# Lyapunov functions for nabla discrete fractional order systems

arXiv: https://arxiv.org/abs/1812.11368 · HTML: https://mlsanity.com/paper/1812.11368/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Dec 2018.

## Built on by (most cited first, 6 of 6)

- [A Caputo fractional derivative-based algorithm for optimization](https://mlsanity.com/paper/2104.02259/) (2021)
- [The Novel Adaptive Fractional Order Gradient Decent Algorithms Design via Robust Control](https://mlsanity.com/paper/2303.04328/) (2023)
- [Introducing Fractional Classification Loss for Robust Learning with Noisy Labels](https://mlsanity.com/paper/2508.06346/) (2025)
- [Performance Analysis of Fractional Learning Algorithms](https://mlsanity.com/paper/2110.05201/) (2021)
- [Convergence Analysis of Fractional Gradient Descent](https://mlsanity.com/paper/2311.18426/) (2023)
- [Enhancing Fractional Gradient Descent with Learned Optimizers](https://mlsanity.com/paper/2510.18783/) (2025)

## Abstract (arXiv)

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.