## 3 Global Practical Mittag-Leffler Stability Results
 \tag{16}\]  
From (15) and (16), according to Lemma 4, it follows that  
\[V(\eta,\mathcal{U}(\eta))\leq\Bigl{(}V(0,\mathcal{U}^{0})-\frac{\sigma}{ \mathcal{L}}\Bigr{)}\mathbb{E}_{\mathbb{F}}(-\mathcal{L},\eta)+\frac{\sigma}{ \mathcal{L}},\;\eta\in\mathbb{N}_{0}. \tag{17}\]  
Hence,  
\[\sum_{h=1}^{n}\mathcal{U}_{h}^{2}(\eta)\leq\left(\sum_{h=1}^{n}\mathcal{U}_{h} ^{2}(0)-\frac{2\sigma}{\mathcal{L}}\right)\mathbb{E}_{\mathbb{F}}(-\mathcal{ L},\eta)+\frac{2\sigma}{\mathcal{L}},\;\eta\in\mathbb{N}_{0},\]  
which implies that  
\[||\mathcal{U}(\eta)||<||\mathcal{U}^{0}||[\mathbb{E}_{\mathbb{F}}(-\mathcal{ L},\eta)]^{1/2}+\left(\frac{2\sigma}{\mathcal{L}}\right)^{1/2},\;\eta\in \mathbb{N}_{0}.\]  
Therefore, Definition 3 is satisfied for \(\mathcal{M}(\mathcal{U}^{0})=||\mathcal{U}^{0}||^{2}\), \(\beta=1/2\) and \(A=\left(\frac{2\sigma}{\mathcal{L}}\right)^{1/2}\), and, thus, the impulsive control discrete fractional-order neural network model (5) is globally practically Mittag-Leffler stable. Theorem 1 offers global practical Mittag-Leffler stability results for the generalized impulsive fractional-order neural network model (5). The fact that the stability criteria are in the form of inequalities between the model parameters make them easy to implement. The practical significance of the proposed stability analysis is as follows: if the activation functions and the impulsive controllers satisfy hypotheses \(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\), and the model parameters satisfy conditions (9)-(11), then the impulsive control discrete fractional-order neural network model (5) is globally practically Mittag-Leffler stable. **Remark 13**. : _Note that condition (11) in Theorem 1 imposes strict restrictions. Although these limitations are justified from a practical point of view, weaker criteria for the global practical Mittag-Leffler stability of model (5) will be offered in the next result._  
**Theorem 2**. : _Suppose that hypotheses \(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\) hold. If the following inequalities_  
\[0<\hat{\mathcal{L}}=\min_{1\leq h\leq n}\left[a_{h}-\mathcal{F}_{h}\sum_{j=1}^{n }|w_{jh}|\right]<1, \tag{18}\]  
\[\sum_{h=1}^{n}|\Theta_{h}|<\hat{\sigma},\,\hat{\sigma}>0 \tag{19}\]  
_hold, then the impulsive control discrete fractional-order neural network model (5) is globally practically Mittag-Leffler stable._  
Proof. : The following Lyapunov function is defined:  
\[V(\eta,\mathcal{U}(\eta))=\sum_{h=1}^{n}|\mathcal{U}_{h}(\eta)|,\,\,\eta\in \mathbb{N}_{0}. \tag{20}\]  
Then, we calculate the \(\alpha\)-order Caputo nabla fractional difference of \(V(\eta,\mathcal{U}(\eta))\) along the trajectories of system (5) for \(\eta\
eq\eta_{l},l\in\mathbb{N}_{1}\). Lemma 2 implies that  
\[\begin{split}{}^{\rm C}\
abla_{0}^{\alpha}V(\eta,\mathcal{U}( \eta))&\leq\sum_{h=1}^{n}  
Using (26) and the Minkowski-type inequality, it follows that  
\[||\mathcal{U}(\eta)||\leq\sum_{h=1}^{n}|\mathcal{U}_{h}(\eta)|\leq\sum_{h=1}^{n}| \mathcal{U}_{h}(0)|\mathbb{E}_{\overline{\pi}}(-\mathcal{L},\eta)+\frac{\hat{ \sigma}}{\mathcal{L}},\ \eta\in\mathbb{N}_{0},\]  
which, by Definition 3 for \(\mathcal{M}(\mathcal{U}^{0})=\sum_{h=1}^{n}|\mathcal{U}_{h}(0)|\), \(\beta=1\) and \(A=\frac{\hat{\sigma}}{\mathcal{L}}\), implies that the impulsive control discrete fractional-order neural network model (5) is globally practically Mittag-Leffler stable. \(\square\)  
**Remark 14**: _The global Mittag-Leffler stability for discrete fractional-order neural networks has been investigated in [28; 30] with the use of the Lyapunov direct method. However, so far, there have been no results regarding the global practical stability of Mittag-Leffler for such neural networks. Considering this, in Theorems 1 and 2, the practical stability case is presented and effective criteria are established. The obtained results can also be used in the study of the global practical Mittag-Leffler synchronization of the states.