Stabilization of asymptotically time-invariant linear time-varying systems

In this paper, we study the stabilization problem of asymptotically time-invariant linear time-varying systems within the framework of nest algebras. The main theorem of the paper establishes that an asymptotically time-invariant system is stabilizable if and only if the time-invariant part is also stabilizable. The proof relies on the theory of Fredholm and compact operators. In particular, when the compact part of systems are strictly causal, we show that a controller (possibly time-varying) stabilizes system if and only if it stabilizes the time-invariant part.