# Quadratic Stability of Non-Linear Systems Modeled with Norm Bounded Linear Differential Inclusions

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## Abstract

**:**

## 1. Introduction

#### Overview of the Article

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

**Definition**

**11.**

**Definition**

**12.**

- (i)
- $V\left(x\right):{\mathbb{R}}^{n,1}\to \mathbb{R},$ a scalar function
- (ii)
- $V\left(x\right)>0,$ positive definite
- (iii)
- $\frac{d}{dt}V\left(x\right)<0,$ dissipativity.

## 3. Linear Time Invariant System with Bounded Perturbation

#### 3.1. Equivalent System

#### 3.2. Quadratic Stability

## 4. A System of ODE’s to Shift Smallest Eigenvalue ${\mathit{\lambda}}_{\mathbf{1}}$

#### 4.1. Formulation of Optimization Problem

#### 4.2. Optimization Problem

#### 4.3. Lemma 4.2.1

#### 4.4. The System of ODE’s

#### 4.5. Characterization of ODE’s

## 5. A System of ODE’s to Shift ${\mathit{\lambda}}_{\mathbf{1}}\left(\mathit{t}\right)$, ${\mathit{\lambda}}_{\mathbf{2}}\left(\mathit{t}\right)$

#### 5.1. Optimization Problem

#### 5.2. System of ODE’s

**Remark**

**1.**

**Remark**

**2.**

#### 5.3. Stability of Equivalent System

## 6. Conclusions

- discuss the stability analysis of non-linear dynamical systems in control model with norm bounded linear differential inclusions.
- enable us to localize the spectrum which is hindrance in the stability such that the perturbed systems has spectrum in the left half plane which results in the stability of non-linear dynamical system.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Rehman, M.-U.; Alzabut, J.; Hyder, A.
Quadratic Stability of Non-Linear Systems Modeled with Norm Bounded Linear Differential Inclusions. *Symmetry* **2020**, *12*, 1432.
https://doi.org/10.3390/sym12091432

**AMA Style**

Rehman M-U, Alzabut J, Hyder A.
Quadratic Stability of Non-Linear Systems Modeled with Norm Bounded Linear Differential Inclusions. *Symmetry*. 2020; 12(9):1432.
https://doi.org/10.3390/sym12091432

**Chicago/Turabian Style**

Rehman, Mutti-Ur, Jehad Alzabut, and Arfan Hyder.
2020. "Quadratic Stability of Non-Linear Systems Modeled with Norm Bounded Linear Differential Inclusions" *Symmetry* 12, no. 9: 1432.
https://doi.org/10.3390/sym12091432