${L}^{\infty}$-norm computation for linear time-invariant systems depending on parameters
Résumé
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.
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. 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.
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