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

Summary: Let $A=\left({a}_{ij}\right)$ be a real square matrix and $1\le p\le \infty$. We present two analogous developments. One for Schur stability and the discrete-time dynamical system $x\left(t+1\right)=Ax\left(t\right)$, and the other for Hurwitz stability and the continuous-time dynamical system $\stackrel{˙}{x}\left(t\right)=Ax\left(t\right)$. Here is a description of the latter development.

For $A$, we define and study “Hurwitz diagonal stability with respect to $p$-norms”, abbreviated “${\text{HDS}}_{p}$”. ${\text{HDS}}_{2}$ is the usual concept of diagonal stability. A is ${\text{HDS}}_{p}$ implies “$\Re \lambda <0$ for every eigenvalue $\lambda$ 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 ${\text{HDS}}_{p}$ for all $p$.

For the dynamical system $\stackrel{˙}{x}\left(t\right)=Ax\left(t\right)$, we define “diagonally invariant exponential stability relative to the $p$-norm”, abbreviated ${\text{DIES}}_{p}$, meaning there exist time-dependent sets, which decrease exponentially and are invariant with respect to the system. We show that ${\text{DIES}}_{p}$ is a special type of exponential stability and the dynamical system has this property iff $A$ is ${\text{HDS}}_{p}$.

##### MSC:
 93D20 Asymptotic stability of control systems 15A45 Miscellaneous inequalities involving matrices 15A60 Applications of functional analysis to matrix theory 15A48 Positive matrices and their generalizations (MSC2000) 15A18 Eigenvalues, singular values, and eigenvectors 34A30 Linear ODE and systems, general 34C14 Symmetries, invariants (ODE) 34D20 Stability of ODE 93C05 Linear control systems