Controllability radii of linear systems with constrained controls under structured perturbations (Q2827485)

This paper is a detailed study about the robustness of the controllability property of linear systems of the formNEWLINENEWLINENEWLINE\[NEWLINE\dot x =Ax+BuNEWLINE\]NEWLINE under perturbation of the entries of \(A\) and \(B\). Here, \(x\in{\mathbb K}^n\), \(u\in{\mathbb K}^m\) (\({\mathbb K}^n={\mathbb R}^n\) or \({\mathbb C}^n\)) and the entries of the matrices \(A\) and \(B\) are, accordingly, real or complex. The main concept involved in this analysis is the controllability radius, which measures the distance of a controllable system to the nearest uncontrollable one. The study is carried out under the following conditions.NEWLINENEWLINE1) The values of the admissible control functions \(u(t)\) are constrained to a subset \(\Omega \subset {\mathbb K}^m\), such that \(0\in\)cl co\(\Omega\). Note that if \(\Omega\) is not the whole of \({\mathbb K}^m\), local and global controllability are not equivalent.NEWLINENEWLINE2) The admissible perturbed systemsNEWLINENEWLINENEWLINE\[NEWLINE\dot x =\tilde Ax+\tilde B uNEWLINE\]NEWLINE are such that \([\tilde A, \tilde B]-[A, B]=D\Delta E\) where \(D\) and \(E\) are fixed and \(\Delta\) represents the perturbations (structured perturbations).NEWLINENEWLINE\smallskip In general, the controllability radius are different when \({\mathbb K}^n={\mathbb R}^n\) or \({\mathbb K}^n={\mathbb C}^n\). Author's aim is to found explicit, computable formulas for the controllability radius in the various cases. Their results cover the complex case (for local and global controllability) and the real case (for local controllability, under the additional hypothesis that \(D\) is invertible).NEWLINENEWLINEOther results are given for more general classes of perturbations and for some particular cases.