# Extended Stability and Control Strategies for Impulsive and Fractional Neural Networks: A Review of the Recent Results

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. CNNs

#### 2.2. Hopfield NNs

#### 2.3. CNNs with Delays

#### 2.4. CNNs with Reaction–Diffusion Terms

#### 2.5. Cohen–Grossberg DCNNs

#### 2.6. DCNNs with Reaction–Diffusion Terms of Cohen–Grossberg Type

#### 2.7. Bidirectional Associative Memory (BAM) Neural Networks

#### 2.8. Impulsive DCNNs

#### 2.9. Fractional-Order Impulsive CNNs

#### 2.10. Extended Stability Concepts

#### 2.10.1. Practical stability

**Definition 1**

#### 2.10.2. Stability of Sets

**Definition 2.**

#### 2.10.3. Stability with Respect to Manifolds

**Definition 3.**

#### 2.10.4. Practical Stability with Respect to Manifolds

**Definition 4.**

#### 2.10.5. Lipschitz stability

**Definition 5.**

#### 2.10.6. Lyapunov Approach

## 3. Results

#### 3.1. Stability of Sets

**Example 1**

#### 3.2. Stability with Respect to Manifolds

**Definition 6.**

**Example 2**

#### 3.3. Practical Stability with Respect to Manifolds

- ${P}_{ik}\left({x}_{i}\left(t\right)\right)=-{\gamma}_{ik}{x}_{i}\left(t\right),\phantom{\rule{4pt}{0ex}}|1-{\gamma}_{ik}|\le \frac{\underline{a}}{\overline{a}},\phantom{\rule{4pt}{0ex}}t={\sigma}_{k}\left(x\left(t\right)\right),$
- $\underline{a}\underset{1\le i\le m}{min}\left({B}_{i}-\sum _{j=1}^{n}\left|{w}_{ji}\right|{L}_{i}\right)>\overline{a}\underset{1\le i\le m}{max}\left(\sum _{j=1}^{m}\left|{h}_{ji}\right|{M}_{i}\right)>0,$

**Example 3.**

**Example 4.**

#### 3.4. Lipschitz Stability

- There exists a continuous for $t\in ({t}_{k-1},{t}_{k}]$ function $\beta \left(t\right)$, $k=1,2,\dots $, such that$${\lambda}_{1}-{\lambda}_{2}\ge \overline{\beta}\left(t\right);$$
- For $\overline{D}}_{i}=\sum _{q=1}^{n}\frac{4n{d}_{iq}}{{B}^{2}$,$$\underset{1\le i\le m}{min}\left(\frac{{\overline{D}}_{i}}{\underline{a}}+{B}_{i}-{L}_{i}\sum _{j=1}^{m}{w}_{ji}^{+}\right)>\frac{\overline{a}}{\underline{a}}\underset{1\le i\le m}{max}\left({M}_{i}\sum _{j=1}^{m}{h}_{ji}^{+}\right)$$$$|1-{\gamma}_{ik}|\le \frac{\underline{a}}{\overline{a}},\phantom{\rule{4pt}{0ex}}i=1,2,\dots ,m,\phantom{\rule{4pt}{0ex}}k=1,2,\dots .\phantom{\rule{4pt}{0ex}}$$

