retrieve articles in all journals with MSC (1991): 34D20, 34A30

Additional Information

L. Hatvani
Affiliation: Bolyai Institute, Aradi vértanúk tere 1, Szeged, Hungary, H–6720

Keywords: Intermittent damping, energy method
Received by editor(s): January 7, 1994
Additional Notes: The author was supported by the Hungarian National Foundation for Scientific Research with grant number 1157
Communicated by: Hal L. Smith
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society