Exponential dichotomy of linear impulsive differential equations in a Banach space

Abstract

A number of processes in physics, chemistry, biology, control theory, and robotics during their evolutionary development are subject to the action of short-time forces in the form of impulses. In most cases the duration of the action of these forces is negligibly small, as a result of which one can assume that the forces act only at certain moments of time. The impulsive differential equations represent a mathematical model of such processes. The work of Mil 'man and Myshkis (1960) marked the beginning of the mathematical theory of these equations, and the work of Bainov et al. (1988, 1989) marked the beginning of the mathematical theory of the same equations in abstract spaces. Samoilenko and Perestyuk (1987) published the first monograph dedicated to this subject. In the present paper the exponential dichotomy of linear impulsive equations in a Banach space is investigated. The exponential dichotomy of linear differential equations in the finite-dimensional case without impulses was investigated by Palmer (1979a, b; 1984a, b) and Eiaydi and Hajek (1988).