#
Exponential Stability of Switched Neural Networks with Partial State Reset and Time-Varying Delays^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

**Remark**

**1.**

**Assumption**

**1.**

**Remark**

**2.**

**Definition**

**1.**

**Assumption**

**2.**

**Lemma**

**1**

**.**Let Q be a positive definite matrix, then the following inequality holds:

**Lemma**

**2**

**.**Let $0\le {\tau}_{i}\left(t\right)\le \tau $, $F\left(t,x,{\overline{x}}_{1},{\overline{x}}_{2},\dots ,{\overline{x}}_{m}\right)$: ${\mathbb{R}}^{+}\times \stackrel{m+1}{\stackrel{\u23de}{\mathbb{R}\times \cdots \times \mathbb{R}}}\to \mathbb{R}$ be nondecreasing in ${\overline{x}}_{i}$ for each fixed $\left(t,x,{\overline{x}}_{1},\dots ,{\overline{x}}_{i-1},{\overline{x}}_{i+1},\dots ,{\overline{x}}_{m}\right)$, $i=1,2,\dots ,m$, and ${I}_{k}\left(x\right):\mathbb{R}\to \mathbb{R}$ be nondecreasing in x. Suppose that

**Lemma**

**3**

**.**Assume that Δ, ${\Psi}_{1}$ and ${\Psi}_{2}$ are appropriate-sized constant matrices, $0\le \rho \left(t\right)\le 1$, then

## 3. Main Results

**Assumption**

**3.**

**Remark**

**3.**

**Theorem**

**1.**

**Proof**

**.**

**Remark**

**4.**

**Remark**

**5.**

**Remark**

**6.**

**Remark**

**7.**

**Corollary**

**1.**

**Proof**

**.**

**Remark**

**8.**

**Remark**

**9.**

**Corollary**

**2.**

**Corollary**

**3.**

## 4. Numerical Example

**Example**

**1.**

**Example**

**2.**

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**State trajectories of the four 2-dimensional subsystems ${\xi}_{i}\left(t\right),i=1,2,3,4$.

**Figure 4.**State trajectories of the four 3-dimensional subsystems ${\xi}_{i}\left(t\right),i=1,2,3,4$.

$\mu $ | $0.6$ | $0.7$ | $0.8$ | $0.9$ |

${\sigma}_{1}$ | $0.044$ | $0.045$ | $0.046$ | $0.046$ |

${\sigma}_{2}$ | $0.052$ | $0.05$ | $0.048$ | $0.047$ |

${\tilde{h}}_{1}$ | $0.0415$ | $0.2010$ | $0.3206$ | $0.4104$ |

${\tilde{h}}_{2}$ | $0.0319$ | $0.0293$ | $0.0238$ | $0.0052$ |

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**MDPI and ACS Style**

Pan, H.; Zhang, W.; Yu, L.
Exponential Stability of Switched Neural Networks with Partial State Reset and Time-Varying Delays. *Mathematics* **2022**, *10*, 3870.
https://doi.org/10.3390/math10203870

**AMA Style**

Pan H, Zhang W, Yu L.
Exponential Stability of Switched Neural Networks with Partial State Reset and Time-Varying Delays. *Mathematics*. 2022; 10(20):3870.
https://doi.org/10.3390/math10203870

**Chicago/Turabian Style**

Pan, Han, Wenbing Zhang, and Luyang Yu.
2022. "Exponential Stability of Switched Neural Networks with Partial State Reset and Time-Varying Delays" *Mathematics* 10, no. 20: 3870.
https://doi.org/10.3390/math10203870