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

DOI:
https://doi.org/10.1090/S0002-9939-96-03266-2

MathSciNet review:
1317039

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The equation is considered under the assumption . It is proved that is sufficient for the asymptotic stability of , and is best possible here. This will be a consequence of a general result on the intermittent damping, which means that is controlled only on a sequence of non-overlapping intervals.

**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*, 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*, 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*, 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**

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

Email:
hatvani@math.u-szeged.hu

DOI:
https://doi.org/10.1090/S0002-9939-96-03266-2

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