Stability/Instability Study and Control of Autonomous Dynamical Systems: Divergence Method

A novel method of instability and stability study of equilibrium points of autonomous dynamical systems using a flow and divergence of the vector field is proposed. A relation between the Lyapunov, Gauss (Ostrogradsky) and Chetaev theorems with the divergence ones is established. The generalizations of Bendixon and Bendixon-Dulac theorems about a lack of periodic solutions in arbitrary order systems are considered. The state feedback control law design based on new divergence conditions is proposed. Examples illustrate the efficiency of the proposed method and comparisons with some existing ones.

Automated summary

A novel method for studying the instability and stability of equilibrium points in autonomous dynamical systems is proposed, establishing a relationship between various theorems and divergence conditions. The article also considers generalizations of certain theorems and suggests a state feedback control law based on new divergence conditions. Examples are used to demonstrate the efficiency of the proposed method, along with comparisons to existing methods.