# Input-to-State Stability of a Scalar Conservation Law with Nonlocal Velocity

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

**nonlocal**quantity (2). The function $\lambda \left(W\right(t\left)\right)$ is a velocity. In production systems, it is natural to assume that the velocity function is positive and decreasing as the total mass is increasing. In the manufacturing system, the initial density of products at production stage x is taken as the initial data

## 2. Asymptotic Stability of a Scalar Conservation Law with Nonlocal Velocity and Measurement Error

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

- (i)
- there exist positive constants ${\alpha}_{1}>0$ and ${\alpha}_{2}>0$ such that for all solutions $\rho \in {C}^{0}([0,\infty );{L}^{2}(0,1))$ and $t\in [0,+\infty )$$${\alpha}_{1}\parallel \rho (t,\xb7)-{\rho}^{*}{\parallel}_{{L}^{2}}^{2}\le \mathbf{L}\left(\rho (t,\xb7)\right)\le {\alpha}_{2}{\parallel \rho (t,\xb7)-{\rho}^{*}\parallel}_{{L}^{2}}^{2},$$
- (ii)
- there exist positive constants $\eta >0$ and $\nu >0$ such that for all solutions $\rho \in {C}^{0}([0,\infty );{L}^{2}(0,1))$ and $t\in [0,+\infty )$$$\frac{d}{dt}\mathbf{L}\left(\rho (t,\xb7)\right)\le -\eta \mathbf{L}\left(\rho (t,\xb7)\right)+\nu {d}^{2}\left(t\right).$$

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

## 3. Numerical Study of Asymptotic Stability of a Scalar Conservation Law with Nonlocal Velocity and Measurement Error

**Definition**

**4.**

**Definition**

**5.**

- (i)
- there exist positive constants ${\alpha}_{1}>0$ and ${\alpha}_{2}>0$ such that for all $n\in \{0,1,\dots \}$$${\alpha}_{1}\parallel {\overrightarrow{\rho}}^{n}-{\rho}^{*}{\parallel}_{{\ell}^{2}}^{2}\le \mathbf{L}\left({\overrightarrow{\rho}}^{n}\right)\le {\alpha}_{2}{\parallel {\overrightarrow{\rho}}^{n}-{\rho}^{*}\parallel}_{{\ell}^{2}}^{2},$$
- (ii)
- there exist positive constants $\eta >0$ and $\nu >0$ such that for all $n\in \{0,1,\dots \}$$$\frac{\mathbf{L}({\overrightarrow{\rho}}^{n+1})-\mathbf{L}({\overrightarrow{\rho}}^{n})}{\Delta t}\le -\eta \mathbf{L}({\overrightarrow{\rho}}^{n})+\nu {\left({d}^{n}\right)}^{2}.$$

**Theorem**

**2.**

**Proof.**

## 4. Numerical Simulations

#### 4.1. Example 1

#### 4.2. Example 2

## 5. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**Comparison of log-scale of $\parallel {\overrightarrow{\rho}}^{n}-{\rho}^{*}\overrightarrow{e}{\parallel}_{{L}_{\Delta x}^{2}}$ with Courant–Friedrichs–Lewy condition (CFL) = 0.75 and ${\rho}^{*}=0$.

**Figure 2.**Comparison of log-scale of $\parallel {\overrightarrow{\rho}}^{n}-{\rho}^{*}\overrightarrow{e}{\parallel}_{{\ell}^{2}}$ with CFL = 0.75 and ${\rho}^{*}=1$.

**Table 1.**Comparison of $\parallel {\overrightarrow{\rho}}^{n}-{\rho}^{*}{\parallel}_{{\ell}^{2}}^{2}$ for different number of grid points J with ${\rho}^{*}=0$, $k=0.3$ and $T=10.$

(a) CFL = 0.5. | ||

J | $\parallel {\overrightarrow{\rho}}^{n}-{\rho}^{*}{\parallel}_{{\ell}^{2}}$ | order |

100 | 1.9171 e-05 | – |

200 | 1.1899 e-05 | 0.6881 e+00 |

400 | 6.9631 e-06 | 0.7730 e+00 |

800 | 3.7638 e-06 | 0.8875 e+00 |

1600 | 1.5902 e-06 | 1.2430 e+00 |

(b) CFL = 0.9. | ||

J | $\parallel {\overrightarrow{\rho}}^{n}-{\rho}^{*}{\parallel}_{{\ell}^{2}}$ | order |

100 | 1.3831 e-05 | – |

200 | 8.1304 e-06 | 0.7665 e+00 |

400 | 4.8604 e-06 | 0.7423 e+00 |

800 | 2.8262 e-06 | 0.7822 e+00 |

1600 | 1.1624 e-06 | 1.2818 e+00 |

**Table 2.**Comparison of $\parallel {\overrightarrow{\rho}}^{n}-{\rho}^{*}{\parallel}_{{\ell}^{2}}$ of the solution for number of grids J with ${\rho}^{*}=1$, $k=0.3$ and $T=20.$

(a) CFL = 0.5. | ||

J | $\parallel {\overrightarrow{\rho}}^{n}-{\rho}^{*}{\parallel}_{{\ell}^{2}}$ | order |

100 | 3.0916 e-04 | – |

200 | 1.5261 e-04 | 1.0185 e+00 |

400 | 7.2438 e-05 | 1.0750 e+00 |

800 | 3.1425 e-05 | 1.2048 e+00 |

1600 | 1.0567 e-05 | 1.5723 e+00 |

(b) CFL = 0.9. | ||

J | $\parallel {\overrightarrow{\rho}}^{n}-{\rho}^{*}{\parallel}_{{\ell}^{2}}$ | order |

100 | 2.8645 e-04 | – |

200 | 1.4299 e-04 | 1.0024 e+00 |

400 | 6.9982 e-05 | 1.0309 e+00 |

800 | 3.0215 e-05 | 1.2117 e+00 |

1600 | 1.0128 e-05 | 1.5769 e+00 |

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Göttlich, S.; Herty, M.; Weldegiyorgis, G. Input-to-State Stability of a Scalar Conservation Law with Nonlocal Velocity. *Axioms* **2021**, *10*, 12.
https://doi.org/10.3390/axioms10010012

**AMA Style**

Göttlich S, Herty M, Weldegiyorgis G. Input-to-State Stability of a Scalar Conservation Law with Nonlocal Velocity. *Axioms*. 2021; 10(1):12.
https://doi.org/10.3390/axioms10010012

**Chicago/Turabian Style**

Göttlich, Simone, Michael Herty, and Gediyon Weldegiyorgis. 2021. "Input-to-State Stability of a Scalar Conservation Law with Nonlocal Velocity" *Axioms* 10, no. 1: 12.
https://doi.org/10.3390/axioms10010012