**Example 5.**

## 4. Discussion

- PS = Practical stability;
- SS = Stability of sets;
- SRhM = Stability with respect to h-manifolds;
- SRIM = Stability with respect to integral manifolds;
- PSRhM = Practical stability with respect to h-manifolds;
- PSRIM = Practical stability with respect to integral manifolds;
- LS = Lipschitz stability;
- DCNNs = Delayed cellular neural networks;
- RDDCNNs = Reaction–diffusion delayed cellular neural networks;
- CGDCNNs = Cohen–Grossberg delayed cellular neural networks;
- RDCGDCNNs = Reaction–diffusion Cohen–Grossberg delayed cellular neural networks;
- BAMDCNNs - BAM delayed cellular neural networks;
- FDCNNs = Fractional delayed cellular neural networks;
- FRDDCNNs = Fractional reaction–diffusion delayed cellular neural networks;
- FCGDCNNs = Fractional Cohen–Grossberg delayed cellular neural networks;
- FRDCGDCNNs = Fractional reaction–diffusion Cohen–Grossberg delayed cellular neural networks;
- FBAMDCNNs = Fractional BAM delayed cellular neural networks.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Güçlü, U.; van Germen, M. Probing human brain function with artificial neural networks. In Computational Models of Brain and Behavior; Moustafa, A.A., Ed.; John Wiley & Sons: Hoboken, NJ, USA, 2017; pp. 413–423. [Google Scholar]
- Kozachkov, L.; Lundqvist, M.; Slotine, J.J.; Miller, E.K. Achieving stable dynamics in neural circuits. PLoS Comput. Biol.
**2020**, 16, e1007659. [Google Scholar] - Parisi, G.I.; Kemker, R.; Part, J.L.; Kanan, C.; Wermer, S. Continual lifelong learning with neural networks: A review. Neural Netw.
**2019**, 113, 54–71. [Google Scholar] [CrossRef] [PubMed] - Zapf, B.; Haubner, J.; Kuchta, M.; Ringstad, G.; Eide, P.K.; Marda, K.A. Investigating molecular transport in the human brain from MRI with physics-informed neural networks. Sci. Rep.
**2022**, 12, 15475. [Google Scholar] [CrossRef] [PubMed] - Moustafa, A.A. Computational Models of Brain and Behavior, 1st ed.; John Wiley & Sons: Hoboken, NJ, USA, 2017; ISBN 978-1119159063. [Google Scholar]
- Arbib, M.A. Brains, Machines, and Mathematics, 2nd ed.; Springer: New York, NY, USA, 1987; ISBN 978-0387965390. [Google Scholar]
- Haykin, S. Neural Networks: A Comprehensive Foundation, 1st ed.; Prentice-Hall: Englewood Cliffs, NJ, USA, 1998; ISBN 9780132733502. [Google Scholar]
- Chua, L.O.; Yang, L. Cellular neural networks: Theory. IEEE Trans. Circuits Syst. CAS
**1988**, 35, 1257–1272. [Google Scholar] [CrossRef] - Chua, L.O.; Yang, L. Cellular neural networks: Applications. IEEE Trans. Circuits Syst. CAS
**1988**, 35, 1273–1290. [Google Scholar] [CrossRef] - Liu, Y.H.; Zhu, J.; Constantinidis, C.; Zhou, X. Emergence of prefrontal neuron maturation properties by training recurrent neural networks in cognitive tasks. iScience
**2021**, 24, 103178. [Google Scholar] [CrossRef] - Al-Darabsah, I.; Chen, L.; Nicola, W.; Campbell, S.A. The impact of small time delays on the onset of oscillations and synchrony in brain networks. Front. Syst. Neurosci.
**2021**, 15, 688517. [Google Scholar] [CrossRef] - Popovych, O.V.; Tass, P.A. Adaptive delivery of continuous and delayed feedback deep brain stimulation—A computational study. Sci. Rep.
**2019**, 9, 10585. [Google Scholar] [CrossRef] [Green Version] - Yu, H.; Wang, J.; Liu, C.; Deng, B.; Wei, X. Delay-induced synchronization transitions in modular scale-free neuronal networks with hybrid electrical and chemical synapses. Physics A
**2014**, 405, 25–34. [Google Scholar] [CrossRef] - Ziaeemehr, A.; Zarei, M.; Valizadeh, A.; Mirasso, C.R. Frequency-dependent organization of the brain’s functional network through delayed-interactions. Neural Netw.
**2020**, 132, 155–165. [Google Scholar] [CrossRef] - Lara, T.; Ecomoviciz, P.; Wu, J. Delayed cellular neural networks: Model, applications, implementations, and dynamics. Differ. Equ. Dyn. Syst.
**2002**, 3, 71–97. [Google Scholar] - Sun, X.; Li, G. Synchronization transitions induced by partial time delay in a excitatory-inhibitory coupled neuronal network. Nonlinear Dyn.
**2017**, 89, 2509–2520. [Google Scholar] [CrossRef] - Xu, Y. Weighted pseudo-almost periodic delayed cellular neural networks. Neural Comput. Appl.
**2018**, 30, 2453–2458. [Google Scholar] [CrossRef] - Lefe´vre, J.; Mangin, J.-F. A reaction-diffusion model of human brain development. PLoS Comput. Biol.
**2010**, 6, e1000749. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jia, Y.; Zhao, Q.; Yin, H.; Guo, S.; Sun, M.; Yang, Z.; Zhao, X. Reaction-diffusion model-based research on formation mechanism of neuron dendritic spine patterns. Front. Neurorobot.
**2021**, 15, 563682. [Google Scholar] [CrossRef] - Ouyang, M.; Dubois, J.; Yu, Q.; Mukherjee, P.; Huang, H. Delineation of early brain development from fetuses to infants with diffusion MRI and beyond. NeuroImage
**2019**, 185, 836–850. [Google Scholar] [CrossRef] - Abdelnour, F.; Voss, H.U.; Raj, A. Network diffusion accurately models the relationship between structural and functional brain connectivity networks. NeuroImage
**2014**, 90, 335–347. [Google Scholar] [CrossRef] [Green Version] - Marinov, T.; Santamaria, F. Modeling the effects of anomalous diffusion on synaptic plasticity. BMC Neurosci.
**2013**, 14, P343. [Google Scholar] [CrossRef] [Green Version] - Alshammari, S.; Al-Sawalha, M.M.; Humaidi, J.R. Fractional view study of the brusselator reaction–diffusion model occurring in chemical reactions. Fractal Fract.
**2023**, 7, 108. [Google Scholar] [CrossRef] - Landge, A.; Jordan, B.; Diego, X.; Mueller, P. Pattern formation mechanisms of self-organizing reaction-diffusion systems. Dev. Biol.
**2020**, 460, 2–11. [Google Scholar] [CrossRef] - Li, A.; Chen, R.; Farimani, A.B.; Zhang, Y.J. Reaction diffusion system prediction based on convolutional neural network. Sci. Rep.
