Computation of the $\mathcal{L} \infty$ -norm of finite-dimensional linear systems

International audience; In this paper, we study the problem of computing the $\mathcal{L}\infty$- norm of finite-dimensional linear time-invariant systems. This problemis first reduced to the computation of the maximal x-projection of the real solutions $(x, y)$ of a bivariate polynomial system $\sum=\{P,{\frac{\partial{P}}{\partial{y}}}\}$, with ${P} \in \mathbb{Z}[x, y]$. Then, we use standard computer algebra methods to solve the problem. In this paper, we alternatively study a method based on rational univariate representations, a method based on root separation, and finally a method first based on the sign variation of the leading coefficients of the signed subresultant sequence and then based on the identification of an isolating interval for the maximal $x$-projection of the real solutions of $\sum$.