New Results for the Stability Analysis of Time-Varying Linear Systems Part I: The Case of Reduced Systems

Abstract

Let A(t) be a real-valued matrix function on [0, + ¿]. A (time-varying) Linear Dynamical Systems (LDS) of the form x=A(t)x is said to be well-defined if A(t) is Lebesgue integrable on every finite subinterval of [0, + ¿], and is called proper if A(t)=f(t,G) for some constant generating matrix G and scalar primitive function f(t, ¿) (see [13], [15]). According to a recent result obtained in [17], every well-defined LDS is reducible to a proper one by a D-similarity transformation. Therefore, it is interesting to study the stability of the reduced (proper) LDS as a means for uncovering new stability information for well-defined LDS. In this paper we use some recent results on proper LDS (see [13], [14], [15], [16]), to derive new necessary and sufficient stability criteria for proper time-varying LDS in terms of the conventional (time-varying) eigenvalues of A(t) and a new entity we have named co-eigenvalues of A(t). The notion of stability index for proper A(t) with Laplace transformable elements is also introduced and serves to unify the well-known stability criteria for time-invariant LDS and periodic proper LDS which are based on the eigenvalues of A and the Floquet characteristic exponents of A(t), respectively.