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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 $1\le p\le\infty$. 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 $\dot x(t)=Ax(t)$. Here is a description of the latter development. For $A$, we define and study “Hurwitz diagonal stability with respect to $p$-norms”, abbreviated “$\text{HDS}_p$”. $\text{HDS}_2$ 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 $\dot x(t)= Ax(t)$, 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 15B48 Positive matrices and their generalizations; cones of matrices 15A18 Eigenvalues, singular values, and eigenvectors 34A30 Linear ODE and systems, general 34C14 Symmetries, invariants (ODE) 34D20 Stability of ODE 93C05 Linear control systems
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