Stability and Instability Divergence Conditions for Dynamical Systems

TL;DR

This paper introduces a new approach for analyzing the stability and instability of autonomous dynamical systems using flow and divergence, extending classical theorems and designing control laws.

Contribution

It proposes a novel method based on flow and divergence, extends Bendixon theorems to n-dimensional systems, and designs state feedback control laws from differential inequalities.

Findings

The method effectively analyzes stability and instability.

Extended Bendixon and Bendixon-Dulac theorems to higher dimensions.

Demonstrated control law design via examples.

Abstract

A novel method for stability and instability study of autonomous dynamical systems using the flow and divergence of the vector field is proposed. A relation between the method of Lyapunov functions and the proposed method is established. Bendixon and Bendixon-Dulac theorems for $n$th dimensional systems are extended. Based on the proposed method, the state feedback control law is designed. The control signal is obtained from the partial differential inequality. The examples illustrate the application of the proposed method and the existing ones.