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Stability radii for real linear Hamiltonian systems with perturbed dissipation

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Abstract

We study linear dissipative Hamiltonian (DH) systems with real constant coefficients that arise in energy based modeling of dynamical systems. We analyze when such a system is on the boundary of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues, or how much the dissipation term has to be perturbed to be on this boundary. For unstructured systems the explicit construction of the real distance to instability (real stability radius) has been a challenging problem. We analyze this real distance under different structured perturbations to the dissipation term that preserve the DH structure and we derive explicit formulas for this distance in terms of low rank perturbations. We also show (via numerical examples) that under real structured perturbations to the dissipation the asymptotical stability of a DH system is much more robust than for unstructured perturbations.

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Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Str. des 17. Juni 136, 10623, Berlin, Germany

    Christian Mehl, Volker Mehrmann & Punit Sharma

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  1. Christian Mehl
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  2. Volker Mehrmann
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  3. Punit Sharma
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Correspondence to Volker Mehrmann.

Additional information

Communicated by Daniel Kressner.

Supported by Einstein Stiftung Berlin through the Research Center Matheon Mathematics for key technologies in Berlin.

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Mehl, C., Mehrmann, V. & Sharma, P. Stability radii for real linear Hamiltonian systems with perturbed dissipation. Bit Numer Math 57, 811–843 (2017). https://doi.org/10.1007/s10543-017-0654-0

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  • DOI: https://doi.org/10.1007/s10543-017-0654-0

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