# Lyapunov functions for nabla discrete fractional order systems

## TL;DR

This paper develops new Lyapunov function methods for nabla discrete fractional order systems, deriving inequalities to establish stability criteria, and demonstrates their effectiveness through illustrative examples.

## Contribution

Introduces a novel approach to constructing Lyapunov functions for nabla discrete fractional systems and derives inequalities for stability analysis.

## Key findings

- Derived five inequalities for each fractional derivative definition
- Established sufficient conditions for asymptotic stability
- Validated results with three illustrative examples

## Abstract

This paper focuses on the fractional difference of Lyapunov functions related to Riemann-Liouville, Caputo and Grunwald-Letnikov definitions. A new way of building Lyapunov functions is introduced and then five inequalities are derived for each definition. With the help of the developed inequalities, the sufficient conditions can be obtained to guarantee the asymptotic stability of the nabla discrete fractional order nonlinear systems. Finally, three illustrative examples are presented to demonstrate the validity and feasibility of the proposed theoretical results.