A necessary condition for quantitative exponential stability of linear state-space systems

Abstract

For an arbitrary n×n constant matrix A the two following facts are well known: • (1/n) Re(trace A) − max j=1,…,n Re λ j(A)⩽0 ; • If U is a unitary matrix, one can always find a skew-Hermitian matrix A so that U=e A . In this note we present the extension of these two facts to the context of linear time-varying dynamical systems x ̇ =A(t)x, t⩾0. As a by-product, this result suggests that, the notion of “slowly varying state-space systems”, commonly used in literature, is mathematically not natural to the problem of exponential stability.