On the Distance to Singular Descriptor Dynamical Systems With Impulsive Initial Conditions

In this paper, we study the problem of computing the distance between a given singular descriptor system <inline-formula><tex-math notation="LaTeX">$ (E,A)$</tex-math></inline-formula>, and a nearest descriptor system that has <italic>impulsive</italic> initial conditions. The link between existence of impulsive initial conditions and zeros at infinity for the associated matrix pencil <inline-formula><tex-math notation="LaTeX">$sE - A$</tex-math></inline-formula> is well-known. Much of the literature focusses on the case when only one of <inline-formula><tex-math notation="LaTeX">$E$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$A$</tex-math></inline-formula> is perturbed. We give a closed form expression of the distance to a nearest descriptor system having impulsive solutions via rank-1 perturbations when both <inline-formula><tex-math notation="LaTeX">$E$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$A$</tex-math></inline-formula> are perturbed. Next, for the case of perturbations without rank restrictions, we propose and evaluate the bounds for the distance. In the context of structured perturbations, we formulate and obtain an explicit expression for the distance, when <inline-formula><tex-math notation="LaTeX">$E$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$A$</tex-math></inline-formula> are <italic>Hermitian</italic> and are perturbed by Hermitian matrices. For a suitable class of systems, we also show that upper and lower bounds are within a factor of <inline-formula><tex-math notation="LaTeX">$\sqrt{2}$</tex-math></inline-formula>. We finally construct examples and compare the bounds obtained from our results with those from the literature as well as with computed values of the distance obtained via three numerical optimization techniques such as the structured low rank approximation, the Broyden–Fletcher–Goldfarb–Shanno algorithm, and direct optimization tools like <monospace>globalsearch</monospace>.

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