# Construction of Lyapunov functions using Helmholtz-Hodge decomposition

## TL;DR

This paper introduces a novel method for constructing Lyapunov functions using Helmholtz-Hodge decomposition, demonstrating its effectiveness and limitations in analyzing stability of vector fields.

## Contribution

It presents a new approach to construct Lyapunov functions via HHD and explores the properties of orthogonal HHD vector fields, extending gradient field concepts.

## Key findings

- Potential functions from HHD can serve as Lyapunov functions under stability conditions.
- Orthogonal HHD vector fields generalize gradient vector fields with similar stability properties.
- The method's limitations are examined through planar vector field analysis.

## Abstract

The Helmholtz-Hodge decomposition (HHD) is applied to the construction of Lyapunov functions. It is shown that if a stability condition is satisfied, such a decomposition can be chosen so that its potential function is a Lyapunov function. In connection with the Lyapunov function, vector fields with strictly orthogonal HHD are analyzed. It is shown that they are a generalization of gradient vector fields and have similar properties. Finally, to examine the limitations of the proposed method, planar vector fields are analyzed.