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Asymptotic Stability of Coupled Oscillators with Time-Dependent Damping

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Abstract

The present paper is devoted to an investigation on the asymptotic stability for the damped oscillators with multiple degrees of freedom,

$$\begin{aligned} {\mathbf {x}}'' + h(t)\,{\mathbf {x}}' + A\,{\mathbf {x}} = {\mathbf {0}} \end{aligned}$$

and its generalization

$$\begin{aligned} M\,{\mathbf {x}}'' + C(t)\,{\mathbf {x}}' + K\,{\mathbf {x}} = {\mathbf {0}}, \end{aligned}$$

where \(h: [0,\infty ) \rightarrow [0,\infty )\) is a function, A, M and K are \(n \times n\) real constant matrices. and C is an \(n \times n\) matrix whose elements are real-valued functions. The functions h and C correspond to the damping coefficient and the damping matrix, respectively. The origin \(({\mathbf {x}},{\mathbf {x}}') = ({\mathbf {0}},{\mathbf {0}})\) is the only equilibrium of the above-mentioned damped oscillators. Necessary and sufficient conditions are presented for the equilibrium of these oscillators to be asymptotically stable. The obtained conditions are given by the forms of certain growth conditions concerning the damping h and C, respectively.

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Acknowledgments

This work was supported in part by Grant-in-Aid for Scientific Research, No. 25400165, from the Japan Society for the Promotion of Science. The author thanks the anonymous reviewer for their valuable comments.

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  1. Department of Mathematics and Computer Science, Shimane University, Matsue, 690-8504, Japan

    Jitsuro Sugie

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  1. Jitsuro Sugie
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Sugie, J. Asymptotic Stability of Coupled Oscillators with Time-Dependent Damping. Qual. Theory Dyn. Syst. 15, 553–573 (2016). https://doi.org/10.1007/s12346-015-0175-7

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