# Eigenvalue Based Approach for Assessment of Global Robustness of Nonlinear Dynamical Systems

## Abstract

**:**

## 1. Notations, Motivation and Introduction

#### 1.1. Notations

#### 1.2. Motivation and Introduction

## 2. Results

#### 2.1. Auxiliary Lemma

**Lemma**

**1.**

**Proof.**

**Corollary**

**1.**

#### 2.2. Main Result

**Theorem**

**1.**

- (A1)
- ${\lambda}_{max}(t)\le {\lambda}_{0}<0$ in some left neighborhood of $+\infty ,$ where ${\lambda}_{max}(t)$ is the largest pointwise eigenvalue of $\frac{1}{2}[A(t)+{A}^{T}(t)],$ $A(t)={J}_{x}f(0,t);$
- (A2)
- for all $(x,t)\in {\mathbb{R}}^{n}\times [{t}_{0},\infty )$ is ${\u2225\delta (x,t)\u2225}_{2}\le {\u2225\tilde{\mathsf{\Delta}}(t)\u2225}_{2}-{\u2225f(x,t)-[{J}_{x}f(0,t)]x\u2225}_{2},$ where function $\tilde{\mathsf{\Delta}}(t)$ is continuous on $[{t}_{0},\infty )$ and satisfies
- (A3)
- $\underset{t\to \infty}{lim}\left({\u2225\tilde{\mathsf{\Delta}}(t)\u2225}_{2}/{\lambda}_{max}(t)\right)=0.$

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

- (i)
- When A is negative definite, then Assumption A1 of Theorem 1 is automatically satisfied because $\frac{1}{2}[A+{A}^{T}]$ is also negative definite ([34] Corollary 14.2.7), and
- (ii)
- in connection with Assumption A3, it is worth noting that Assumption A3 reduces to ${\u2225\tilde{\mathsf{\Delta}}(t)\u2225}_{2}\to 0$ as $t\to \infty ,$ ensuring the vanishing at infinity of all solutions of perturbed system $\dot{x}=f(x,t)+\delta (x,t),$ cf. Example 2 below, where non-constant ${\lambda}_{max}(t)$ allows convergence to zero of all solutions of perturbed system for a wider class of perturbations, where even unbounded perturbations are admissible.

## 3. Simulation Experiments in MATLAB^{®}

**Example**

**1.**

**Remark**

**3.**

**Example**

**2.**

^{®}environment (the source code Listing 2).

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Solution $x(t)={\left({x}_{1}(t),{x}_{2}(t)\right)}^{T}$ of (7) for $\lambda =2,\phantom{\rule{0.166667em}{0ex}}$ $\delta (x,t)={\left(\frac{arctan({x}_{1}+{x}_{2})}{(t+1)},\phantom{\rule{0.166667em}{0ex}}\frac{\mathrm{exp}[-t]}{({x}_{1}^{2}+1)}\right)}^{T}$ and initial state $x(0)={(50,\phantom{\rule{4pt}{0ex}}-20)}^{T}.$ Obviously, ${\u2225\tilde{\mathsf{\Delta}}(t)\u2225}_{2}=\sqrt{{\left(\frac{\pi /2}{t+1}\right)}^{2}+\mathrm{exp}[-2t]}=o(1)$ as $t\to \infty .$

**Figure 2.**Solution $x(t)={\left({x}_{1}(t),{x}_{2}(t)\right)}^{T}$ of (8) for $b=5,\phantom{\rule{0.166667em}{0ex}}$ $\delta (x,t)={\left({t}^{3/2},\phantom{\rule{0.166667em}{0ex}}3cos\left(t{x}_{1}-{x}_{2}\right)\right)}^{T}$ and initial state $x(0)={(10,\phantom{\rule{4pt}{0ex}}-5)}^{T}.$ Obviously, ${\u2225\tilde{\mathsf{\Delta}}(t)\u2225}_{2}=\sqrt{{t}^{3}+9}=o({t}^{2})$ as $t\to \infty .$

**Figure 3.**The solution component ${x}_{1}(t)$ of (8) for $b=5,$ initial state $x(0)=0$ and with (

**a**) $\delta (x,t)={\left(50{t}^{1.95},\phantom{\rule{0.166667em}{0ex}}{\delta}_{2}(x,t)\right)}^{T}$ satisfying Assumption 3 of Theorem 1, (

**b**) $\delta (x,t)={\left(50{t}^{2.05},\phantom{\rule{0.166667em}{0ex}}{\delta}_{2}(x,t)\right)}^{T}$ that does not satisfy Assumption 3 of Theorem 1 and (

**c**) the borderline case, $\delta (x,t)={\left(50{t}^{2.00},\phantom{\rule{0.166667em}{0ex}}{\delta}_{2}(x,t)\right)}^{T},$ ${\delta}_{2}(x,t)=t{x}_{1}/({x}_{1}^{2}+{x}_{2}^{2}+1)$.

**Listing 1.**MATLAB

^{®}code for Figure 1 (The MathWorks, Inc., 3 Apple Hill Drive, Natick, Massachusetts 01760 USA).

f = @(t,x) [(-2)*x(1)+(exp(t))*x(2)+(atan(x(1)+x(2))/(t+1));(-exp(t))*x(1)+(-2)*x(2)+(exp(-t)/(x(1)^2+1))][t,xa] = ode45(f,[0 4],[50 -20]);hold^{~}onpbaspect([2 1 1])plot(t,xa(:,1), ’k’, ’LineWidth’,1.5) % 1 or 2grid onxlabel(’t’)ylabel(’x_1’) % 1 or 2print(’example_first_x_1’,’-deps’) % 1 or 2 |

syms bb=5;f = @(t,x) [(-t^2+sin(t))*x(1)+(b)*x(2)+t^(1.5);(0)*x(1)+(1-t^2+sin(t))*x(2)+3*cos(t*x(1)-x(2))][t,xa] = ode45(f,[0 4],[10 -5]);hold^{~}onpbaspect([2 1 1])plot(t,xa(:,1), ’k’, ’LineWidth’,1.5) % 1 or 2grid onxlabel(’t’)ylabel(’x_1’) % 1 or 2print(’example_second_x_1’,’-deps’) % 1 or 2 |

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Vrabel, R.
Eigenvalue Based Approach for Assessment of Global Robustness of Nonlinear Dynamical Systems. *Symmetry* **2019**, *11*, 569.
https://doi.org/10.3390/sym11040569

**AMA Style**

Vrabel R.
Eigenvalue Based Approach for Assessment of Global Robustness of Nonlinear Dynamical Systems. *Symmetry*. 2019; 11(4):569.
https://doi.org/10.3390/sym11040569

**Chicago/Turabian Style**

Vrabel, Robert.
2019. "Eigenvalue Based Approach for Assessment of Global Robustness of Nonlinear Dynamical Systems" *Symmetry* 11, no. 4: 569.
https://doi.org/10.3390/sym11040569