## 4 Discussion and Examples
Moreover, conditions (9) and (10) of Theorem 1 hold, since_  
\[7=\mathcal{L}_{1}=\min_{1\leq h\leq n}(2a_{h})>\max_{1\leq h\leq n}\sum_{j=1} ^{n}\Bigl{(}\mathcal{F}_{j}|w_{hj}|+\mathcal{F}_{h}|w_{jh}|\Bigr{)}=\mathcal{ L}_{2}=6.3>0,\]  
_and_  
\[0<\mathcal{L}=\mathcal{L}_{1}-\mathcal{L}_{2}=0.7<1.\] \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Discrete Fractional-Order Neural Networks** & **[**28**]** & **[**30**]** & **[**31**]** & **This Research** \\ \hline Mittag-Leffler stability and synchronization & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) \\ Practical stability & \(\times\) & \(\times\) & \(\surd\) & \(\surd\) \\ Practical Mittag-Leffler stability and synchronization & \(\times\) & \(\times\) & \(\times\) & \(\surd\) \\ Impulsive controllers & \(\times\) & \(\times\) & \(\times\) & \(\surd\) \\ Applied in product form design & \(\times\) & \(\times\) & \(\surd\) & \(\surd\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison with existing stability, synchronization and control results on discrete fractional-order neural networks. According to Theorem 1, the impulsive control discrete fractional-order neural network model (5) is globally practically Mittag-Leffler stable. The global practical Mittag-Leffler stable behavior of the states is demonstrated in Figure 4.  
However, condition (18) of Theorem 2 is not satisfied since  
\[\hat{\mathcal{L}}=\min_{1\leq h\leq n}\left[a_{h}-\mathcal{F}_{h}\sum_{j=1}^{n }|w_{jh}|\right]=1.2>1.\]  
Hence, for the model constructed in Example 1, the criteria established in Theorem 2 cannot be applied. **Example 2**. : _Consider the impulsive discrete fractional-order neural network model (5) for \(n=3\), \(\alpha=0.97\), \(a_{h}=2.7\), \(\Theta_{h}=0.2\), \(h=1,2,3\), \(f_{j}(u_{j})=\frac{1}{2}(|u_{j}+1|-|u_{j}-1|)\), \(u_{j}\in\mathbb{R}\), \(j=1,2,3\), with the connection weights given as_  
\[W=\left[ \begin{array}{ccc}w_{11}&w_{12}&w_{13}\\ w_{21}&w_{22}&w_{23}\\ w_{31}&w_{32}&w_{33}\end{array} \right]=\left[ \begin{array}{ccc}0.7&0.6&-0.7 \\ -0.5&0.4&0.9\\ 0.8&1&-0.4\end{array} \right].\]  
_and the impulsive controllers defined as_  
\[V_{hl}(\mathcal{U}_{h}(\eta_{l}-1))=\left(1-\frac{1}{2h}\right)\mathcal{U}_{ h}(\eta_{l}-1), \tag{28}\]  
_at the impulsive instances \(\eta_{l}\) that satisfy (4)._  
_For the choice of the model's parameters, hypotheses \(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\) hold for \(\mathcal{F}_{j}=1\), \(j=1,2,3\) and \(\rho_{hl}=\frac{1}{2h}\), \(h=1,2,3,l\in\mathbb{N}_{1}\). Moreover, condition (18) of Theorem 2 is satisfied for_  
\[0<\hat{\mathcal{L}}=0.7=\min_{1\leq h\leq n}\left[a_{h}-\mathcal{F}_{h}\sum_{ j=1}^{n}|w_{jh}|\right]<1,\]  
_and condition (19) holds for_  
\[\sum_{h=1}^{3}|\Theta_{h}|=0.6<\hat{\sigma}<\infty.\]  
Figure 4: The global practical Mittag-Leffler stability behavior of the states of the model discussed in Example 1. According to Theorem 2, the impulsive control discrete fractional-order neural network model (5) is globally practically Mittag-Leffler stable. The global practical Mittag-Leffler stable behavior of the states is demonstrated in Figure 5.  
In addition, it is clear that condition (10) of Theorem 1 is not satisfied, since  
\[5.4=\mathcal{L}_{1}=\min_{1\leq h\leq n}(2a_{h})>\max_{1\leq h\leq n}\sum_{j=1 }^{n}\left(\mathcal{F}_{j}|w_{hj}|+\mathcal{F}_{h}|w_{jh}|\right)=\mathcal{L} _{2}=4.2>0,\]  
but  
\[\mathcal{L}=\mathcal{L}_{1}-\mathcal{L}_{2}=1.2>1.\]  
Therefore, for the model constructed in Example 2, the criteria established in Theorem 1 cannot be applied. In this example, consider first the impulse-free model (3) for \(n=4\), \(\alpha=0.96\), \(a_{h}=3.8\), \(h=1,2,3,4\), \(f_{j}(u_{j})=\frac{1}{2}(|u_{j}+1|-|u_{j}-1|)\), \(u_{j}\in\mathbb{R}\), \(j=1,2,3,4\), with the connection weights given as  
\[W=\left[ \begin{array}{cccc}w_{11}&w_{12}&w_{13}&w_{14}\\ w_{21}&w_{22}&w_{23}&w_{24}\\ w_{31}&w_{32}&w_{33}&w_{34}\\ w_{41}&w_{42}&w_{43}&w_{44}\end{array} \right]=\left[