# Finite-Time Stability Analysis of Linear Differential Systems with Pure Delay

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**.**The two-parameter Mittag–Leffler function is given by

**Definition**

**2**

**.**The system (1) is finite-time stable with respect to $\left(\right)$, $\delta <\rho $ if and only if $\varrho <\delta $ implies $\left(\right)open="\parallel "\; close="\parallel ">y\left(x\right)$ for all $x\in W$, where $\varrho =max\left(\right)open="\{"\; close="\}">{\u2225\psi \u2225}_{C},{\left(\right)}_{{\psi}^{\prime}}C$ and δ, ρ are real positive numbers.

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

## 3. Main Results

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Remark**

**1.**

## 4. An Example

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Theorem | L | $\u2225\mathit{B}\u2225$ | $\mathit{\delta}$ | $\left(\right)open="\parallel "\; close="\parallel ">\mathit{y}\left(\mathit{x}\right)$ | $\mathit{\rho}$ | h | Finite-Time Stability |
---|---|---|---|---|---|---|---|

1 | 1 | 2 | $0.31$ | ≤3.29158 | $3.3$ (optimal) | $0.5$ | Yes |

2 | 1 | 2 | $0.31$ | ≤4.30474 | $4.31$ | $0.5$ | Yes |

3 | 1 | 2 | $0.31$ | ≤3.48949 | $3.49$ | $0.5$ | Yes |

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**MDPI and ACS Style**

Elshenhab, A.M.; Wang, X.; Bazighifan, O.; Awrejcewicz, J.
Finite-Time Stability Analysis of Linear Differential Systems with Pure Delay. *Mathematics* **2022**, *10*, 1359.
https://doi.org/10.3390/math10091359

**AMA Style**

Elshenhab AM, Wang X, Bazighifan O, Awrejcewicz J.
Finite-Time Stability Analysis of Linear Differential Systems with Pure Delay. *Mathematics*. 2022; 10(9):1359.
https://doi.org/10.3390/math10091359

**Chicago/Turabian Style**

Elshenhab, Ahmed M., Xingtao Wang, Omar Bazighifan, and Jan Awrejcewicz.
2022. "Finite-Time Stability Analysis of Linear Differential Systems with Pure Delay" *Mathematics* 10, no. 9: 1359.
https://doi.org/10.3390/math10091359