A Treatise on Stability of Autonomous and Non-autonomous Systems: Theory and Illustrative Practical Applications

TL;DR

This paper provides a comprehensive overview of stability theory for autonomous and non-autonomous systems, emphasizing Lyapunov's method and addressing the complexities of non-autonomous systems with practical insights.

Contribution

It offers a systematic study of stability analysis techniques, especially Lyapunov's method, for both autonomous and non-autonomous systems, including practical applications.

Findings

Lyapunov's method is the most effective for stability analysis.

Non-autonomous systems exhibit complex behaviors requiring careful analysis.

The book emphasizes systematic approaches to stability in practical systems.

Abstract

Stability is a very important property of any physical system. By a stable system, we broadly mean that small disturbances either in the system inputs or in the initial conditions do not lead to large changes in the overall behavior of the system. To be of practical use, a system must have to be stable. The theory of stability is a vast, rapidly growing subject with prolific and innovative contributions from numerous researchers. As such, an introductory book that covers the basic concepts and minute details about this theory is essential. The primary aim of this book is to make the readers familiar with the various terminologies and methods related to the stability analysis of time-invariant (autonomous) and time-varying (non-autonomous) systems. A special treatment is given to the celebrated Liapunov's direct method which is so far the most widely used and perhaps the best method for determining the stability nature of both autonomous as well as non-autonomous systems. After discussing autonomous systems to a considerable extent, the book concentrates on the non-autonomous systems. From stability point of view, these systems are often quite difficult to manage. Also, unlike their autonomous counterparts, the non-autonomous systems often behave in peculiar manners which can make the analysts arrive at misleading conclusions. Due to these issues, this book attempts to present a careful and systematic study about the stability properties of non-autonomous systems.