# An Accelerated Double-Integral ZNN with Resisting Linear Noise for Dynamic Sylvester Equation Solving and Its Application to the Control of the SFM Chaotic System

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## Abstract

**:**

## 1. Introduction

- Based on the novel ZNN design formula, an innovative ADIZNN is constructed for settling the dynamic Sylvester equation under the linear noise.
- The ADIZNN model has a novel double integral structure and activation function, which guarantees accelerated convergence and enhanced anti-noise capacity.
- Theoretical analyses and simulation results are provided to ensure that the ADIZNN model can handle the DSE with excellent convergence and robustness.
- Chaos control schemes of the TFM chaotic system are established to display that the controller based on the ADIZNN has superior performance than that based on the OZNN and IEZNN.

## 2. DSE Description and Models Design

#### 2.1. Description of DSE

#### 2.2. Relevant Models Design

#### 2.3. ADIZNN Model Design

**Remark**

**1.**

**Remark**

**2.**

- Based on the novel ZNN design formula, an innovative ADIZNN is constructed for settling the DSE under the linear noise.
- The novel double integral structure and activation function, which guarantees accelerated convergence and enhanced anti-noise capacity.

## 3. Theoretical Analyses

#### 3.1. Convergence

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

#### 3.2. Robustness

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

## 4. Examples Verification

**Remark**

**3.**

#### 4.1. Experiment 1

**Remark**

**4.**

#### 4.2. Experiment 2

## 5. Application to the Control of the Sine Function Memristor Chaotic System

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DSE | Dynamic Sylvester equation |

ZNN | Zeroing neural network |

OZNN | Original zeroing neural network |

ADIZNN | Accelerated double integral ZNN |

SFM | Sine function memristor |

RNNs | recurrent neural networks |

GNN | Gradient neural network |

IEZNN | integral enhanced ZNN model |

FTAF | fixed-time activation function |

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**Figure 1.**State trajectories of OZNN (9), IEZNN (12) and ADIZNN (19) for the DSE with (35) in the absence of the noise. (

**a**) State trajectory of ${p}_{11}\left(t\right)$. (

**b**) State trajectory of ${p}_{12}\left(t\right)$. (

**c**) State trajectory of ${p}_{21}\left(t\right)$. (

**d**) State trajectory of ${p}_{22}\left(t\right)$.

**Figure 2.**State trajectories of OZNN (9), IEZNN (12) and ADIZNN (19) for the DSE with (35) under the linear noise ${z}_{i}\left(t\right)=t/4+4$. (

**a**) State trajectory of ${p}_{11}\left(t\right)$. (

**b**) State trajectory of ${p}_{12}\left(t\right)$. (

**c**) State trajectory of ${p}_{21}\left(t\right)$. (

**d**) State trajectory of ${p}_{22}\left(t\right)$.

**Figure 3.**Error norms ${\u2225W\left(t\right)\u2225}_{\mathrm{F}}$ of OZNN (9), IEZNN (12) and ADIZNN (19) for the DSE with (35) in different noise environments. (

**a**) No noise ${z}_{i}\left(t\right)=0$. (

**b**) Linear noise ${z}_{i}\left(t\right)=t/4+4$. (

**c**) Linear noise ${z}_{i}\left(t\right)=4t+4$. (

**d**) Linear noise ${z}_{i}\left(t\right)=16t+4$.

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**MDPI and ACS Style**

Han, L.; He, Y.; Liao, B.; Hua, C.
An Accelerated Double-Integral ZNN with Resisting Linear Noise for Dynamic Sylvester Equation Solving and Its Application to the Control of the SFM Chaotic System. *Axioms* **2023**, *12*, 287.
https://doi.org/10.3390/axioms12030287

**AMA Style**

Han L, He Y, Liao B, Hua C.
An Accelerated Double-Integral ZNN with Resisting Linear Noise for Dynamic Sylvester Equation Solving and Its Application to the Control of the SFM Chaotic System. *Axioms*. 2023; 12(3):287.
https://doi.org/10.3390/axioms12030287

**Chicago/Turabian Style**

Han, Luyang, Yongjun He, Bolin Liao, and Cheng Hua.
2023. "An Accelerated Double-Integral ZNN with Resisting Linear Noise for Dynamic Sylvester Equation Solving and Its Application to the Control of the SFM Chaotic System" *Axioms* 12, no. 3: 287.
https://doi.org/10.3390/axioms12030287