Sheldon P. Gordon

Copyright © 1979 Sheldon P. Gordon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The problem of determining the behavior of the solutions of a perturbed differential equation with respect to the solutions of the original unperturbed differential equation is studied. The general differential equation considered is X′=f(t,X) and the associated perturbed differential equation is Y′=f(t,Y)+g(t,Y).

The approach used is to examine the difference between the respective solutions F(t,t0,x0) and G(t,t0,y0) of these two differential equations. Definitions paralleling the usual concepts of stability, asymptotic stability, eventual stability, exponential stability and instability are introduced for the difference G(t,t0,y0)−F(t,t0,x0) in the case where the initial values y0 and x0 are sufficiently close. The principal mathematical technique employed is a new modification of Liapunov's Direct Method which is applied to the difference of the two solutions. Each of the various stabillty-type properties considered is then shown to be guaranteed by the existence of a Liapunov-type function with appropriate properties.