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

For an arbitrary n×n constant matrix A the two following facts are well known:

• (1/n)Re(traceA)−max_j=1,…,nRe 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 systemsAs 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.