Lyapunov differential equations and inequalities for stability and stabilization of linear time-varying systems

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.