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

Hatvani, L.

doi:10.1090/S0002-9939-96-03266-2

Integral conditions on the asymptotic stability for the damped linear oscillator with small damping
HTML articles powered by AMS MathViewer

by L. Hatvani PDF

Proc. Amer. Math. Soc. 124 (1996), 415-422 Request permission

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

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. MR 466789, DOI 10.1090/S0033-569X-1976-0466789-0
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. 65 (1978), no. 2, 321–332. MR 506309, DOI 10.1016/0022-247X(78)90183-X
T. A. Burton and J. W. Hooker, On solutions of differential equations tending to zero, J. Reine Angew. Math. 267 (1974), 151–165. MR 348204, DOI 10.1515/crll.1974.267.151
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
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).
László Hatvani and Vilmos Totik, Asymptotic stability of the equilibrium of the damped oscillator, Differential Integral Equations 6 (1993), no. 4, 835–848. MR 1222304
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), no. 2-4, 385–393. MR 874506
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. MR 890560
Viktor Kertész, Stability investigations by indefinite Lyapunov functions, Alkalmaz. Mat. Lapok 8 (1982), no. 3-4, 307–322 (Hungarian, with English summary). MR 709152
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). MR 119524, DOI 10.1007/BF00252903
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. MR 1226530, DOI 10.1007/BF02392788
—, Precise damping conditions for global asymptotic stability for non-linear second order systems, II, J. Differential Equations 113 (1994), 505-534.
Patrizia Pucci and James Serrin, Asymptotic stability for intermittently controlled nonlinear oscillators, SIAM J. Math. Anal. 25 (1994), no. 3, 815–835. MR 1271312, DOI 10.1137/S0036141092240679
R. A. Smith, Asymptotic stability of $x^{\prime \prime }+a(t)x^{\prime } +x=0$, Quart. J. Math. Oxford Ser. (2) 12 (1961), 123–126. MR 124582, DOI 10.1093/qmath/12.1.123
A. G. Surkov, Asymptotic stability of certain two-dimensional linear systems, Differentsial′nye Uravneniya 20 (1984), no. 8, 1452–1454 (Russian). MR 759607