# American Institute of Mathematical Sciences

September  2020, 25(9): 3597-3607. doi: 10.3934/dcdsb.2020074

## Assignability of dichotomy spectra for discrete time-varying linear control systems

 1 Department of Information Technology, National University of Civil Engineering, 55 Giai Phong str., Hanoi, Vietnam 2 Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam 3 Thang Long Institute of Mathematics and Applied Sciences, Thang Long University, Nghiem Xuan Yem Road, Hoang Mai, Hanoi, Vietnam

Received  August 2019 Revised  November 2019 Published  September 2020 Early access  April 2020

Fund Project: This research is funded by Vietnam National University of Civil Engineering (NUCE) under grant number 202-2018/KHXD-TD

In this paper, we show that for discrete time-varying linear control systems uniform complete controllability implies arbitrary assignability of dichotomy spectrum of closed-loop systems. This result significantly strengthens the result in [5] about arbitrary assignability of Lyapunov spectrum of discrete time-varying linear control systems.

Citation: Le Viet Cuong, Thai Son Doan. Assignability of dichotomy spectra for discrete time-varying linear control systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3597-3607. doi: 10.3934/dcdsb.2020074
##### References:
 [1] L. Ya. Adrianova, Introduction to Linear Systems of Differential Equations, Translated from the Russian by Peter Zhevandrov. Translations of Mathematical Monographs, 146. American Mathematical Society, Providence, RI, 1995. [2] B. Aulbach, C. Pötzsche and S. Siegmund, A smoothness theorem for invariant fiber bundles, J. Dynam. Differential Equations, 14 (2002), 519-547.  doi: 10.1023/A:1016383031231. [3] B. Aulbach and S. Siegmund, A spectral theory for nonautonomous difference equations, Proc. of 5th Int. Conference on Difference Equations and Applications, Temuco/Chile, 2002, 45–55. [4] A. Babiarz, I. Banshchikova, A. Czornik, E. K. Makarov, M. Niezabitowski and S. Popova, Necessary and sufficient conditions for assignability of the Lyapunov spectrum of discrete linear time-varying systems, IEEE Trans. Automat. Control, 63 (2018), 3825-3837.  doi: 10.1109/TAC.2018.2823086. [5] A. Babiarz, A. Czornik, E. Makarov, M. Niezabitowski and S. Popova, Pole placement theorem for discrete time-varying linear systems, SIAM J. Control Optim., 55 (2017), 671-692.  doi: 10.1137/15M1033666. [6] L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, Lecture Notes in Mathematics, 1926. Springer, Berlin, 2008. doi: 10.1007/978-3-540-74775-8. [7] F. Battelli and K. J. Palmer, Criteria for exponential dichotomy for triangular systems, J. Math. Anal. Appl., 428 (2015), 525-543.  doi: 10.1016/j.jmaa.2015.03.029. [8] L. V. Cuong, T. S. Doan and S. Siegmund, A Sternberg theorem for nonautonomous differential equations, J. Dynam. Differential Equations, 31 (2019), 1279-1299.  doi: 10.1007/s10884-017-9629-8. [9] R. A. Johnson, K. J. Palmer and G. R. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 18 (1987), 1-33.  doi: 10.1137/0518001. [10] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176, American Mathematical Society, 2011. doi: 10.1090/surv/176. [11] K. J. Palmer, A generalization of Hartman's linearization theorem, J. Math. Anal. Appl., 41 (1973), 753-758.  doi: 10.1016/0022-247X(73)90245-X. [12] C. Pötzsche and S. Siegmund, $C^m$-smoothness of invariant fiber bundles, Topol. Methods Nonlinear Anal., 24 (2004), 107-145.  doi: 10.12775/TMNA.2004.021. [13] S. Popova, On the global controllability of Lyapunov exponents of linear systems, Differential Equations, 43 (2007), 1072-1078.  doi: 10.1134/S0012266107080058. [14] C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Notes in Mathematics, 2002. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14258-1. [15] C. Pötzsche, Dichotomy spectra of triangular equations, Discrete & Continuous Dynamical Systems, 36 (2016), 423-450.  doi: 10.3934/dcds.2016.36.423. [16] M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems, Lecture Notes in Mathematics, 1907. Springer, Berlin, 2007. [17] R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358.  doi: 10.1016/0022-0396(78)90057-8. [18] A. L. Sasu and B. Sasu, On the dichotomic behavior of discrete dynamical systems on the half-line, Discrete & Continuous Dynamical Systems, 33 (2013), 3057-3084.  doi: 10.3934/dcds.2013.33.3057. [19] A. L. Sasu and B. Sasu, Discrete admissibility and exponential trichotomy of dynamical systems, Discrete & Continuous Dynamical Systems, 34 (2014), 2929-2962.  doi: 10.3934/dcds.2014.34.2929. [20] S. Siegmund, Dichotomy spectrum for nonautonomous differential equations, J. Dynam. Differential Equations, 14 (2002), 243-258.  doi: 10.1023/A:1012919512399. [21] S. Siegmund, Normal forms for nonautonomous differential equations, J. Differential Equations, 178 (2002), 541-573.  doi: 10.1006/jdeq.2000.4008.

