The structured distance to non-surjectivity and its application to calculating the controllability radius of descriptor systems

The classical Eckart–Young formula for square matrices identifies the distance to singularity of a matrix. The main purpose of this paper is to get generalizations of this formula. We characterize the distance to non-surjectivity of a linear operator W∈L(X,Y) in finite-dimensional normed spaces X, Y, under the assumption that the operator W is surjective (i.e. WX=Y) and subjected to structured perturbations of the form W+MΔN. As an application of these results, we shall derive formulas of the controllability radius for a descriptor controllable system , t⩾0, under the assumption that systems matrices E, A, B are subjected to structured perturbations and to multi-perturbations.