**2020**, 10, 3894. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cohen, M.A.; Grossberg, S. Absolute stability of global pattern formation and parallel memory storage by competitive neural networks. IEEE Trans. Syst. Man Cybern.
**1983**, 13, 815–826. [Google Scholar] [CrossRef] - Aouiti, C.; Assali, E.A. Nonlinear Lipschitz measure and adaptive control for stability and synchronization in delayed inertial Cohen–Grossberg-type neural networks. Int. J. Adapt. Control
**2019**, 33, 1457–1477. [Google Scholar] [CrossRef] - Lu, W.; Chen, T. Dynamical behaviors of Cohen–Grossberg neural networks with discontinuous activation functions. Neural Netw.
**2005**, 18, 231–242. [Google Scholar] [CrossRef] - Ozcan, N. Stability analysis of Cohen—Grossberg neural networks of neutral-type: Multiple delays case. Neural Netw.
**2019**, 113, 20–27. [Google Scholar] [CrossRef] [PubMed] - Peng, D.; Li, J.; Xu, W.; Li, X. Finite-time synchronization of coupled Cohen-Grossberg neural networks with mixed time delays. J. Frankl. Inst.
**2020**, 357, 11349–11367. [Google Scholar] [CrossRef] - Hopfield, J.J. Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA
**1982**, 79, 2554–2558. [Google Scholar] [CrossRef] [Green Version] - Dong, T.; Gong, X.; Huang, T. Zero-Hopf bifurcation of a memristive synaptic Hopfield neural network with time delay. Neural Netw.
**2022**, 149, 146–156. [Google Scholar] [CrossRef] [PubMed] - Ma, T.; Mou, J.; Li, B.; Banerjee, S.; Yan, H.Z. Study on complex dynamical behavior of the fractional-order Hopfield neural network system and its implementation. Fractal Fract.
**2022**, 6, 637. [Google Scholar] [CrossRef] - Kosko, B. Adaptive bi-directional associative memories. Appl. Opt.
**1987**, 26, 4947–4960. [Google Scholar] [CrossRef] [Green Version] - Kosko, B. Bidirectional associative memories. IEEE Trans. Syst. Man Cybern.
**1988**, 18, 49–60. [Google Scholar] [CrossRef] [Green Version] - Kosko, B. Neural Networks and Fuzzy Systems: A Dynamical System Approach to Machine Intelligence; Prentice-Hall: Englewood Cliffs, NJ, USA, 1992; ISBN 0136114350, 9780136114352. [Google Scholar]
- Wang, H.; Song, Q.; Duan, C. LMI criteria on exponential stability of BAM neural networks with both time-varying delays and general activation functions. Math. Comput. Simul.
**2010**, 81, 837–850. [Google Scholar] [CrossRef] - Chau, F.T.; Cheung, B.; Tam, K.Y.; Li, L.K. Application of a bi-directional associative memory (BAM) network in computer assisted learning in chemistry. Comput. Chem.
**1994**, 18, 359–362. [Google Scholar] [CrossRef] [PubMed] - Palm, G. Neural associative memories and sparse coding. Neural Netw.
**2013**, 37, 165–171. [Google Scholar] [CrossRef] - Tryon, W.W. A bidirectional associative memory explanation of posttraumatic stress disorder. Clin. Psychol. Rev.
**1999**, 19, 789–818. [Google Scholar] [CrossRef] [PubMed] - Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J.J. Fractional Calculus: Models and Numerical Methods, 1st ed.; World Scientific: Singapore, 2012; ISBN 978-981-4355-20-9. [Google Scholar]
- Magin, R. Fractional Calculus in Bioengineering, 1st ed.; Begell House: Redding, CA, USA, 2006; ISBN 978-1567002157. [Google Scholar]
- Petra´s˘, I. Fractional-Order Nonlinear Systems, 1st ed.; Springer: Heidelberg, Germany; Dordrecht, The Netherlands; London, UK; New York, NY, USA, 2011; ISBN 978-3-642-18101-6. [Google Scholar]
- Sandev, T.; Tomovski, Z. Fractional Equations and Models, Theory and Applications, 1st ed.; Springer: Cham, Switzerland, 2019; ISBN 978-3-030-29616-2. [Google Scholar]
- Teka, W.; Marinov, T.M.; Santamaria, F. Neuronal spike timing adaptation described with a fractional leaky integrate-and-fire model. PLoS Comput. Biol.
**2014**, 10, e1003526. [Google Scholar] [CrossRef] - Coronel-Escamilla, A.; Gomez-Aguilar, J.F.; Stamova, I.; Santamaria, F. Fractional order controllers increase the robustness of closed-loop deep brain stimulation systems. Chaos Solitons Fract.
**2020**, 17, 110149. [Google Scholar] [CrossRef] - Mondal, A.; Sharma, S.K.; Upadhyay, R.K.; Mondal, A. Firing activities of a fractional-order FitzHugh-Rinzel bursting neuron model and its coupled dynamics. Sci. Rep.
**2019**, 9, 15721. [Google Scholar] [CrossRef] [Green Version] - Lundstrom, B.N.; Higgs, M.H.; Spain, W.J.; Fairhall, A.L. Fractional differentiation by neocortical pyramidal neurons. Nat. Neurosci.
**2008**, 11, 1335–1342. [Google Scholar] [CrossRef] - Ganji, R.M.; Jafari, H.; Moshokoa, S.P.; Nkimo, N.S. A mathematical model and numerical solution for brain tumor derived using fractional operator. Results Phys.
**2021**, 28, 104671. [Google Scholar] [CrossRef] - Jun, D.; Guang-Jun, Z.; Yong, X.; Hong, Y.; Jue, W. Dynamic behavior analysis of fractional-order Hindmarsh–Rose neuronal model. Cogn. Neurodyn.
**2014**, 8, 167–175. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Li, P.; Lu, Y.; Xu, C.; Ren, J. Bifurcation phenomenon and control technique in fractional BAM neural network models concerning delays. Fractal Fract.
**2023**, 7, 7. [Google Scholar] [CrossRef] - Ramakrishnan, B.; Parastesh, F.; Jafari, S.; Rajagopal, K.; Stamov, G.; Stamova, I. Synchronization in a multiplex network of nonidentical fractional-order neurons. Fractal Fract.
**2022**, 6, 169. [Google Scholar] [CrossRef] - Weinberg, S.H.; Santamaria, F. History dependent neuronal activity modeled with fractional order dynamics. In Computational Models of Brain and Behavior; Moustafa, A.A., Ed.; John Wiley & Sons: Hoboken, NJ, USA, 2017; pp. 531–548. [Google Scholar]
- Guan, Z.H.; Lam, J.; Chen, G. On impulsive autoassociative neural networks. Neural Netw.
