This paper focuses on representing the $L^{\infty}$-norm of finite-dimensional linear time-invariant systems with parameter-dependent coefficients. Previous studies tackled the problem in a non-parametric scenario by simplifying it to finding the maximum $y$-projection of real solutions $(x, y)$ of a system of the form $\Sigma=\{P=0, \, \partial P/\partial x=0\}$, where $P \in \Z[x, y]$. To solve this problem, standard computer algebra methods were employed and analyzed \cite{bouzidi2021computation}. In this paper, we extend our approach to address the parametric case. We aim to represent the "maximal" $y$-projection of real solutions of $\Sigma$ as a function of the given parameters. %a set of parameters $\alpha$. To accomplish this, we utilize cylindrical algebraic decomposition. This method allows us to determine the desired value as a function of the parameters within specific regions of parameter space.

翻译：本文聚焦于具有参数依赖系数的有限维线性时不变系统的${L}^{\infty}$范数表示。先前的研究在非参数化场景中，通过将该问题简化为求解形如$\Sigma=\{P=0, \, \partial P/\partial x=0\}$（其中$P \in \Z[x, y]$）的方程组实解$(x, y)$的最大$y$投影来处理。为求解此问题，经典计算机代数方法被采用并加以分析\cite{bouzidi2021computation}。本文扩展了已有方法以处理参数化情形，旨在将$\Sigma$实解的“最大”$y$投影表示为给定参数的函数。为此，我们利用柱形代数分解方法，从而能够在参数空间的特定区域内，将目标值确定为参数的函数。