be minimized) on the input $v$ can be written as $z = F(G,K) v$. The $\mathcal{H}_{\infty}$ control problem consists of finding the best controller $K$ such that the $\mathcal{H}_{\infty}$ norm of the matrix $F(G,K)$ is minimized: $\left\|F(G,K) \right\|_{\infty} = \sup_{\omega \in {\mathbb{R}}} \overline{\sigma}(F(G,K)(j \omega))$. It can be interpreted as the maximum gain from the input $v$ to the output $z$. This criterion was introduced in order to deal with uncertainties on the model (on the plant $G$). In [@Zames1981], the author introduced the notion and highlighted the connection with robustness. In [@Doyle1989], a link is shown between the existence of such a controller and conditions on the solutions of two Riccati equations. Following a notion introduced in [@Gahinet1992], the linear matrix inequality (LMI) approach was introduced in [@Gahinet1994], and used in [@apkarian2004; @noll2006] to solve the $\mathcal{H}_{\infty}$ synthesis. The Riccati equations are replaced with Riccati inequalities, whose set of solutions parameterizes the $\mathcal{H}_{\infty}$ controllers (see also [@boyd1994] for the use of LMIs in control theory). The papers [@Doyle1981; @McFarlane1992; @Xie1992] present design procedures in this context to elaborate the feedback controller $K$. In [@Ge1996], the theory is extended to systems with parameters uncertainties and state delays, as well as in 