A Survey of the Recent Results on Characterizations of Exponential Stability and Dichotomy over Finite Dimensional Spaces

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 \( t \in \mathbb{R}_+ \).