**2000**, 13, 63–69. [Google Scholar] [CrossRef] [PubMed] - Hu, B.; Guan, Z.-H.; Chen, G.; Lewis, F.L. Multistability of delayed hybrid impulsive neural networks with application to associative memories. IEEE Trans. Neural Netw. Learn. Syst.
**2019**, 30, 1537–1551. [Google Scholar] - Liu, C.; Liu, X.; Yang, H.; Zhang, G.; Cao, Q.; Huang, J. New stability results for impulsive neural networks with time delays. Neural Comput. Appl.
**2019**, 31, 6575–6586. [Google Scholar] [CrossRef] - Stamov, G.; Stamova, I.; Martynyuk, A.; Stamov, T. Design and practical stability of a new class of impulsive fractional-like neural networks. Entropy
**2020**, 22, 337. [Google Scholar] [CrossRef] [Green Version] - Stamov, G.; Stamova, I.; Spirova, C. Impulsive reaction-diffusion delayed models in biology: Integral manifolds approach. Entropy
**2021**, 23, 1631. [Google Scholar] [PubMed] - Xu, B.; Liu, Z.; Teo, K.L. Global exponential stability of impulsive high-order Hopfield type neural networks with delays. Comput. Math. Appl.
**2009**, 57, 1959–1967. [Google Scholar] [CrossRef] [Green Version] - Benchohra, M.; Henderson, J.; Ntouyas, S. Impulsive Differential Equations and Inclusions, 1st ed.; Hindawi Publishing Corporation: New York, NY, USA, 2006; ISBN 9789775945501. [Google Scholar]
- Li, X.; Song, S. Impulsive Systems with Delays: Stability and Control, 1st ed.; Science Press & Springer: Singapore, 2022; ISBN 978-981-16-4687-4. [Google Scholar]
- Stamova, I.M.; Stamov, G.T. Applied Impulsive Mathematical Models, 1st ed.; Springer: Cham, Switzerland, 2016; ISBN 978-3-319-28060-8/978-3-319-28061-5. [Google Scholar]
- Stamova, I.M.; Stamov, G.T. Functional and Impulsive Differential Equations of Fractional Order: Qualitative Analysis and Applications, 1st ed.; CRC Press, Taylor and Francis Group: Boca Raton, MA, USA, 2017; ISBN 9781498764834. [Google Scholar]
- Yang, T. Impulsive Control Theory, 1st ed.; Springer: Berlin, Germany, 2001; ISBN 978-3-540-47710-5. [Google Scholar]
- Yang, X.; Peng, D.; Lv, X.; Li, X. Recent progress in impulsive control systems. Math. Comput. Simulation
**2019**, 155, 244–268. [Google Scholar] [CrossRef] - Cacace, F.; Cusimano, V.; Palumbo, P. Optimal impulsive control with application to antiangiogenic tumor therapy. IEEE Trans. Control Syst. Technol.
**2020**, 28, 106–117. [Google Scholar] [CrossRef] - Cao, J.; Stamov, T.; Sotirov, S.; Sotirova, E.; Stamova, I. Impulsive control via variable impulsive perturbations on a generalized robust stability for Cohen—Grossberg neural networks with mixed delays. IEEE Access
**2020**, 8, 222890–222899. [Google Scholar] [CrossRef] - Li, M.; Li, X.; Han, X.; Qiu, J. Leader-following synchronization of coupled time-delay neural networks via delayed impulsive control. Neurocomputing
**2019**, 357, 101–107. [Google Scholar] [CrossRef] - Lv, X.; Li, X.; Cao, J.; Perc, M. Dynamical and static multisynchronization of coupled multistable neural networks via impulsive control. IEEE Trans. Neural Netw. Learn. Syst.
**2018**, 29, 6062–6072. [Google Scholar] [CrossRef] - Li, X.; Rakkiyappan, R. Impulsive controller design for exponential synchronization of chaotic neural networks with mixed delays. Commun. Nonlinear Sci. Numer. Simul.
**2013**, 18, 1515–1523. [Google Scholar] [CrossRef] - Xu, Z.; Li, X.; Duan, P. Synchronization of complex networks with time-varying delay of unknown bound via delayed impulsive control. Neural Netw.
**2020**, 125, 224–232. [Google Scholar] [CrossRef] - Stamov, T.; Stamova, I. Design of impulsive controllers and impulsive control strategy for the Mittag—Leffler stability behavior of fractional gene regulatory networks. Neurocomputing
**2021**, 424, 54–62. [Google Scholar] [CrossRef] - Gatto, E.M.; Aldinio, V. Impulse control disorders in Parkinson’s disease. A brief and comprehensive review. Front. Neurol.
**2019**, 10, 351. [Google Scholar] [CrossRef] [Green Version] - Dlala, M.; Saud Almutairi, A. Rapid exponential stabilization of nonlinear wave equation derived from brain activity via event-triggered impulsive control. Mathematics
**2021**, 9, 516. [Google Scholar] [CrossRef] - Hammad, H.A.; De la Sen, M. Stability and controllability study for mixed integral fractional delay dynamic systems endowed with impulsive effects on time scales. Fractal Fract.
**2023**, 7, 92. [Google Scholar] [CrossRef] - Ditzler, G.; Roveri, M.; Alippi, C.; Polikar, R. Learning in nonstationary environments: A survey. IEEE Comput. Intell. Mag.
**2015**, 10, 12–25. [Google Scholar] [CrossRef] - Popa, C.-A. Neutral-type and mixed delays in fractional-order neural networks: Asymptotic stability analysis. Fractal Fract.
**2023**, 7, 36. [Google Scholar] [CrossRef] - Yang, Z.; Zhang, J.; Hu, J.; Mei, J. New results on finite-time stability for fractional-order neural networks with proportional delay. Neurocomputing
**2021**, 442, 327–336. [Google Scholar] [CrossRef] - Stamova, I. Stability Analysis of Impulsive Functional Differential Equations, 1st ed.; De Gruyter: Berlin, Germany, 2009; ISBN 9783110221817. [Google Scholar]
- McCulloch, W.; Pitts, W. A logical calculus of the ideas imminent in nervous activity. Bull. Math. Biol.
**1943**, 5, 115–133. [Google Scholar] - Abiodun, O.I.; Jantan, A.; Omolara, A.E.; Dada, K.V.; Umar, A.M.; Linus, O.U.; Arshad, H.; Kazaure, A.A.; Gana, U.; Kiru, M.U. Comprehensive review of artificial neural network applications to pattern recognition. IEEE Access
**2019**, 7, 158820–158846. [Google Scholar] [CrossRef] - Kasai, H.; Ziv, N.E.; Okazaki, H.; Yagishita, S.; Toyoizumi, T. Spine dynamics in the brain, mental disorders and artificial neural networks. Nat. Rev. Neurosci.
**2021**, 22, 407–422. [Google Scholar] [CrossRef] - Shehab, M.; Abualigan, L.; Omari, M.; Shambour, M.K.Y.; Alshinwan, M.; Abuaddous, H.Y.; Khasawneh, A.M. Artificial neural networks for engineering applications: A review. In Artificial Neural Networks for Renewable Energy Systems and Real-World Applications; Elsheikh, A., Elaziz, M.E.A., Eds.; Academic Press: London, UK, 2022; pp. 189–206. [Google Scholar]
- Gao, Z.; Ure, K.; Ables, J.L.; Lagace, D.C.; Nave, K.A.; Goebbels, S.; Eisch, A.J.; Hsieh, J. Neurod1 is essential for the survival and maturation of adult-born neurons. Nat. Neurosci.
