×

zbMATH — the first resource for mathematics

Integral conditions on the asymptotic stability for the damped linear oscillator with small damping. (English) Zbl 0844.34051
The author considers the equation (1) \(x'' + h(t)x' + k^2x = 0\) under the assumption (2) \(0 \leq h(t) \leq \overline h < \infty\). He proves that the condition \(\limsup_{t \to \infty} (t^{- 2/3} \int^t_0 h(s) ds) > 0\) is sufficient for the asymptotic stability of \(x = x' = 0\), and the exponent \(2/3\) is best possible. He obtains this as a corollary of a general result on intermittent damping which reads as follows: Suppose that (2) is satisfied. If there exists a sequence \(\{I_n\}\) of non-overlapping intervals such that \[ \sum^\infty_{n = 1} {1 \over 1 + |I_n |^2} \left( \int^t_0 h(s)ds \right)^3 = \infty, \] then the zero solution of (1) is asymptotically stable. Moreover, the exponent 3 in the statement is best possible.
Reviewer: W.Müller (Berlin)

MSC:
34D20 Stability of solutions to ordinary differential equations
34A30 Linear ordinary differential equations and systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Zvi Artstein and E. F. Infante, On the asymptotic stability of oscillators with unbounded damping, Quart. Appl. Math. 34 (1976/77), no. 2, 195 – 199. · Zbl 0336.34048
[2] R. J. Ballieu and K. Peiffer, Attractivity of the origin for the equation \?+\?(\?,\?, \?)\?^{\?}\?+\?(\?)=0, J. Math. Anal. Appl. 65 (1978), no. 2, 321 – 332. · Zbl 0387.34038
[3] T. A. Burton and J. W. Hooker, On solutions of differential equations tending to zero, J. Reine Angew. Math. 267 (1974), 151 – 165. · Zbl 0298.34042
[4] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1952. · Zbl 0047.05302
[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). · Zbl 0831.34052
[6] László Hatvani and Vilmos Totik, Asymptotic stability of the equilibrium of the damped oscillator, Differential Integral Equations 6 (1993), no. 4, 835 – 848. · Zbl 0777.34036
[7] J. Karsai, On the global asymptotic stability of the zero solution of the equation \?+\?(\?,\?,\?)\?+\?(\?)=0, Studia Sci. Math. Hungar. 19 (1984), no. 2-4, 385 – 393. · Zbl 0528.34050
[8] J. Karsai, On the asymptotic stability of the zero solution of certain nonlinear second order differential equations, Differential equations: qualitative theory, Vol. I, II (Szeged, 1984) Colloq. Math. Soc. János Bolyai, vol. 47, North-Holland, Amsterdam, 1987, pp. 495 – 503. · Zbl 0624.34042
[9] Viktor Kertész, Stability investigations by indefinite Lyapunov functions, Alkalmaz. Mat. Lapok 8 (1982), no. 3-4, 307 – 322 (Hungarian, with English summary). Viktor Kertész, Stability investigation of the differential equation of damped oscillation, Alkalmaz. Mat. Lapok 8 (1982), no. 3-4, 323 – 339 (Hungarian, with English summary).
[10] 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 (1960). · Zbl 0094.06402
[11] Patrizia Pucci and James Serrin, Precise damping conditions for global asymptotic stability for nonlinear second order systems, Acta Math. 170 (1993), no. 2, 275 – 307. · Zbl 0797.34059
[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] Patrizia Pucci and James Serrin, Asymptotic stability for intermittently controlled nonlinear oscillators, SIAM J. Math. Anal. 25 (1994), no. 3, 815 – 835. · Zbl 0809.34067
[14] R. A. Smith, Asymptotic stability of \?\(^{\prime}\)\(^{\prime}\)+\?(\?)\?\(^{\prime}\)+\?=0, Quart. J. Math. Oxford Ser. (2) 12 (1961), 123 – 126. · Zbl 0103.05604
[15] A. G. Surkov, Asymptotic stability of certain two-dimensional linear systems, Differentsial\(^{\prime}\)nye Uravneniya 20 (1984), no. 8, 1452 – 1454 (Russian). · Zbl 0562.34043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.