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##### References:
 [1] L. Ya. Adrianova, Introduction to Linear Systems of Differential Equations, Translated from the Russian by Peter Zhevandrov. Translations of Mathematical Monographs, 146. American Mathematical Society, Providence, RI, 1995. [2] B. Aulbach, C. Pötzsche and S. Siegmund, A smoothness theorem for invariant fiber bundles, J. Dynam. Differential Equations, 14 (2002), 519-547.  doi: 10.1023/A:1016383031231. [3] B. Aulbach and S. Siegmund, A spectral theory for nonautonomous difference equations, Proc. of 5th Int. Conference on Difference Equations and Applications, Temuco/Chile, 2002, 45–55. [4] A. Babiarz, I. Banshchikova, A. Czornik, E. K. Makarov, M. Niezabitowski and S. Popova, Necessary and sufficient conditions for assignability of the Lyapunov spectrum of discrete linear time-varying systems, IEEE Trans. Automat. Control, 63 (2018), 3825-3837.  doi: 10.1109/TAC.2018.2823086. [5] A. Babiarz, A. Czornik, E. Makarov, M. Niezabitowski and S. Popova, Pole placement theorem for discrete time-varying linear systems, SIAM J. Control Optim., 55 (2017), 671-692.  doi: 10.1137/15M1033666. [6] L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, Lecture Notes in Mathematics, 1926. Springer, Berlin, 2008. doi: 10.1007/978-3-540-74775-8. [7] F. Battelli and K. J. Palmer, Criteria for exponential dichotomy for triangular systems, J. Math. Anal. Appl., 428 (2015), 525-543.  doi: 10.1016/j.jmaa.2015.03.029. [8] L. V. Cuong, T. S. Doan and S. Siegmund, A Sternberg theorem for nonautonomous differential equations, J. Dynam. Differential Equations, 31 (2019), 1279-1299.  doi: 10.1007/s10884-017-9629-8. [9] R. A. Johnson, K. J. Palmer and G. R. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 18 (1987), 1-33.  doi: 10.1137/0518001. [10] P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176, American Mathematical Society, 2011. doi: 10.1090/surv/176. [11] K. J. Palmer, A generalization of Hartman's linearization theorem, J. Math. Anal. Appl., 41 (1973), 753-758.  doi: 10.1016/0022-247X(73)90245-X. [12] C. Pötzsche and S. Siegmund, $C^m$-smoothness of invariant fiber bundles, Topol. Methods Nonlinear Anal., 24 (2004), 107-145.  doi: 10.12775/TMNA.2004.021. [13] S. Popova, On the global controllability of Lyapunov exponents of linear systems, Differential Equations, 43 (2007), 1072-1078.  doi: 10.1134/S0012266107080058. [14] C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Notes in Mathematics, 2002. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14258-1. [15] C. Pötzsche, Dichotomy spectra of triangular equations, Discrete & Continuous Dynamical Systems, 36 (2016), 423-450.  doi: 10.3934/dcds.2016.36.423. [16] M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems, Lecture Notes in Mathematics, 1907. Springer, Berlin, 2007. [17] R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358.  doi: 10.1016/0022-0396(78)90057-8. [18] A. L. Sasu and B. Sasu, On the dichotomic behavior of discrete dynamical systems on the half-line, Discrete & Continuous Dynamical Systems, 33 (2013), 3057-3084.  doi: 10.3934/dcds.2013.33.3057. [19] A. L. Sasu and B. Sasu, Discrete admissibility and exponential trichotomy of dynamical systems, Discrete & Continuous Dynamical Systems, 34 (2014), 2929-2962.  doi: 10.3934/dcds.2014.34.2929. [20] S. Siegmund, Dichotomy spectrum for nonautonomous differential equations, J. Dynam. Differential Equations, 14 (2002), 243-258.  doi: 10.1023/A:1012919512399. [21] S. Siegmund, Normal forms for nonautonomous differential equations, J. Differential Equations, 178 (2002), 541-573.  doi: 10.1006/jdeq.2000.4008.
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