This paper studies exponential stability and stabilization of linear time-varying systems without assuming that the coefficient matrix is bounded, which is generally necessary in the existing literature. First, by introducing the notions of uniformly exponentially stable and uniformly exponentially expanding functions, some Lyapunov differential inequalities based characterizations for exponential stability of linear time-varying systems are established. Second, it is shown that the linear time-varying system is both uniformly exponentially stable and uniformly exponentially expanding if and only if the corresponding Lyapunov differential equation has a unique positive definite solution. Third, some degenerated Lyapunov differential equations, where the $Q$ matrix is only positive semi-definite, are also re-examined. Finally, a weighted controllability Gramian, which contains a weighting time-varying function that can be properly designed to reveal a trade-off between the convergence rate/regulation time of the closed-loop system and the control effort, based stabilization approach is established to ensure that the closed-loop system is both uniformly exponentially stable and uniformly exponentially expanding. The effectiveness of the proposed approach is illustrated by the design of the spacecraft attitude control system with magnetic actuation.

本文在不假设系数矩阵有界的情况下研究了线性时变系统的指数稳定性和镇定性，这在现有文献中是普遍需要的。首先，通过引入一致指数稳定和一致指数展开函数的概念，建立了线性时变系统指数稳定性的一些基于李雅普诺夫微分不等式的表征。其次，证明了线性时变系统既是一致指数稳定的，又是一致指数扩张的，当且仅当相应的李雅普诺夫微分方程具有唯一的正定解。第三，一些退化的 Lyapunov 微分方程，其中$问$矩阵只是半正定的，也被重新检查。最后，一个加权可控性 Gramian，它包含一个加权时变函数，可以适当地设计以揭示闭环系统的收敛速度/调节时间与控制努力之间的权衡，基于稳定方法被建立确保闭环系统既一致指数稳定又一致指数扩张。所提出方法的有效性通过具有磁驱动的航天器姿态控制系统的设计来说明。