Construction of the instability theory of two-dimensional nonautonomous systems and its application

已知的数学模型在许多情况下是用具有周期或常系数的微分方程组来描述的。在本研究中，我们针对的是具有时变系数的微分系统(我们称之为非自治微分系统)，这可能对应于其他情况。特别地，对于线性系统和含阻尼振子的半线性系统，给出了一致渐近稳定及其不稳定的一些充分条件。所得到的一致渐近稳定性的结果可望用于构造一个具有良好性质的Lyapunov函数。因此，本研究对系统的摄动问题和鲁棒稳定性问题具有重要的意义。

The known mathematical models are described by differential equations with periodic or constant coefficients in many cases. In this study, we targeted differential systems with time-varying coefficients (we call non-autonomous differential systems), which may correspond to other cases. In particular, I gave some sufficient conditions of uniform asymptotic stability and its instability for the linear systems and half-linear systems including a damped oscillator. The obtained results for uniform asymptotic stability can be expected strongly to be applied to giving a Lyapunov function with good properties. Therefore, the present study can contribute significantly to the perturbation problem and the robust stability problem.