**2009**, 12, 1090–1092. [Google Scholar] [CrossRef] [Green Version] - Huang, H.; Cao, J. On global asymptotic stability of recurrent neural networks with time-varying delays. Appl. Math. Comput.
**2003**, 142, 143–154. [Google Scholar] [CrossRef] - Ensari, T.; Arik, S. Global stability of a class of neural networks with time-varying delay. IEEE Trans. Circuits Syst. I
**2005**, 52, 126–130. [Google Scholar] [CrossRef] - Lee, T.H.; Trinh, H.M.; Park, J.H. Stability analysis of neural networks with time-varying delay by constructing novel Lyapunov functionals. IEEE Trans. Neural Netw. Learn. Syst.
**2018**, 29, 4238–4247. [Google Scholar] [CrossRef] - Manivannan, R.; Samidurai, R.; Cao, J.; Alsaedi, A.; Alsaadi, F.E. Stability analysis of interval time-varying delayed neural networks including neutral time-delay and leakage delay. Chaos Soliton Fract.
**2018**, 114, 433–445. [Google Scholar] [CrossRef] - Cao, J.; Li, J. The stability of neural networks with interneuronal transmission delay. Appl. Math. Mech.
**1998**, 19, 457–462. [Google Scholar] - Zhang, F.Y.; Li, W.T.; Huo, H.F. Global stability of a class of delayed cellular neural networks with dynamical thresholds. Int. J. Appl. Math.
**2003**, 13, 359–368. [Google Scholar] - Li, R.; Cao, J.; Alsaedi, A.; Ahmad, B. Passivity analysis of delayed reaction-diffusion Cohen–Grossberg neural networks via Hardy–Poincare inequality. J. Franklin Inst.
**2017**, 354, 3021–3038. [Google Scholar] [CrossRef] - Lu, J.G. Global exponential stability and periodicity of reaction-diffusion delayed recurrent neural networks with Dirichlet boundary conditions. Chaos Solitons Fract.
**2008**, 35, 116–125. [Google Scholar] [CrossRef] - Qiu, J. Exponential stability of impulsive neural networks with time-varying delays and reaction-diffusion terms. Neurocomputing
**2007**, 70, 1102–1108. [Google Scholar] [CrossRef] - Gan, Q. Adaptive synchronization of Cohen—Grossberg neural networks with unknown parameters and mixed time-varying delays. Commun. Nonlinear Sci. Numer. Simul.
**2012**, 17, 3040–3049. [Google Scholar] [CrossRef] - Song, Q.; Cao, J. Stability analysis of Cohen—Grossberg neural network with both time-varying and continuously distributed delays. Comput. Appl. Math.
**2006**, 197, 188–203. [Google Scholar] [CrossRef] - Yuan, K.; Cao, J.; Li, H. Robust stability of switched Cohen—Grossberg neural networks with mixed time-varying delays. IEEE Trans. Syst. Man Cybern.
**2006**, 36, 1356–1363. [Google Scholar] [CrossRef] - Chen, W.; Huang, Y.; Ren, S. Passivity and robust passivity of delayed Cohen—Grossberg neural networks with and without reaction-diffusion terms. Circuits Syst. Signal Process.
**2018**, 37, 2772–2804. [Google Scholar] [CrossRef] - Wang, Z.; Zhang, H. Global asymptotic stability of reaction-diffusion Cohen—Grossberg neural networks with continuously distributed delays. IEEE Trans. Neral Netw.
**2010**, 21, 39–49. [Google Scholar] [CrossRef] [PubMed] - Zhao, H.; Wang, K. Dynamical behaviors of Cohen—Grossberg neural networks with delays and reaction–diffusion terms. Neurocomputing
**2006**, 70, 536–543. [Google Scholar] [CrossRef] - Song, Q.K.; Cao, J.D. Global exponential stability of bidirectional associative memory neural networks with distributed delays. J. Comput. Appl. Math.
**2007**, 202, 266–279. [Google Scholar] [CrossRef] [Green Version] - Ali, M.S.; Saravanan, S.; Rani, M.E.; Elakkia, S.; Cao, J.; Alsaedi, A.; Hayat, T. Asymptotic stability of Cohen–Grossberg BAM neutral type neural networks with distributed time varying delays. Neural Process. Lett.
**2017**, 46, 991–1007. [Google Scholar] [CrossRef] - Du, Y.; Zhong, S.; Zhou, N.; Shi, K.; Cheng, J. Exponential stability for stochastic Cohen—Grossberg BAM neural networks with discrete and distributed time-varying delays. Neurocomputing
**2014**, 127, 144–151. [Google Scholar] [CrossRef] - Wang, J.; Tian, L.; Zhen, Z. Global Lagrange stability for Takagi-Sugeno fuzzy Cohen—Grossberg BAM neural networks with time-varying delays. Int. J. Control Autom.
**2018**, 16, 1603–1614. [Google Scholar] [CrossRef] - Stamov, G.; Tomasiello, S.; Stamova, I.; Spirova, C. Stability of sets criteria for impulsive Cohen–Grossberg delayed neural networks with reaction-diffusion terms. Mathematics
**2020**, 8, 27. [Google Scholar] [CrossRef] [Green Version] - Chatterjee, A.N.; Al Basir, F.; Takeuchi, Y. Effect of DAA therapy in hepatitis C treatment–an impulsive control approach. Math. Biosci. Eng.
**2021**, 18, 1450–1464. [Google Scholar] [CrossRef] - Rao, R. Impulsive control and global stabilization of reaction-diffusion epidemic model. Math. Methods Appl. Sci.
**2021**. [Google Scholar] [CrossRef] - Rao, X.B.; Zhao, X.P.; Chu, Y.D.; Zhang, J.G.; Gao, J.S. The analysis of mode-locking topology in an SIR epidemic dynamics model with impulsive vaccination control: Infinite cascade of Stern-Brocot sum trees. Chaos Solitons Fractals
**2020**, 139, 110031. [Google Scholar] [CrossRef] - Stamov, G.; Gospodinova, E.; Stamova, I. Practical exponential stability with respect to h-manifolds of discontinuous delayed Cohen–Grossberg neural networks with variable impulsive perturbations. Math. Model. Control
**2021**, 1, 26–34. [Google Scholar] [CrossRef] - Stamov, G.; Stamova, I.; Venkov, G.; Stamov, T.; Spirova, C. Global stability of integral manifolds for reaction-diffusion Cohen-Grossberg-type delayed neural networks with variable impulsive perturbations. Mathematics
**2020**, 8, 1082. [Google Scholar] [CrossRef] - Benchohra, M.; Henderson, J.; Ntouyas, S.K.; Ouahab, A. Impulsive functional differential equations with variable times. Comput. Math. Appl.
