1

In fact I want to write this function definition with the leftbrace : enter image description here

And here is the code that I have written:

\begin{equation}
\begin{aligned}
Si \,\,\,\,\,\, \varepsilon Q_s \geq \Delta Q_s\\
Si \,\,\,\,\,\,  \varepsilon Q_s \leq -\Delta Q_s\\
Si \,\,\,\,\,\,  \varepsilon P_s \geq \Delta P_s\\
Si \,\,\,\,\,\,  -\Delta P_s \leq \varepsilon P_s \leq \Delta P_s\\
Si \,\,\,\,\,\,  \varepsilon P_s \leq -\Delta P_s
\end{aligned}
\end{equation}
1

It can be done with cases and alignedat. Since you seem to write in French, I took the liberty to use the French inequality signs. I propose also a second alignment which looks lighter, in my opinion: \documentclass[9pt]{extbook}

\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}

\usepackage{array}
\usepackage{amsfonts,amssymb}
\usepackage{empheq}

\usepackage[francais]{babel}
\def\fgeq{\geqslant}
\def\fleq{\geqslant}

\begin{document}

\begin{equation}
\begin{cases}
\begin{alignedat}{3}
 &\text{Si} & \enspace & \varepsilon Q_s \fgeq \Delta Q_s & \quad\implies & H_Q = 1\\
 & \text{Si} & & \varepsilon Q_s \fleq -\Delta Q_s & \implies & H_Q = -1\\
 & \text{Si} & & \varepsilon P_s \fgeq \Delta P_s & \implies & H_P = 1\\
 & \text{Si} & & -\Delta P_s \fleq \varepsilon P_s \fleq \Delta P_s & \implies & H_P = 0\\
 & \text{Si} & & \varepsilon P_s \fleq -\Delta P_s & \implies & H_P = -1
\end{alignedat}
\end{cases}
\end{equation}

\end{document} 

enter image description here

2

A single cases environment (provided by the amsmath package) is all you need:

enter image description here

\documentclass{article}
\usepackage{amsmath} % for 'cases' environment
\begin{document}

\begin{equation}
\begin{cases}
\text{Si } \varepsilon Q_s \geq \Delta Q_s   & \Rightarrow H_Q=\phantom{-}1 \\
\text{Si } \varepsilon Q_s \leq -\Delta Q_s  & \Rightarrow H_Q=-1  \\
\text{Si } \varepsilon P_s \geq \Delta P_s   & \Rightarrow H_P=\phantom{-}1 \\
\text{Si }{-}\Delta P_s\leq\varepsilon P_s\leq\Delta P_s 
                                             & \Rightarrow H_P=\phantom{-}0\\
\text{Si } \varepsilon P_s \leq -\Delta P_s  & \Rightarrow H_P=-1
\end{cases}
\end{equation}

\end{document}

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