Generalized matrix diagonal stability and linear dynamical systems

Let A = (aij ) be a real square matrix and 1 p ∞. We present two analogous developments. One for Schur stability and the discrete-time dynamical system x(t + 1) = Ax(t), and the other for Hurwitz stability and the continuous-time dynamical system ẋ(t) = Ax(t). Here is a description of the latter development. For A, we define and study “Hurwitz diagonal stability with respect to p-norms”, abbreviated as “HDSp”. HDS2 is the usual concept of diagonal stability. A is HDSp implies “Re λ < 0 for every eigenvalue λ of A”, which means A is “Hurwitz stable”, abbreviated as “HS”. When the off-diagonal elements of A are nonnegative, A is HS iff A is HDSp for all p. For the dynamical system ẋ(t) = Ax(t), we define “diagonally invariant exponential stability relative to the p-norm”, abbreviated as DIESp, meaning there exist time-dependent sets, which decrease exponentially and are invariant with respect to the system. We show that DIESp is a special type of exponential stability and the dynamical system has this property iff A is HDSp. © 2006 Elsevier Inc. All rights reserved. AMS classification: 15A45; 15A60; 15A48; 15A18; 34A30; 34C14; 34D20; 93C05; 93D20