# 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

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**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