# General divergent stability conditions of dynamic systems

## TL;DR

This paper introduces new necessary and sufficient stability conditions for dynamical systems based on flow divergence, extending classical results and linking to Lyapunov methods, with applications to control law synthesis.

## Contribution

It generalizes existing stability criteria using divergence-based conditions and connects them with Lyapunov methods, providing new tools for control law design.

## Key findings

- Conditions are applicable to linear and nonlinear systems.
- New control laws are synthesized based on the conditions.
- Examples demonstrate the method's effectiveness and comparison with existing approaches.

## Abstract

New necessary and sufficient conditions are proposed for the stability investigation of dynamical systems using the flow and the divergence of the phase vector velocity. The obtained conditions generalize the well-known results of V.P. Zhukov and A. Rantzer. The relation of Lyapunov methods with the proposed methods is established. The application of the obtained results to study the stability of linear systems goes to the problem of matrix inequality solvability. The new control laws are synthesized for linear and nonlinear systems. Examples illustrate the applicability of the proposed method and show the comparison results with some existing ones.