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Computing nearest stable matrix pairs

Gillis, Nicolas; Mehrmann, Volker; Sharma, Punit

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

In this paper, we study the nearest stable matrix pair problem: given a square matrix pair (E,A), minimize the Frobenius norm of (∆E,∆A) such that (E+\∆E,A+∆A) is a stable matrix pair. We propose a reformulation of the problem with a simpler feasible set byintroducing dissipative Hamiltonian (DH) matrix pairs: A matrix pair (E,A) is DH if A=(J-R)Q with skew-symmetric J, positive semidefinite R, and an invertible Q such that Q^TE is positive semidefinite. This reformulation has a convex feasible domain onto which it is easy to project. This allows us to employ a fast gradient method to obtain a nearby stable approximation of a given matrix pair.