**2004**, 47, 1659–1665. [Google Scholar] [CrossRef] [Green Version] - Li, X. Exponential stability of Cohen—Grossberg-type BAM neural networks with time-varying delays via impulsive control. Neurocomputing
**2009**, 73, 525–530. [Google Scholar] [CrossRef] - Maharajan, C.; Raja, R.; Cao, J.; Rajchakit, G.; Alsaedi, A. Impulsive Cohen–Grossberg BAM neural networks with mixed time-delays: An exponential stability analysis issue. Neurocomputing
**2018**, 275, 2588–2602. [Google Scholar] [CrossRef] - Stamov, G.; Stamova, I.; Simeonov, S.; Torlakov, I. On the stability with respect to h-manifolds for Cohen–Grossberg-type bidirectional associative memory neural networks with variable impulsive perturbations and time-varying delays. Mathematics
**2020**, 8, 335. [Google Scholar] [CrossRef] [Green Version] - Podlubny, I. Fractional Differential Equations, 1st ed.; Academic Press: San Diego, CA, USA, 1999; ISBN 558840-2. [Google Scholar]
- Delavari, H.; Baleanu, D.; Sadati, J. Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dyn.
**2012**, 67, 2433–2439. [Google Scholar] [CrossRef] - Stamova, I.M. Global Mittag—Leffler stability and synchronization of impulsive fractional-order neural networks with time-varying delays. Nonlinear Dyn.
**2014**, 77, 1251–1260. [Google Scholar] [CrossRef] - Stamova, I.; Stamov, G. Impulsive control strategy for the Mittag—Leffler synchronization of fractional-order neural networks with mixed bounded and unbounded delays. AIMS Math.
**2021**, 6, 2287–2303. [Google Scholar] [CrossRef] - Cao, J.; Stamov, G.; Stamova, I.; Simeonov, S. Almost periodicity in reaction-diffusion impulsive fractional neural networks. IEEE Trans. Cybern.
**2021**, 51, 151–161. [Google Scholar] [CrossRef] - Stamov, G.; Stamov, T.; Stamova, I. On the stability with respect to manifolds of reaction-diffusion impulsive control fractional-order neural networks with time-varying delays. AIP Conf. Proc.
**2021**, 2333, 060004. [Google Scholar] - Stamova, I.; Stamov, G. Mittag—Leffler synchronization of fractional neural networks with time-varying delays and reaction-diffusion terms using impulsive and linear controllers. Neural Netw.
**2017**, 96, 22–32. [Google Scholar] [CrossRef] [PubMed] - Stamova, I.; Sotirov, S.; Sotirova, E.; Stamov, G. Impulsive fractional Cohen-Grossberg neural networks: Almost periodicity analysis. Fractal Fract.
**2021**, 5, 78. [Google Scholar] [CrossRef] - Zhang, L.; Yang, Y.; Xu, X. Synchronization analysis for fractional order memristive Cohen-Grossberg neural networks with state feedback and impulsive control. Physics A
**2018**, 506, 644–660. [Google Scholar] [CrossRef] - Stamov, G.T.; Stamova, I.M.; Spirova, C. Reaction-diffusion impulsive fractional-order bidirectional neural networks with distributed delays: Mittag-Leffler stability along manifolds. AIP Conf. Proc.
**2019**, 2172, 050002. [Google Scholar] - Stamova, I.; Stamov, G.; Simeonov, S.; Ivanov, A. Mittag-Leffler stability of impulsive fractional-order bi-directional associative memory neural networks with time-varying delays. Trans. Inst. Meas. Control.
**2018**, 40, 3068–3077. [Google Scholar] [CrossRef] - Ren, F.; Cao, F.; Cao, J. Mittag—Leffler stability and generalized Mittag—Leffler stability of fractional-order gene regulatory networks. Neurocomputing
**2015**, 160, 185–190. [Google Scholar] [CrossRef] - Qiao, Y.; Yan, H.; Duan, L.; Miao, J. Finite-time synchronization of fractional-order gene regulatory networks with time delay. Neural Netw.
**2020**, 126, 1–10. [Google Scholar] [CrossRef] - Wu, Z.; Wang, Z.; Zhou, T. Global stability analysis of fractional-order gene regulatory networks with time delay. Int. J. Biomath.
**2019**, 12, 1950067. [Google Scholar] [CrossRef] - Stamova, I.; Stamov, G. Lyapunov approach for almost periodicity in impulsive gene regulatory networks of fractional order with time-varying delays. Fractal Fract.
**2021**, 5, 268. [Google Scholar] [CrossRef] - Ballinger, G.; Liu, X. Practical stability of impulsive delay differential equations and applications to control problems. In Optimization Methods and Applications, Applied Optimization; Yang, X., Teo, K.L., Caccetta, L., Eds.; Kluwer: Dordrecht, The Netherlands, 2001; Volume 52, pp. 3–21. [Google Scholar]
- Lakshmikantham, V.; Leela, S.; Martynyuk, A.A. Practical Stability of Nonlinear Systems; World Scientific: Teaneck, NJ, USA, 1990; ISBN 981-02-0351-9/981-02-0356-X. [Google Scholar]
- Tian, Y.; Sun, Y. Practical stability and stabilisation of switched delay systems with non-vanishing perturbations. IET Control Theory Appl.
**2019**, 13, 1329–1335. [Google Scholar] [CrossRef] - Stamova, I.; Henderson, J. Practical stability analysis of fractional-order impulsive control systems. ISA Trans.
**2016**, 64, 77–85. [Google Scholar] [CrossRef] [PubMed] - Yao, Q.; Lin, P.; Wang, L.; Wang, Y. Practical exponential stability of impulsive stochastic reaction-diffusion systems with delays. IEEE Trans. Cybern.
**2022**, 52, 2687–2697. [Google Scholar] [CrossRef] [PubMed] - Chen, F.C.; Chang, C.H. Practical stability issues in CMAC neural network control systems. IEEE Trans. Control Syst. Technol.
**1996**, 4, 86–91. [Google Scholar] [CrossRef] [Green Version] - Jiao, T.; Zong, G.; Ahn, C.K. Noise-to-state practical stability and stabilization of random neural networks. Nonlinear Dyn.
**2020**, 100, 2469–2481. [Google Scholar] [CrossRef] - Stamov, T. Neural networks in engineering design: Robust practical stability analysis. Cybern. Inf. Technol.
**2021**, 21, 3–14. [Google Scholar] [CrossRef] - Hale, J.K.; Verduyn Lunel, S.M. Introduction to Functional Differential Equations, 1st ed.; Springer: New York, NY, USA, 1993; ISBN 978-0-387-94076-2/978-1-4612-8741-4/978-1-4612-4342-7. [Google Scholar]
- Parshad, R.D.; Kouachi, S.; Gutierrez, J.B. Global existence and asymptotic behavior of a model for biological control of invasive species via supermale introduction. Commun. Math. Sci.
