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Stabilizing the Metzler matrices with applications to dynamical systems

Abstract

Real matrices with non-negative off-diagonal entries play a crucial role for modelling the positive linear dynamical systems. In the literature, these matrices are referred to as Metzler matrices or negated Z-matrices. Finding the closest stable Metzler matrix to an unstable one (and vice versa) is an important issue with many applications. The stability considered here is in the sense of Hurwitz, and the distance between matrices is measured in \(l_\infty ,\,l_1\), and in the max norm. We provide either explicit solutions or efficient algorithms for obtaining the closest (un)stable matrix. The procedure for finding the closest stable Metzler matrix is based on the recently introduced selective greedy spectral method for optimizing the Perron eigenvalue. Originally intended for non-negative matrices, here is generalized to Metzler matrices. The efficiency of the new algorithms is demonstrated in examples and numerical experiments for the dimension of up to 2000. Applications to dynamical systems, linear switching systems, and sign-matrices are considered.

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Notes

  1. 1.

    See the beginning of the next subsection.

  2. 2.

    Norms without a subscript will be regarded as an \(l_\infty \) norms (i.e \(\Vert \cdot \Vert \ \equiv \Vert \cdot \Vert _\infty \)).

  3. 3.

    For the reminder of the text, each time we mention Metzler product families or positive product families, or similar, it will be in the context of product families of Metzler matrices, product families of positive matrices, etc.

  4. 4.

    See Definition 11.

  5. 5.

    \(\gamma _i\) percent of non-zero entries are chosen from (0,1) interval, and then randomly distributed as entries of a d-dimensional vector; if the ith entry is non-zero, then it can take a corresponding negative value. All other entries are set to zero.

  6. 6.

    See Examples below.

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Acknowledgements

Countless thanks to prof V. Yu. Protasov for the hours of inspiring discussions, all the valuable remarks and the support he was selflessly giving during the writing of this paper.

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Correspondence to Aleksandar Cvetković.

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Cvetković, A. Stabilizing the Metzler matrices with applications to dynamical systems. Calcolo 57, 1 (2020). https://doi.org/10.1007/s10092-019-0350-3

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Keywords

  • Metzler matrix
  • Spectral abscissa
  • Spectral radius
  • Perron eigenvalue
  • Sign-matrix
  • Dynamical system
  • Hurwitz stability

Mathematics Subject Classification

  • 15A42
  • 15B35
  • 15B48
  • 90C26
  • 93C30