We study computational methods for computing the distance to singularity, the distance to the nearest high-index problem, and the distance to instability for linear differential-algebraic systems (DAEs) with dissipative Hamiltonian structure. While for general unstructured DAEs the characterization of these distances is very difficult and partially open, it has been shown in [C. Mehl, V. Mehrmann, and M. Wojtylak, Distance problems for dissipative Hamiltonian systems and related matrix polynomials, Linear Algebra Appl., 623 (2021), pp. 335–366] that for dissipative Hamiltonian systems and related matrix pencils there exist explicit characterizations. We will use these characterizations for the development of computational methods to approximate these distances via methods that follow the flow of a differential equation converging to the smallest perturbation that destroys the property of regularity, index one, or stability.
Keywords: dissipative Hamiltonian systems, structured distance to singularity, structured distance to high-index problem, structured distance to instability, low-rank perturbation, differential-algebraic system
