*(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 $x(t+1)=Ax\left(t\right)$, and the other for Hurwitz stability and the continuous-time dynamical system $\dot{x}\left(t\right)=Ax\left(t\right)$. 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 $\dot{x}\left(t\right)=Ax\left(t\right)$, 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 |