On \(t_{\infty}\) quasi-similarity of linear systems (*)
Let the linear homogeneous system (1) \(x'=A(t)x\), (t\(\geq a)\) have certain properties in connection with stability and asymptotic behaviour of solutions. The old problem of finding conditions connecting the matrix A and B which imply that the second system (2) \(y'=B(t)y\), (t\(\geq a)\) has the same or related properties, is considered. This problem has been investigated e.g. by Conti who defined \(t_{\infty}\) similarity of systems (1) and (2) and showed that it is an equivalence relation which preserves uniform and strict stability. Here this definition is weakened by imposing less stringent integrability conditions, some in terms of perhaps conditionally convergent improper integrals, on the matrix function relating A and B. Integrability conditions of this kind are rather unusual [see e.g. the author's paper in Proc. Amer. Math. Soc. 87, 263-270 (1983; Zbl 0514.34031)]. The extended relation \(t_{\infty}\) quasi-similarity, is not symmetric or transitive. However, it is shown that if (2) is \(t_{\infty}\) quasi-similar to (1) and (1) is uniformly, uniformly asymptotically, or strictly stable, then so is (2). Results are also given concerning linear asymptotic equilibrium of (2) in the case where (1) is strictly stable or has linear asymptotic equilibrium. (Based on the author's summary.)
scientific article; zbMATH DE number 4032114
scientific article