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. W X = Y ) and subjected to structured perturbations of the form W + M Δ N . As an application of these results, we shall derive formulas of the controllability radius for a descriptor controllable system [ E , A , B ] : E x ˙ = A x + B u , t ⩾ 0 , under the assumption that systems matrices E, A, B are subjected to structured perturbations and to multi-perturbations.