# Finite-Time Boundedness of Linear Uncertain Switched Positive Time-Varying Delay Systems with Finite-Time Unbounded Subsystems and Exogenous Disturbance

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## Abstract

**:**

## 1. Introduction

## 2. System Descriptions and Preliminaries

**Definition**

**1**

**.**System (1) is said to be positive if for any initial function $\psi \left(\theta \right)\u2ab00,\phantom{\rule{3.33333pt}{0ex}}\theta \in [-\widehat{d},0]$, for any exogenous disturbance $\omega \left(t\right)\u2ab00$ and for any switching signal $\sigma \left(t\right)$, the corresponding trajectory $x\left(t\right)\u2ab00$ holds for all $t\ge {t}_{0}$.

**Definition**

**2**

**Lemma**

**1**

**Assumption**

**1**

**.**For each ${A}_{i}$, ${D}_{i}$ and ${G}_{i}$ in system (1), there are the known Metzler matrices ${\underline{A}}_{i}$ and the matrices ${\underline{D}}_{i}\u2ab00,\phantom{\rule{3.33333pt}{0ex}}{\underline{G}}_{i}\u2ab00$ such that ${A}_{i}\in [{\underline{A}}_{i},{\overline{A}}_{i}]$, ${D}_{i}\in [{\underline{D}}_{i},{\overline{D}}_{i}]$, and ${G}_{i}\in [{\underline{G}}_{i},{\overline{G}}_{i}]$, where ${\underline{A}}_{i},\phantom{\rule{3.33333pt}{0ex}}{\underline{D}}_{i},\phantom{\rule{3.33333pt}{0ex}}{\underline{G}}_{i},\phantom{\rule{3.33333pt}{0ex}}{\overline{A}}_{i},\phantom{\rule{3.33333pt}{0ex}}{\overline{D}}_{i},\phantom{\rule{3.33333pt}{0ex}}{\overline{G}}_{i}$ are the given constant system matrices with appropriate dimensions for all $i\in \underline{N}$.

**Definition**

**3**

**.**Given two constants ${c}_{2}>{c}_{1}>0$, a time constant ${T}_{f}$, two vectors ${l}_{1}\succ {l}_{2}\succ 0$, and a switching signal $\sigma \in \Omega $. System (1) is said to be finite-time bounded with respect to $({c}_{1},{c}_{2},{T}_{f},{l}_{1},{l}_{2},\rho ,\sigma )$ if the solution $x\left(t\right)$ of the system satisfies the condition:

**Definition**

**4**

**.**For any $T\ge t\ge 0$ and a switching signal $\sigma \in \Omega $, let ${N}_{\sigma p}(T,t)$ be the numbers of the pth subsystem being activated and ${T}_{p}(T,t)$ be the total running time of the pth subsystem, $p\in \underline{B}$. We say that σ has the SMDADT ${\tau}_{ap}$ if there exist two constants ${N}_{0p}\ge 0$ and ${\tau}_{ap}>0$ such that

**Definition**

**5**

**.**For any $T\ge t\ge 0$ and a switching signal $\sigma \in \Omega $, let ${N}_{\sigma q}(T,t)$ be the numbers of the qth subsystem being activated and ${T}_{q}(T,t)$ be the total running time of the qth subsystem, $q\in \underline{U}$. We say that σ has the FMDADT ${\tau}_{aq}$ if there exist two constants ${N}_{0q}\ge 0$ and ${\tau}_{aq}>0$ such that

## 3. Main Results

**Theorem**

**1.**

**Proof.**

**Step 1.**We will prove that system (1) is positive.

**Step 2.**We will prove the FTB for system (1) under the switching signals with SMDADT satisfying condition (14) and FMDADT satisfying condition (15).

**Remark**

**1.**

**Corollary**

**1.**

**Proof.**

## 4. Numerical Simulations

**Example**

**1.**

**Example**

**2.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**The value of ${x}^{T}\left(t\right){l}_{2}$ of the system in Example 1 under the corresponding switching signal.

**Figure 6.**The value of ${x}^{T}\left(t\right){l}_{2}$ of the system in Example 2 under the corresponding switching signal.

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**MDPI and ACS Style**

Mouktonglang, T.; Yimnet, S.
Finite-Time Boundedness of Linear Uncertain Switched Positive Time-Varying Delay Systems with Finite-Time Unbounded Subsystems and Exogenous Disturbance. *Mathematics* **2022**, *10*, 65.
https://doi.org/10.3390/math10010065

**AMA Style**

Mouktonglang T, Yimnet S.
Finite-Time Boundedness of Linear Uncertain Switched Positive Time-Varying Delay Systems with Finite-Time Unbounded Subsystems and Exogenous Disturbance. *Mathematics*. 2022; 10(1):65.
https://doi.org/10.3390/math10010065

**Chicago/Turabian Style**

Mouktonglang, Thanasak, and Suriyon Yimnet.
2022. "Finite-Time Boundedness of Linear Uncertain Switched Positive Time-Varying Delay Systems with Finite-Time Unbounded Subsystems and Exogenous Disturbance" *Mathematics* 10, no. 1: 65.
https://doi.org/10.3390/math10010065