Matrix Stabilization Using Differential Equations
Nicola Guglielmi, Christian Lubich · SIAM Journal on Numerical Analysis · 2017
We consider the problem of stabilizing a matrix by a correction of minimal norm: Given a square matrix that has some eigenvalues with positive real part, find the nearest matrix having no eigenvalue with positive real part. It can be further required that the correction have a prescribed structure, e.g., be real, have a prescribed sparsity pattern, or have a given maximal rank. We propose and study a novel approach to this nonconvex and nonsmooth optimization problem, based on the solution of low-rank matrix differential equations. This enables us to compute locally optimal solutions in a fast way, also for higher-dimensional problems. Illustrative numerical experiments provide evidence of the efficiency of the method. It is further shown that the approach applies equally to the related problems of closed-loop stabilization of control systems and to the stabilization of gyroscopic systems.