Geodesic
On a linear autonomous descriptor equation with discrete time. II.~Canonical representation and structural properties
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 56 (2020), pp. 102-121.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider a linear homogeneous autonomous descriptor equation with discrete time $$B_0g(k+1)+\sum_{i=1}^mB_ig(k+1-i)=0,\,k=m,m+1,\ldots,$$ with rectangular (in general case) matrices $ B_i. $ Such an equation arises in the study of the most important control problems for systems with many commensurate delays in control: the $0$-controllability problem, the synthesis problem of the feedback-type regulator, which provides calming to the solution of the original system, the modal controllability problem (controllability to the coefficients of characteristic quasipolynomial), the spectral reduction problem and the synthesis problem observers for dual surveillance system. The main method of the presented study is based on replacing the original equation with an equivalent equation in the “expanded” state space, which allows one to match the new equation of the beam of matrices. This made it possible to study a number of structural properties of the original equation by using the canonical form of the beam of matrices, and express the results in terms of minimal indices and elementary divisors. In the article, a criterion is obtained for the existence of a nontrivial admissible initial condition for the original equation, the verification of which is based on the calculation of the minimum indices and elementary divisors of the beam of matrices. The following problem was studied: it is required to construct a solution to the original equation in the form $g (k + 1) = T \psi (k + 1)$, $k = 1,2 \ldots, $ where $ T $ is some matrix, the sequence of vectors $ \psi (k + 1)$, $k = 1,2, \ldots, $ satisfies the equation $ \psi (k + 1) = S \psi (k)$, $k = 1,2,\ldots,$ and the square matrix $ S $ has a predetermined spectrum (or part of the spectrum). The results obtained make it possible to construct solutions of the initial descriptor equation with predetermined asymptotic properties, for example, uniformly asymptotically stable.
Keywords: linear descriptor autonomous equation with discrete time, the subspace of initial conditions, representation of the solution, beam of matrixes. 
@article{IIMI_2020_56_a8,
     author = {V. E. Khartovskii},
     title = {On a linear autonomous descriptor equation with discrete time. {II.~Canonical} representation and structural properties},
     journal = {Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta},
     pages = {102--121},
     publisher = {mathdoc},
     volume = {56},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IIMI_2020_56_a8/}
}
TY  - JOUR
AU  - V. E. Khartovskii
TI  - On a linear autonomous descriptor equation with discrete time. II.~Canonical representation and structural properties
JO  - Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
PY  - 2020
SP  - 102
EP  - 121
VL  - 56
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IIMI_2020_56_a8/
LA  - ru
ID  - IIMI_2020_56_a8
ER  - 
%0 Journal Article
%A V. E. Khartovskii
%T On a linear autonomous descriptor equation with discrete time. II.~Canonical representation and structural properties
%J Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
%D 2020
%P 102-121
%V 56
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IIMI_2020_56_a8/
%G ru
%F IIMI_2020_56_a8
V. E. Khartovskii. On a linear autonomous descriptor equation with discrete time. II.~Canonical representation and structural properties. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 56 (2020), pp. 102-121. http://geodesic.mathdoc.fr/item/IIMI_2020_56_a8/

[1] Khartovskii V. E., “On a linear autonomous descriptor equation with discrete time. I. Appendix to the 0-controllability problem”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 30:2 (2020), 290–311 (in Russian) | DOI | MR

[2] Khartovskii V. E., “A generalization of the problem of complete controllability for differential systems with commensurable delays”, Journal of Computer and Systems Sciences International, 48:6 (2009), 847–855 | DOI | MR | Zbl

[3] Khartovskii V. E., “Criteria for modal controllability of completely regular differential-algebraic systems with aftereffect”, Differential Equations, 54:4 (2018), 509–524 | DOI | DOI | MR | Zbl

[4] Khartovskii V. E., “Finite spectrum assignment for completely regular differential-algebraic systems with aftereffect”, Differential Equations, 54:6 (2018), 823–838 | DOI | DOI | MR | Zbl

[5] Khartovskii V. E., “Synthesis of observers for linear systems of neutral type”, Differential Equations, 55:3 (2019), 404–417 | DOI | DOI | MR | MR | Zbl

[6] Khartovskii V. E., “Controlling the spectrum of linear completely regular differential-algebraic systems with delay”, Journal of Computer and Systems Sciences International, 59:1 (2020), 19–38 | DOI | DOI | MR | Zbl

[7] Belov A. A., Kurdyukov A. P., Descriptor systems and control problems, Fizmatlit, M., 2015

[8] Boyarintsev Yu. E., Linear and nonlinear algebro-differential systems, Nauka, Novosibirsk, 2000

[9] Riaza R., Differential-algebraic systems: Analytical aspects and circuit applications, World Scientific, Hackensack, NY, 2008 | DOI | MR | Zbl

[10] Gantmakher R. F., Matrix theory, Nauka, M., 1988 | MR

[11] Rodina L. I., Tyuteev I. I., “About asymptotical properties of solutions of difference equations with random parameters”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 26:1 (2016), 79–86 | DOI | MR | Zbl

[12] Zaitsev V. A., Kim I. G., Khartovskii V. E., “Finite spectrum assignment problem for bilinear systems with several delays”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 29:3 (2019), 319–331 | DOI | MR | Zbl