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

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

Le Viet Cuong 1, and  Thai Son Doan 2,3,

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.

Keywords: Time-varying systems, stability, dichotomy spectrum, control systems, Lyapunov spectrum.
Mathematics Subject Classification: Primary: 34A30, 34D09, 34D08, 34H05.
Citation: Le Viet Cuong, Thai Son Doan. Assignability of dichotomy spectra for discrete time-varying linear control systems. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176, American Mathematical Society, 2011. doi: 10.1090/surv/176.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

S. Popova, On the global controllability of Lyapunov exponents of linear systems, Differential Equations, 43 (2007), 1072-1078.  doi: 10.1134/S0012266107080058.  Google Scholar

[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.  Google Scholar

[15]

C. Pötzsche, Dichotomy spectra of triangular equations, Discrete & Continuous Dynamical Systems, 36 (2016), 423-450.  doi: 10.3934/dcds.2016.36.423.  Google Scholar

[16]

M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems, Lecture Notes in Mathematics, 1907. Springer, Berlin, 2007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

S. Siegmund, Dichotomy spectrum for nonautonomous differential equations, J. Dynam. Differential Equations, 14 (2002), 243-258.  doi: 10.1023/A:1012919512399.  Google Scholar

[21]

S. Siegmund, Normal forms for nonautonomous differential equations, J. Differential Equations, 178 (2002), 541-573.  doi: 10.1006/jdeq.2000.4008.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176, American Mathematical Society, 2011. doi: 10.1090/surv/176.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

S. Popova, On the global controllability of Lyapunov exponents of linear systems, Differential Equations, 43 (2007), 1072-1078.  doi: 10.1134/S0012266107080058.  Google Scholar

[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.  Google Scholar

[15]

C. Pötzsche, Dichotomy spectra of triangular equations, Discrete & Continuous Dynamical Systems, 36 (2016), 423-450.  doi: 10.3934/dcds.2016.36.423.  Google Scholar

[16]

M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems, Lecture Notes in Mathematics, 1907. Springer, Berlin, 2007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

S. Siegmund, Dichotomy spectrum for nonautonomous differential equations, J. Dynam. Differential Equations, 14 (2002), 243-258.  doi: 10.1023/A:1012919512399.  Google Scholar

[21]

S. Siegmund, Normal forms for nonautonomous differential equations, J. Differential Equations, 178 (2002), 541-573.  doi: 10.1006/jdeq.2000.4008.  Google Scholar

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