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$L^{\infty}$范数表示问题。既有研究通过将非参数情形下的问题简化为求解形如$\Sigma=\{P=0, \, \partial P/\partial x=0\}$系统中实数解$(x, y)$的最大$y$投影，并采用标准计算机代数方法进行分析与求解\cite{bouzidi2021computation}。本文将此方法扩展至参数化情形，旨在将$\Sigma$系统实数解的“最大”$y$投影表示为给定参数的函数。为此，我们采用柱形代数分解方法，从而能够在参数空间的特定区域内，将所需值表示为参数的函数。