Computing nearest stable matrix pairs

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 by introducing 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 QE 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.