# Stability for coupled systems on networks with Caputo-Hadamard fractional derivative

Document Type : Research Article

Authors

1 Laboratoire de Mathematiques et Sciences appliquees, University of Ghardaia, Algeriaa

2 Faculty of Sciences, Saad Dahlab University, Blida, Algeria

Abstract

This paper discusses stability and uniform asymptotic stability of the trivial solution of the following coupled systems of fractional differential equations on networks
\begin{equation*}
\left\{
\begin{array}{l l l}
^{cH}D^{\alpha} x_{i}=f_{i}(t,x_{i})+\sum\limits_{j=1}^{n}g_{ij}(t,x_{i},x_{j}),&t> t_{0}, \\
x_{i}(t_{0})=x_{i0},
\end{array}
\right.
\end{equation*}
where $^{cH}D^{\alpha}$ denotes the Caputo-Hadamard fractional derivative of order $\alpha$, $1<\alpha\leq 2$,   $i=1,2,\dots,n$, and $f_{i}:\mathbb{R}_{+}\times\mathbb{R}^{m_i} \to \mathbb{R}^{m_i}$,   $g_{ij} : \mathbb{R}_{+}\times \mathbb{R}^{m_i}\times \mathbb{R}^{m_j} \to \mathbb{R}^{m_i}$ are given functions. Based on graph theory and the classical Lyapunov technique, we prove stability and uniform asymptotic stability under suitable sufficient conditions. We also provide an example to illustrate the obtained results.

Keywords