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Singular renormalization group approach to SIS problems
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 |
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 [
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. |
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. |
[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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