Computation of the $$\mathcal {L}_{\infty }$$-norm of Finite-Dimensional Linear Systems

In this paper, we study the problem of computing the \(\mathcal {L}_{\infty }\)-norm of finite-dimensional linear time-invariant systems. This problem is first reduced to the computation of the maximal x-projection of the real solutions \((x, \, y)\) of a bivariate polynomial system \({\varSigma = \,}\{P,\frac{\partial P}{\partial y}\}\), 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 \(\varSigma \).