**2013**, 11, 971–992. [Google Scholar] - Li, Z.; Yan, L.; Zhou, X. Global attracting sets and stability of neutral stochastic functional differential equations driven by Rosenblatt process. Front. Math. China
**2018**, 13, 87–105. [Google Scholar] [CrossRef] - Ruiz del Portal, F.R. Stable sets of planar homeomorphisms with translation preudo-arcs. Discret. Contin. Dynam. Syst.
**2019**, 12, 2379–2390. [Google Scholar] [CrossRef] [Green Version] - Skjetne, R.; Fossen, T.I.; Kokotovic, P.V. Adaptive output maneuvering, with experiments, for a model ship in a marine control laboratory. Automatica
**2005**, 41, 289–298. [Google Scholar] [CrossRef] - Stamova, I.M.; Stamov, G.T. On the stability of sets for delayed Kolmogorov-type systems. Proc. Amer. Math. Soc.
**2014**, 142, 591–601. [Google Scholar] [CrossRef] - Bohner, M.; Stamova, I.; Stamov, G. Impulsive control functional differential systems of fractional order: Stability with respect to manifolds. Eur. Phys. J. Spec. Top.
**2017**, 226, 3591–3607. [Google Scholar] [CrossRef] - Smale, S. Stable manifolds for differential equations and diffeomorphisms. Ann. Sc. Norm. Sup. Pisa
**1963**, 3, 97–116. [Google Scholar] - Burby, J.W.; Hirvijoki, E. Normal stability of slow manifolds in nearly periodic Hamiltonian systems. J. Math. Phys.
**2021**, 62, 093506. [Google Scholar] [CrossRef] - Moura, A.; Feudel, U.; Gouillart, E. Mixing and chaos in open flows. Adv. Appl. Mech.
**2012**, 45, 1–50. [Google Scholar] - Stamov, G.; Stamova, I. On stable integral manifolds for impulsive Kolmogorov systems of fractional order. Mod. Phys. Lett. B
**2017**, 31, 1750168. [Google Scholar] [CrossRef] - Gallego, J.A.; Perich, M.G.; Chowdhury, R.H.; Solla, S.A.; Miller, L.E. Long-term stability of cortical population dynamics underlying consistent behavior. Nat. Neurosci.
**2020**, 23, 260–270. [Google Scholar] [CrossRef] [PubMed] - Gallego, J.A.; Perich, M.G.; Miller, L.E.; Solla, S.A. Neural manifolds for the control of movement. Neuron
**2017**, 94, 978–984. [Google Scholar] [CrossRef] [Green Version] - Sadtler, P.T.; Quick, K.M.; Golub, M.D.; Chase, S.M.; Ryu, S.I.; Tyler-Kabara, E.C.; Yu, B.M.; Batista, A.P. Neural constraints on learning. Nature
**2014**, 512, 423–426. [Google Scholar] [CrossRef] [Green Version] - Ionescu, C.; Lopes, A.; Copot, D.; Machado, J.A.T.; Bates, J.H.T. The role of fractional calculus in modeling biological phenomena: A review. Commun. Nonlinear Sci. Numer. Simul.
**2017**, 51, 141–159. [Google Scholar] [CrossRef] - Martynyuk, A.A.; Stamov, G.T.; Stamova, I.M. Impulsive fractional-like differential equations: Practical stability and boundedness with respect to h-manifolds. Fractal Fract.
**2019**, 3, 50. [Google Scholar] - Stamov, T. Discrete bidirectional associative memory neural networks of the Cohen–Grossberg type for engineering design symmetry related problems: Practical stability of sets analysis. Symmetry
**2022**, 14, 216. [Google Scholar] [CrossRef] - Dannan, F.M.; Elaydi, S. Lipschitz stability of nonlinear systems of differential equations. J. Math. Anal. Appl.
**1986**, 113, 562–577. [Google Scholar] [CrossRef] [Green Version] - Harrach, B.; Meftahi, H. Global uniqueness and Lipschitz stability for the inverse Robin transmission problem. SIAM J. Appl. Math.
**2019**, 79, 525–550. [Google Scholar] [CrossRef] [Green Version] - Imanuvilov, O.; Yamamoto, M. Global Lipschitz stability in an inverse hyperbolic problem by interior observations. Inverse Probl.
**2001**, 17, 717–728. [Google Scholar] [CrossRef] - Kawamoto, A.; Machida, M. Global Lipschitz stability for a fractional inverse transport problem by Carleman estimates. Appl. Anal.
**2021**, 100, 752–771. [Google Scholar] [CrossRef] [Green Version] - Ru¨land, A.; Sincich, E. Lipschitz stability for finite dimensional fractional Calderón problem with finite Cauchy data. Inverse Probl. Imaging
**2019**, 13, 1023–1144. [Google Scholar] [CrossRef] [Green Version] - Kulev, G.K.; Bainov, D.D. Lipschitz stability of impulsive systems of differential equations. Int. J. Theor. Phys.
**1991**, 30, 737–756. [Google Scholar] [CrossRef] - Stamova, I.; Stamov, G. Lipschitz stability criteria for functional differential systems of fractional order. J. Math. Phys.
**2013**, 54, 043502. [Google Scholar] [CrossRef] - Gouk, H.; Frank, E.; Pfahringer, B.; Cree, M.J. Regularisation of neural networks by enforcing Lipschitz continuity. Mach. Learn.
**2021**, 110, 393–416. [Google Scholar] [CrossRef] - Stamova, I.; Stamov, T.; Stamov, G. Lipschitz stability analysis of fractional-order impulsive delayed reaction-diffusion neural network models. Chaos Solitons Fract.
**2022**, 162, 112474. [Google Scholar] [CrossRef] - Martynyuk, A.A.; Martynyuk-Chernienko, Y.A. Uncertain Dynamical Systems: Stability and Motion Control, 1st ed.; Chapman and Hall/CRC: Boca Raton, MA, USA, 2011; ISBN 9781439876855. [Google Scholar]
- Stamov, G.; Stamova, I. Uncertain impulsive differential systems of fractional order: Almost periodic solutions. Int. J. Sys. Sci.
**2018**, 49, 631–638. [Google Scholar] [CrossRef] - Zecevic, A.I.; Siljak, D.D. Control of Complex Systems: Structural Constraints and Uncertainty; Springer: New York, NY, USA, 2010; ISBN 978-1-4419-1216-9. [Google Scholar]

