## Date of Award

Spring 1968

## Document Type

Dissertation - Restricted

## Degree Name

Doctor of Philosophy (PhD)

## Department

Electrical and Computer Engineering

## First Advisor

Wu, Sherman H.

## Abstract

In many cases of practical interest, there is concern with the behavior of dynamical systems only over a finite time interval. This concern may arise in one of two ways: in one case the system under consideration is defined over a fixed and finite interval of time while in the second case, the system in question is defined for all time; however the behavior of the system is of interest only over a finite time interval. The problem of finite time stability of dynamical systems was first considered by investigators who were interested in obtaining good estimates of the maximum value of one or more components of the trajectory of a differential system with fixed initial conditions and under bounded forcing functions. Recently, stronger and more general results were obtained by Weiss and Infante (1965, 1967) who utilized Lyapunov-like functions to treat the problem of finite time stability of continuous dynamical systems. The purpose of the present investigation.is to extend certain existing results for the case of continuous systems, to develop a theory of finite time stability of discrete systems, to establish a connection between finite time stability and classical stability, and to apply the developed theory to problems which are especially well treated by means.of finite time stability. both the continuous and the discrete case, the dynamical systems under consideration are general enough so as to include unforced systems, systems under the influence of perturbing forces, linear systems, non-linear systems, time-invariant systems, time-varying system, simple systems, and composite systems. In many cases where the finite time interval in question is very large, classical and finite time stability would appear to be overlapping concepts. However, this is not true, since there still remains the important distinction between unspecified bounds in the case of classical stability and specific trajectory bounds in the case of finite time stability. Hence, in all cases the theory presented in this investigation may serve well as a supplement to the classical theory of stability.