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. Read More: https://epubs.siam.org/doi/10.1137/16M1105840

Matrix stabilization using differential equations

Guglielmi, N.;
2017

Abstract

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. Read More: https://epubs.siam.org/doi/10.1137/16M1105840
matrix stabilization, stabilizing feedback, matrix nearness problems, low-rank matrix manifolds, gradient flow
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12571/384
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