**Figure 1.**The trajectory of a mature state ${x}_{i}\left(t\right)$ of the impulsive control model (9).

**Figure 4.**The unstable trajectory of the state variable ${x}_{2}\left(t\right)$ of the CNN in Example 4.

NNs | PS | SS | SRhM | SRIM | PSRhM | PSRIM | LS |
---|---|---|---|---|---|---|---|

DCNNs | √ | √ | √ | √ | √ | √ | √ |

RDDCNNs | √ | √ | √ | √ | √ | × | √ |

CGDCNNs | × | √ | × | √ | × | × | √ |

RDCGDCNNs | × | √ | × | √ | × | × | √ |

BAMDCNNs | √ | × | √ | × | √ | √ | × |

FDCNNs | √ | × | √ | √ | √ | × | √ |

FRDDCNNs | × | × | √ | √ | √ | × | √ |

FCGDCNNs | × | × | × | × | × | × | √ |

FRDCGDCNNs | × | × | × | × | × | × | √ |

FBAMRDDCNNs | × | × | √ | × | × | × | × |

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**MDPI and ACS Style**

Stamov, G.; Stamova, I.
Extended Stability and Control Strategies for Impulsive and Fractional Neural Networks: A Review of the Recent Results. *Fractal Fract.* **2023**, *7*, 289.
https://doi.org/10.3390/fractalfract7040289

**AMA Style**

Stamov G, Stamova I.
Extended Stability and Control Strategies for Impulsive and Fractional Neural Networks: A Review of the Recent Results. *Fractal and Fractional*. 2023; 7(4):289.
https://doi.org/10.3390/fractalfract7040289

**Chicago/Turabian Style**

Stamov, Gani, and Ivanka Stamova.
2023. "Extended Stability and Control Strategies for Impulsive and Fractional Neural Networks: A Review of the Recent Results" *Fractal and Fractional* 7, no. 4: 289.
https://doi.org/10.3390/fractalfract7040289