## 2 The Proposed Model and Strategies
### Impulsive Controllers  
At present, there are various control approaches adopted to discrete fractional-order neural network models to ensure the stability, synchronization, quazi-synchronization and stabilization of the states [23; 24; 25; 26; 27; 28; 29; 30].  
Since, in response to external or internal noise, ANN models are often subject to short-term perturbations, the present paper will consider impulsive controllers. Such controllers can be also used to artificially control the behavior of the process modeled.  
Let \(\eta_{l}\),  
\[0<\eta_{1}<\eta_{2}<\ldots,\ \lim_{l\to\infty}\eta_{l}=\infty \tag{4}\]  
be the control times.  
Consider the impulsive control discrete fractional neural network model defined by  
\[\left\{\begin{array}{l}\mathcal{C}\
abla_{0}^{\alpha}\mathcal{U}_{h}(\eta)= -a_{h}\mathcal{U}_{h}(\eta)+\sum_{j=1}^{n}w_{hj}f_{j}(\mathcal{U}_{j}(\eta))+ \Theta_{h},\ \eta\in\mathbb{N}_{0},\ \eta\
eq\eta_{l},\\ \\ \mathcal{U}_{h}(\eta_{l})=V_{hl}(\mathcal{U}_{h}(\eta_{l}-1)),\ l\in\mathbb{N} _{1},\\ \\ \mathcal{U}(0)=\mathcal{U}^{0},\end{array}\right. \tag{5}\]  
where \(0<\alpha<1\), \(V_{hl}\), determine the control law at the instants \(\eta_{l}\), \(l\in\mathbb{N}_{1}\), \(h=1,2,\ldots,n\) and \(\mathcal{U}^{0}=(\mathcal{U}_{1}^{0},\ldots,\mathcal{U}_{n}^{0})^{T}\) represent the initial data.  
The impulsive controllers \(V_{hl}\), \(h=1,2,\ldots,n\) are applied only at moments \(\eta_{l}\), \(l\in\mathbb{N}_{1}\) and these control gains can appropriately control the states of the impulse-free discrete fractional-order neural network model (3).  
In fact, short-term (impulsive) perturbations exist in industrial product form design problems and they can greatly affect the stability of the model. The relationship between the input variables and output variables in a neural network product form design model can be impulsively changed in response to internal transient stimuli such as exterior supporting forces that result in a structural change in the form diagram [11], personal preferences, changing rules, updates to the weights, innovative impulses [66; 67] or external noise in the threshold values. For example, the use of pulse control design is crucial in the form design of many products used in industrial and power electronics, such as power converters and other switching-mode devices [68]. Moreover, impulses can be used as a control mechanism at the fixed moments \(\eta_{l}\), \(l\in\mathbb{N}_{1}\) with the control actions determined by the functions \(V_{hl}\), which represent the functional relationships between the states at the \(\eta_{l}-1\) and \(\eta_{l}\) instances, in order to develop the product form variables at the next stage and identify the most stable settings for the output values. The fact that the fixed instances of impulsive controllers are predetermined is beneficial in reducing the resource requirements for control updates and for the application of the transmitted information.  
**Remark 3**.: _A hybrid discrete fractional and impulsive strategy is used in defining model (5). Several authors have studied the dynamic properties and the potential applications of discrete fractional neural network models [23; 24; 25; 26; 27; 28; 29; 30]. The paper [31] proposed such a neural network modeling approach for product form design. Different from [23; 24; 25; 26; 27; 28; 29; 30], impulsive controllers are considered in model (5). The combined modeling technique will increase the power of the ANN model and will extend the area of its applicability in engineering design._  
**Remark 4**.: _There are a few papers about impulsive discrete fractional-order systems [46,47]. However, none of the reported results consider practical stability concepts._