Unifying matrix stability concepts with a view to applications

Abstract

Multiplicative and additive $D$-stability, diagonal stability, Schur $D$-stability, $H$-stability are classical concepts which arise in studying linear dynamical systems. We unify these types of stability, as well as many others, in one concept of $({\mathfrak D}, {\mathcal G}, \circ)$-stability, which depends on a stability region ${\mathfrak D} \subset {\mathbb C}$, a matrix class ${\mathcal G}$ and a binary matrix operation $\circ$. This approach allows us to unite several well-known matrix problems and to consider common methods of their analysis. In order to collect these methods, we make a historical review, concentrating on diagonal and $D$-stability. We prove some elementary properties of $({\mathfrak D}, {\mathcal G}, \circ)$-stable matrices, uniting the facts that are common for many partial cases. Basing on the properties of a stability region $\mathfrak D$ which may be chosen to be a concrete subset of $\mathbb C$ (e.g. the unit disk) or to belong to a specified type of regions (e.g. LMI regions) we briefly describe the methods of further development of the theory of $({\mathfrak D}, {\mathcal G}, \circ)$-stability. We mention some applications of the theory of $({\mathfrak D}, {\mathcal G}, \circ)$-stability to the dynamical systems of different types.