## 3 Global Practical Mittag-Leffler Stability Results  
The following hypotheses will be important in the following. \(\mathcal{H}_{1}\). The activation functions \(f_{j}\) in model (5) are such that \(f_{j}(0)=0,\ j=1,2,\ldots,n\), and, for any \(u_{1},u_{2}\in\mathbb{R}\), there exist positive constants \(\mathcal{F}_{j}\) such that  
\[|f_{j}(u_{1})-f_{j}(u_{2})|\leq\mathcal{F}_{j}|u_{1}-u_{2}|,\ j=1,2,\ldots,n.\]  
\(\mathcal{H}_{2}\). The impulsive controllers \(V_{hl}\) are such that  
\[V_{hl}(\mathcal{U}_{h}(\eta_{l}-1))=(1-\rho_{hl})\mathcal{U}_{h}(\eta_{l}-1),\]  
where \(\rho_{hl}\) are constants such that \(0<\rho_{hl}\leq 2\), \(h=1,2,\ldots,n,l\in\mathbb{N}_{1}\). **Theorem 1**. : _Suppose that hypotheses \(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\) hold. If the following inequalities_  
\[\mathcal{L}_{1}=\min_{1\leq h\leq n}(2a_{h})>\max_{1\leq h\leq n}\sum_{j=1}^{ n}\Bigl{(}\mathcal{F}_{j}|w_{hj}|+\mathcal{F}_{h}|w_{jh}|\Bigr{)}=\mathcal{L}_{2 }>0, \tag{9}\]  
\[\mathcal{L}=\mathcal{L}_{1}-\mathcal{L}_{2},\,0<\mathcal{L}<1, \tag{10}\]  
\[\sum_{h=1}^{n}|\mathcal{U}_{h}(\eta)||\Theta_{h}|<\sigma,\,\sigma>0,\,\eta\in \mathbb{N}_{0} \tag{11}\]  
_hold, then the impulsive control discrete fractional-order neural network model (5) is globally practically Mittag-Leffler stable._  
Proof. : Consider a Lyapunov function of the type  
\[V(\eta,\mathcal{U}(\eta))=\frac{1}{2}\sum_{h=1}^{n}\mathcal{U}_{h}^{2}(\eta), \ \eta\in\mathbb{N}_{0}. \tag{12}\]  
First, the \(\alpha\)-order Caputo nabla fractional difference of \(V(\eta,\mathcal{U}(\eta))\) will be calculated along the trajectories of system (5) for \(\eta\
eq\eta_{l}\), \(l\in\mathbb{N}_{1}\). From Lemma 1, it follows that  
\[ \begin{split}{}^{C}\
abla_{0}^{\alpha}V(\eta,\mathcal{U}(\eta))& \leq\sum_{h=1}^{n}\mathcal{U}_{h}(\eta)^{C}\
abla_{0}^{\alpha} \mathcal{U}_{h}(\eta)\\ &=\sum_{h=1}^{n}\mathcal{U}_{h}(\eta)\left[-a_{h}\mathcal{U}_{h} (\eta)+\sum_{j=1}^{n}w_{hj}f_{j}(\mathcal{U}_{j}(\eta))+\Theta_{h}\right]. \end{split}  \tag{13}\]  
According to hypothesis \(\mathcal{H}_{1}\), there exist positive constants \(\mathcal{F}_{j}\) such that  
\[ \begin{split}{}^{C}\
abla_{0}^{\alpha}V(\eta,\mathcal{U}(\eta))& \leq\sum_{h=1}^{n}\left[-a_{h}\mathcal{U}_{h}^{2}(\eta)+\sum_{j=1}^{n}|w_{hj} |\mathcal{F}_{j}|\mathcal{U}_{h}(\eta)||\mathcal{U}_{j}(\eta)|+|\mathcal{U}_{ h}(\eta)||\Theta_{h}|\right]\\ &\leq\sum_{h=1}^{n}\left[-a_{h}\mathcal{U}_{h}^{2}(\eta)+\frac{1}{ 2}\sum_{j=1}^{n}|w_{hj}|\mathcal{F}_{j}\Bigl{(}\mathcal{U}_{h}^{2}(\eta)+ \mathcal{U}_{j}^{2}(\eta)\Bigr{)}\right.\\ &+\left.|\mathcal{U}_{h}(\eta)||\Theta_{h}|\right].\end{split}  \tag{14}\]Then, from (11) and (9), it follows that  
\[ \begin{split}{}^{\text{C}}\
abla_{0}^{a}V(\eta,\mathcal{U}(\eta))& \leq-\min_{1\leq h\leq n}(2a_{h})V(\eta,\mathcal{U}(\eta))\\ &+\max_{1\leq h\leq n}\sum_{j=1}^{n}\Bigl{(}\mathcal{F}_{j}|w_{hj }|+\mathcal{F}_{h}|w_{hj}|\Bigr{)}V(\eta,\mathcal{U}(\eta))+\sigma\\ &=-(\mathcal{L}_{1}-\mathcal{L}_{2})V(\eta,\mathcal{U}(\eta))+ \sigma\\ &=-\mathcal{L}V(\eta,\mathcal{U}(\eta))+\sigma,\;\eta\
eq\eta_{l},\end{split}  \tag{15}\]  
where \(0<\mathcal{L}<1\). Next, for \(\eta=\eta_{l},l\in\mathbb{N}_{1}\), hypothesis \(\mathcal{H}_{2}\) implies that  
\[ \begin{split} V(\eta_{l},\mathcal{U}(\eta_{l}))=\frac{1}{2}\sum _{h=1}^{n}\mathcal{U}_{h}^{2}(\eta_{l})&=\frac{1}{2}\sum_{h=1}^ {n}(1-\rho_{hl})^{2}\mathcal{U}_{h}^{2}(\eta_{l}-1)\\ &\leq\frac{1}{2}\sum_{h=1}^{n}\mathcal{U}_{h}^{2}(\eta_{l}-1)=V( \eta_{l}-1,\mathcal{U}(\eta_{l}-1)).\end{split}