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Generalized matrix diagonal stability and linear dynamical systems. (English) Zbl 1125.93058

Summary: Let A=(a ij ) be a real square matrix and 1p. 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 “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 .

MSC:
93D20Asymptotic stability of control systems
15A45Miscellaneous inequalities involving matrices
15A60Applications of functional analysis to matrix theory
15A48Positive matrices and their generalizations (MSC2000)
15A18Eigenvalues, singular values, and eigenvectors
34A30Linear ODE and systems, general
34C14Symmetries, invariants (ODE)
34D20Stability of ODE
93C05Linear control systems