Geodesic
On the conditions of proportional local assignability of the Lyapunov spectrum of a linear discrete-time system
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 29 (2019) no. 3, pp. 301-311
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We consider a problem of assigning the Lyapunov spectrum for a linear control discrete-time system \begin{equation} x(m+1)=A(m)x(m)+B(m)u(m),\quad m\in\mathbb N,\ x\in\mathbb R^{n},\ u\in\mathbb R^{k}, \tag{1} \end{equation} in a small neighborhood of the Lyapunov spectrum of the free system \begin{equation} x(m+1)=A(m)x(m),\quad m\in\mathbb N,\ x\in\mathbb R^{n}, \tag{2} \end{equation} by means of linear feedback $u(m)=U(m)x(m)$. We assume that the norm of the feedback matrix $U(\cdot)$ satisfies the Lipschitz estimate with respect to the required shift of the Lyapunov spectrum. This property is called proportional local assignability of the Lyapunov spectrum of the closed-loop system \begin{equation} x(m+1)=\bigl(A(m)+B(m)U(m)\bigr)x(m),\quad m\in\mathbb N,\ x\in\mathbb R^{n}. \tag{3} \end{equation} We previously proved that uniform complete controllability of system (1) and stability of the Lyapunov spectrum of free system (2) are sufficient conditions for proportional local assignability of the Lyapunov spectrum of closed-loop system (3). In this paper we give an example demonstrating that these conditions are not necessary.
Keywords: linear discrete-time system, Lyapunov exponents, сontrollability, stabilizability. 
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     title = {On the conditions of proportional local assignability of the {Lyapunov} spectrum of a linear discrete-time system},
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I. N. Banshchikova; E. K. Makarov; S. N. Popova. On the conditions of proportional local assignability of the Lyapunov spectrum of a linear discrete-time system. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 29 (2019) no. 3, pp. 301-311. http://geodesic.mathdoc.fr/item/VUU_2019_29_3_a0/

[1] V. B. Demidovich, “A certain criterion for the stability of difference equations”, Differ. Uravn., 5:7 (1969), 1247–1255 (in Russian) | MR | Zbl

[2] I. V. Gaishun, Discrete-time systems, Institute of Mathematics of the National Academy of Sciences of Belarus, Minsk, 2001, 400 pp.

[3] A. Babiarz, I. Banshchikova, A. Czornik, E. Makarov, M. Niezabitowski, S. Popova, “On assignability of Lyapunov spectrum of discrete linear time-varying system with control”, 2016 21st International Conference on Methods and Models in Automation and Robotics (MMAR), IEEE, 2016, 697–701 | DOI | MR

[4] A. Babiarz, A. Czornik, E. Makarov, M. Niezabitowski, S. Popova, “Pole placement theorem for discrete time-varying linear systems”, SIAM Journal on Control and Optimization, 55:2 (2017), 671–692 | DOI | MR | Zbl

[5] A. Babiarz, I. Banshchikova, A. Czornik, E. Makarov, M. Niezabitowski, S. Popova, “Necessary and sufficient conditions for assignability of the Lyapunov spectrum of discrete linear time-varying systems”, IEEE Transactions on Automatic Control, 63:11 (2018), 3825–3837 | DOI | MR | Zbl

[6] S. N. Popova, “Assignability of certain Lyapunov invariants for linear discrete-time systems”, IFAC-PapersOnLine, 51:32 (2018), 40–45 | DOI

[7] A. Babiarz, I. Banshchikova, A. Czornik, E. Makarov, M. Niezabitowski, S. Popova, “Proportional local assignability of Lyapunov spectrum of linear discrete time-varying systems”, SIAM Journal on Control and Optimization, 57:2 (2019), 1355–1377 | DOI | MR | Zbl

[8] A. Halanay, V. Ionescu, Time-varying discrete linear systems: input-output operators, Riccati equations, disturbance attenuation, Springer, Basel, 1994, 230 pp. | DOI | MR

[9] I. N. Banshchikova, S. N. Popova, “On the spectral set of a linear discrete system with stable Lyapunov exponents”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 26:1 (2016), 15–26 (in Russian) | DOI | MR

[10] B. F. Bylov, R. E. Vinograd, D. M. Grobman, V. V. Nemytskii, Theory of Lyapunov exponents and its application to problems of stability, Nauka, M., 1966, 576 pp. | MR

[11] I. N. Banshchikova, “An example of a linear discrete system with unstable Lyapunov exponents”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 26:2 (2016), 169–176 (in Russian) | DOI | MR