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On computing the distance to stability for matrices using linear dissipative Hamiltonian systems. (English) Zbl 1375.93110
Summary: In this paper, we consider the problem of computing the nearest stable matrix to an unstable one. We propose new algorithms to solve this problem based on a reformulation using linear dissipative Hamiltonian systems: we show that a matrix \(A\) is stable if and only if it can be written as \(A=(J-R)Q\), where \(J=-J^T\), \(R \succeq 0\) and \(Q\succ 0\) (that is, \(R\) is positive semidefinite and \(Q\) is positive definite). This reformulation results in an equivalent optimization problem with a simple convex feasible set. We propose three strategies to solve the problem in variables \((J,R,Q)\): (i) a block coordinate descent method, (ii) a projected gradient descent method, and (iii) a fast gradient method inspired from smooth convex optimization. These methods require \(\mathcal{O}(n^3)\) operations per iteration, where \(n\) is the size of \(A\). We show the effectiveness of the fast gradient method compared to the other approaches and to several state-of-the-art algorithms.

93D21 Adaptive or robust stabilization
93C15 Control/observation systems governed by ordinary differential equations
93C05 Linear systems in control theory
93B60 Eigenvalue problems
Full Text: DOI
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