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
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
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.
- J. J. Levin and J. A. Nohel, Global asymptotic stability for nonlinear systems of differential equations and applications to reactor dynamics, Bol. Soc. Mat. Mexicana (2) 5 (1960), 152–157. MR 133529
- 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 https://doi.org/10.1515/crll.1974.267.151
- Nelson Onuchic, Invariance properties in the theory of ordinary differential equations with applications to stability problems, SIAM J. Control 9 (1971), 97–104. MR 0279394
- J. P. LaSalle, The extent of asymptotic stability, Proc. Nat. Acad. Sci. U.S.A. 46 (1960), 363–365. MR 113014, DOI https://doi.org/10.1073/pnas.46.3.363
- Joseph P. LaSalle, An invariance principle in the theory of stability, Differential Equations and Dynamical Systems (Proc. Internat. Sympos., Mayaguez, P.R., 1965) Academic Press, New York, 1967, pp. 277–286. MR 0226132
- J. P. LaSalle, Stability theory for ordinary differential equations, J. Differential Equations 4 (1968), 57–65. MR 222402, DOI https://doi.org/10.1016/0022-0396%2868%2990048-X
- David R. Wakeman, An application of topological dynamics to obtain a new invariance property for nonautonomous ordinary differential equations, J. Differential Equations 17 (1975), 259–295. MR 385250, DOI https://doi.org/10.1016/0022-0396%2875%2990044-3
- Joseph P. LaSalle, Stability theory and invariance principles, Dynamical systems (Proc. Internat. Sympos., Brown Univ., Providence, R.I., 1974) Academic Press, New York, 1976, pp. 211–222. MR 0594977
- Zvi Artstein, The limiting equations of nonautonomous ordinary differential equations, J. Differential Equations 25 (1977), no. 2, 184–202. MR 442381, DOI https://doi.org/10.1016/0022-0396%2877%2990199-1
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)
N. Onuchic, Invariance properties in the theory of ordinary differential equations with applications to stability problems, SIAM J. Control 9, 97–104 (1971)
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)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
34D05
Retrieve articles in all journals
with MSC:
34D05
Additional Information
Article copyright:
© Copyright 1976
American Mathematical Society
