UNIFYING MATRIX STABILITY CONCEPTS WITH A VIEW TO APPLICATIONS Olga Y. Kushel Shanghai University, Department of Mathematics, Shangda Road 99, 200444 Shanghai, China
[email protected] 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 (D, G, ◦)-stability, which depends on a stability region D ⊂ C, a matrix class G and a binary matrix operation ◦. This ap- proach allows us to unite several well-known matrix problems and to con- sider 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 (D, G, ◦)-stable matrices, uniting the facts that are common for many partial cases. Basing on the proper- ties of a stability region D which may be chosen to be a concrete subset of 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 (D, G, ◦)-stability. We mention some applications of the theory of (D, G, ◦)-stability to the dynamical systems of different types. Hurwitz stability, D-stability, diagonal stability, eigenvalue clustering, LMI regions, Lyapunov equation. MSC[2010] 15A48, 15A18, 15A75 Contents I Motivation and history 3 1 Introduction 3 1.1 OverviewofPaper .......................... 4 1.2 Unifyingconcept ........................... 5 1.3 Multiplicative (D, G)-stability.................... 6 arXiv:1907.07089v1 [math.SP] 14 Jul 2019 1.4 Hadamard (D, G)-stability.....................