## 1 Introduction
Recent advancements in fractional-order ANN modeling have led to significant progress in understanding complex real-world phenomena, studied in many areas of science and engineering [20; 21; 22]. In particular, the applicability of fractional discrete neural networks models is motivated from the point of view of computer simulation and the actual applications of neural networks. This explains the growing research activity in the design, research and application of discrete fractional neural networks [23; 24; 25; 26; 27; 28; 29; 30]. Despite the many reported results, the topic of using discrete fractional ANNs as a modeling approach in product form design is still unexplored. Unique existing results in the area are reported in [31], where a delta discrete fractional operator is considered. The paper [31] also studies various shape classifications and their shape formation, which supports the use of fractional-order ANNs in product form analysis, since fractal components appear very frequently in industrial design product shapes [32; 33]. In addition, fractional-order modeling methods allow the advantages of fractional calculus to be used in describing the behavior of real materials and allow some restrictions imposed by integer-order models to be overcome [34]. The main purpose of the research conducted in this paper is to contribute to the development of this area. In addition, the design and functioning of the neural network models used in product form design require the application of appropriate control techniques in the modeling process. Such control mechanisms also ensure that the final designed product form achieves the desired performance. Numerous researchers in the industrial design area have studied the importance of different design control strategies [35; 36; 37]. In particular, nonlinear controllers [23], linear controllers [28; 31], feedback controllers [29] and adaptive controllers [30] are widely applied in the reported discrete fractional-order ANN models. The impulsive control technique is one of the simplest to implement and involves less expensive control mechanisms [38]. As such, intensive research has been performed on this control strategy. It has been intensively applied to integer-order and fractional-ordercontinuous models [39; 40; 41; 42; 43; 44; 45]. However, unfortunately, very little research has been done on the dynamic behavior of fractional-order discrete-time neural network models under impulsive controllers [46; 47]. Impulsive controllers, moreover, have not been applied to ANN models used in the field of product form design. Consequently, the investigation of the dynamics of impulsive control fractional discrete ANN models, as well as their applications, is an attractive and important subject. The impulsive control strategy seems very appropriate in product form design, since the process can be subject to external and internal noise, such as unwanted variations in some product form design characteristics or system failure due to a minor inconvenience or a major accident [35]. Moreover, extensive research has concluded that impulsive controllers can be used to stabilize some unstable ANN models, i.e., as a tool for robust design. Moreover, they allow the use of both industrial design knowledge and ANN approaches, which is an additional focus in this paper. It is relevant to note that in most of the existing ANN models applied in product form design analysis, only training techniques are considered, and the stability behavior is ignored. However, it is well known that the stability of states of ANN models plays a key role in characterizing the qualitative behavior of such models, and that the study of the qualitative properties of the solutions of various types of equations related to science and engineering is crucial to overcome the poor model results when using new initial conditions.