On the distance to uncontrollability and the distance to instability and their relation to some condition numbers in control (Q1360721)

It is recognized since a number of years that for many basic problems in linear algebra and numerical analysis there is a close relation between the condition of a problem and its distance to the set of ill-posed problems, an ill-posed problems being defined in general as one with an infinite condition number. Of course, details vary with the norm used to define distance functions etc. The present author discusses four interesting cases of this kind, namely the computation of the Hessenberg form, the solution of the algebraic Riccati equation, the pole assignment problem and the matrix exponential. The ill-posed problems are those which are uncontrollable or unstable in the appropriate setting. The paper contains several numerical examples in MATLAB. Also, it has a very good list of references to the literature on this fundamental topic.