On the asymptotic stability of oscillators with unbounded damping
Authors:
Zvi Artstein and E. F. Infante
Journal:
Quart. Appl. Math. 34 (1976), 195-199
MSC:
Primary 34D05
DOI:
https://doi.org/10.1090/qam/466789
MathSciNet review:
466789
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Abstract: Through a technique inspired by the invariance principle of LaSalle, a general growth condition on the damping coefficient $h\left ( t \right )$ of the equation \[ \ddot x + h\left ( t \right )\dot x + kx = 0, k > 0,h(t) \ge \epsilon > 0\], is given which is sufficient for the global asymptotic stability of the origin yet permits this coefficient to grow to infinity with time. The methods used do not depend on linearity, and are applied to obtain similar results to the nonlinear analogue of this equation.
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J. J. Levin and J. A. Nohel, Global asymptotic stability of nonlinear systems of differential equations and applications to reactor dynamics, Arch. Rat. Mech. Anal. 5, 194–211 (1960)
T. A. Burton and J. W. Hooker, On solutions of differential equations tending to zero, J. Reine Angew. Math. 267, 151–165 (1974)
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J. P. LaSalle, The extent of asymptotic stability, Proc. Nat. Acad. Sci. 46, 363–365 (1960)
J. P. LaSalle, An invariance principle in the theory of stability, in Differential equations and dynamical systems, eds. J. K. Hale, and J. P. LaSalle, Academic Press, 1967, pp. 277–286
J. P. LaSalle, Stability theory for ordinary differential equations, J. Diff. Equations 9, 57–65 (1968)
D. R. Wakeman, An application to topological dynamics to obtain a new invariance property for non-autonomous ordinary differential equations, J. Diff. Equations 17, 259–295 (1975)
J. P. LaSalle, Stability theory and invariance principles, in Proc. international symposium on dynamical systems, Brown University, August 1974, Academic Press (to appear)
Z. Artstein, The limiting equations of nonautonomous ordinary differential equations, J. Diff. Eqs. (to appear)
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© Copyright 1976
American Mathematical Society