Summary: Let be a real square matrix and . We present two analogous developments. One for Schur stability and the discrete-time dynamical system , and the other for Hurwitz stability and the continuous-time dynamical system . Here is a description of the latter development.
For , we define and study “Hurwitz diagonal stability with respect to -norms”, abbreviated “”. is the usual concept of diagonal stability. A is implies “ for every eigenvalue of ”, which means is “Hurwitz stable”, abbreviated “HS”. When the off-diagonal elements of are nonnegative, is HS iff is for all .
For the dynamical system , we define “diagonally invariant exponential stability relative to the -norm”, abbreviated , meaning there exist time-dependent sets, which decrease exponentially and are invariant with respect to the system. We show that is a special type of exponential stability and the dynamical system has this property iff is .