A survey of the recent results on characterizations of exponential stability and dichotomy over finite dimensional spaces

A. Zada^a, T. Li^b, R. Amin^a, G. Rahmat<sup>c

a</sup> Department of Mathematics, University of Peshawar, Peshawar, Pakistan
^b School of Informatics, Linyi University, Linyi, Shandong 276005, P. R. China
^c Islamia College University Peshawar, Peshwar, Pakistan

Abstract: The main purpose of this article is the investigation of the recent advances on the exponential stability and dichotomy of autonomous and nonautonomous linear differential systems, in both continuous and discrete cases i.e. $\dot x(t)=Ax(t)$, $\dot x(t)=A(t)x(t)$, $x_{n+1}=Ax_n$ and $x_{n+1}=A_nx_n$ in terms of the boundedness of solutions of some Cauchy problems, where $A,A_n$, and $A(t)$ are square matrices, for any $n\in\mathbb Z_+$ and $t\in\mathbb R_+$.

Keywords and phrases: autonomous and nonautonomous systems, spectrum, spectral decomposition theorem, exponential stability and dichotomy.

MSC: 34D09, 34D23

Received: 25.09.2013

Language: English