# An Overview on Irreversible Port-Hamiltonian Systems

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## Abstract

**:**

## 1. Introduction

## 2. IPHS Defined on Finite Dimensional Spaces

#### 2.1. Port-Hamiltonian Systems and the Second Principle

#### 2.2. Irreversible PHS

**Definition**

**1.**

**Definition**

**2.**

- A pair of functions: the total energy $H:{\mathbb{R}}^{n+1}\to \mathbb{R}$ and the total entropy $s\in \mathbb{R}$,
- A pair of matrices ${P}_{0}=-{P}_{0}^{\top}\in {\mathbb{R}}^{n\times n}$ and ${G}_{0}\in {\mathbb{R}}^{n\times m}$ with $m\le n$ and the positive real-valued functions ${\gamma}_{i}\left(x,s\right),\phantom{\rule{0.222222em}{0ex}}i\in \left\{1,\phantom{\rule{0.166667em}{0ex}}...\phantom{\rule{0.166667em}{0ex}}m\right\}$,

#### 2.3. Examples

#### 2.3.1. The Heat Exchanger

#### 2.3.2. The Gas-Piston System

## 3. IPHS Defined on 1-Dimensional Spatial Domains

#### 3.1. Boundary-Controlled PHS

#### 3.2. Boundary-Controlled IPHS

**Definition**

**3.**

- A pair of matrices ${P}_{0}=-{P}_{0}^{\top}\in {\mathbb{R}}^{n\times n}$ and ${P}_{1}={P}_{1}^{\top}\in {\mathbb{R}}^{n\times n}$;
- A pair of matrices ${G}_{0}\in {\mathbb{R}}^{n\times m}$, ${G}_{1}\in {\mathbb{R}}^{n\times m}$ with $m\le n$ and the strictly positive real-valued functions ${\gamma}_{k,i}\left(x,z,{\textstyle \frac{\delta H}{\delta x}}\right)\phantom{\rule{0.222222em}{0ex}}k=0,\phantom{\rule{0.166667em}{0ex}}1;\phantom{\rule{0.222222em}{0ex}}i\in \left\{1,\phantom{\rule{0.166667em}{0ex}}\dots ,\phantom{\rule{0.166667em}{0ex}}m\right\}$;
- A pair of real-valued functions ${\gamma}_{s}\left(x,z,{\textstyle \frac{\delta H}{\delta x}}\right)>0$ and ${g}_{s}\left(x\right)$

**Definition**

**4.**

**Lemma 1.**

**Lemma 2.**

#### 3.3. Examples

#### 3.3.1. The Heat Equation

#### 3.3.2. The Non-Isentropic Fluid

## 4. Conclusions and Outlook

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Ramirez, H.; Le Gorrec, Y.
An Overview on Irreversible Port-Hamiltonian Systems. *Entropy* **2022**, *24*, 1478.
https://doi.org/10.3390/e24101478

**AMA Style**

Ramirez H, Le Gorrec Y.
An Overview on Irreversible Port-Hamiltonian Systems. *Entropy*. 2022; 24(10):1478.
https://doi.org/10.3390/e24101478

**Chicago/Turabian Style**

Ramirez, Hector, and Yann Le Gorrec.
2022. "An Overview on Irreversible Port-Hamiltonian Systems" *Entropy* 24, no. 10: 1478.
https://doi.org/10.3390/e24101478