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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:
93D20Asymptotic stability of control systems
15A45Miscellaneous inequalities involving matrices
15A60Applications of functional analysis to matrix theory
15B48Positive matrices and their generalizations; cones of matrices
15A18Eigenvalues, singular values, and eigenvectors
34A30Linear ODE and systems, general
34C14Symmetries, invariants (ODE)
34D20Stability of ODE
93C05Linear control systems
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References:
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