Generalized matrix diagonal stability and linear dynamical systems

Abstract

Let A = ( a ij ) 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 x ˙ ( 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 “HDS p ”. HDS 2 is the usual concept of diagonal stability. A is HDS p implies “Re λ < 0 for every eigenvalue λ of A”, which means A is “Hurwitz stable”, abbreviated “HS”. When the off-diagonal elements of A are nonnegative, A is HS iff A is HDS p for all p. For the dynamical system x ˙ ( t ) = Ax ( t ) , we define “diagonally invariant exponential stability relative to the p-norm”, abbreviated DIES p , meaning there exist time-dependent sets, which decrease exponentially and are invariant with respect to the system. We show that DIES p is a special type of exponential stability and the dynamical system has this property iff A is HDS p .