Asymptotic Stability of Coupled Oscillators with Time-Dependent Damping
Tóm tắt
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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