Stability of a Linear Oscillator with Variable Parameters
Stability of a Linear Oscillator with Variable Parameters
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发表时间：
1997-10
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通讯作者：
A. Ignatyev
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作者：
A. Ignatyev
得到了变参数线性振子渐近稳定的一个判据。证明了该判据接近于渐近稳定的一个充要条件。证明了一个不稳定性定理，并考虑了一个力学例子。其中阻尼和刚度系数sf（t）和ndg（t）是时间t的连续有界函数。大多数研究零解稳定性问题的理论都是基于李雅普诺夫稳定性和不稳定性定理，相应的李雅普诺夫函数被假定为能量型函数V = 1c1（t）x2 + 1c2（t）x2，其中c1（t），c2（t）是时变函数。在（6）中，A. P.Merkin考虑了c1（t）= c2（t）= 1的情况，得到了仅对常数f和g的稳定性条件。在（15）中对周期函数f（t）和ndg（t）进行了扩展。利用一个关于x和_ x的二次型李雅普诺夫函数，V. M. Starzhinsky（10）（假设0
A criterion of asymptotic stability for a linear oscillator with variable parameters is obtained. It is shown that this criterion is close to a necessary and sucient conditions of asymptotic stability. An instability theorem is proved, and a mechanical example is considered. where the damping and rigidity coecientsf(t )a ndg (t) are continuous and bounded functions of the time t. Most of the theories examining a stability problem of the zero solution are based on the Lyapunov stability and instability theorems and the corresponding Lyapunov function is assumed as an energy-type function V = 1 c1(t )_ x 2 + 1 c 2 (t )x 2 ; where c1(t);c 2 (t) are time variable functions. In (6), A. P. Merkin considered the case c1(t )= c 2 ( t) = 1 and stability conditions were obtained only for constant f and g. An extension was done in (15) for periodic functions f (t )a ndg (t). By means of a Lyapunov function which is a quadratic form with respect to x and _ x ,V. M. Starzhinsky (10) (assuming that 0