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Transient Response in Matrix Discrete-Time Linear Systems

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Abstract

The behavior of trajectories of multidimensional linear discrete-time systems with nonzero initial conditions is considered in two cases as follows. The first case is the systems with infinite degree of stability (the processes of a finite duration); the second case is the stable systems with a spectral radius close to 1. It is demonstrated that in both cases, large deviations of the trajectories from the equilibrium may occur. These results are applied to accelerated unconstrained optimization methods (such as the Heavy-ball method) for explaining the nonmonotonic behavior of the methods, which is observed in practice.

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Acknowledgments

The work of B.T. Polyak was supported by the Russian Science Foundation, project no. 16-11-10015. We are grateful to P.S. Shcherbakov, M. Danilova and A. Kulakova for careful reading of the manuscript and helpful remarks.

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Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    B. T. Polyak

  2. University of Minho, Braga, Portugal

    G. V. Smirnov

Authors
  1. B. T. Polyak
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  2. G. V. Smirnov
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Corresponding authors

Correspondence to B. T. Polyak or G. V. Smirnov.

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Russian Text © The Author(s), 2019, published in Avtomatika i Telemekhanika, 2019, No. 9, pp. 112–121.

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Polyak, B.T., Smirnov, G.V. Transient Response in Matrix Discrete-Time Linear Systems. Autom Remote Control 80, 1645–1652 (2019). https://doi.org/10.1134/S0005117919090066

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