Sucient, necessary and i conditions of the stability of the coecients-time-varying linear system with time delays

Abstract

Due to appearance of time-varying mechanisms (such as ornithopter, exiblewing micro air vehicles, etc.), time-varying systems become more and more important and ubiquitous. Their bifurcation theory must base on property of their solutions and sucient and necessary conditions of equilibria. In this paper, based on previous few works, rstly sucient conditions of uniform asymptotic stability are successfully obtained for delayed time-varying linear (LTVD) system with any time delay employing Dini derivative, Lozinskii measure and generalized scalar Halanay delayed dierential inequality, especially on estimate of arbitrary solutions but not fundamental solution matrix since its solutions' space is innite. Then sucient, necessary and i conditions of stability, asymptotic stability and uniform asymptotic stability of LTVD system with suciently small time delay are obtained successfully employing Taylor expansion and Dini derivative. It implies that these stabilities can be judged by Lozinskii measure of matrix composing of time delay and coecient matrixes A(t) and B(t).