Geodesic
The stability of differential-difference equations with proportional time delay. I. Linear controlled system
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 16 (2020) no. 3, pp. 316-325
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The article considers a controlled system of linear differential-difference equations with a linearly increasing delay. Sufficient conditions for the asymptotic stability of such systems are known; however, general conditions for the stabilizability of controlled systems and constructive algorithms for constructing stabilizing controls have not yet been obtained. For a linear differential-difference equation of delayed type with linearly increasing delay, the canonical Zubov transformation is applied and conditions for the stabilization of such systems by static control are derived. An algorithm for checking the conditions for the existence of a stabilizing control and for its constructing is formulated. New theorems on stability analysis of systems of linear differential-difference equations with linearly increasing delay are proven. The results obtained can be applied to the case of systems with several proportional delays.
Keywords: system of linear differential-difference equations, linearly increasing time delay, asymptotic stability, stabilizing control, asymptotic evaluation system. 
@article{VSPUI_2020_16_3_a7,
     author = {A. V. Ekimov and A. P. Zhabko and P. V. Yakovlev},
     title = {The stability of differential-difference equations with proportional time delay. {I.} {Linear} controlled system},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {316--325},
     year = {2020},
     volume = {16},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSPUI_2020_16_3_a7/}
}
TY  - JOUR
AU  - A. V. Ekimov
AU  - A. P. Zhabko
AU  - P. V. Yakovlev
TI  - The stability of differential-difference equations with proportional time delay. I. Linear controlled system
JO  - Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
PY  - 2020
SP  - 316
EP  - 325
VL  - 16
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VSPUI_2020_16_3_a7/
LA  - ru
ID  - VSPUI_2020_16_3_a7
ER  - 
%0 Journal Article
%A A. V. Ekimov
%A A. P. Zhabko
%A P. V. Yakovlev
%T The stability of differential-difference equations with proportional time delay. I. Linear controlled system
%J Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
%D 2020
%P 316-325
%V 16
%N 3
%U http://geodesic.mathdoc.fr/item/VSPUI_2020_16_3_a7/
%G ru
%F VSPUI_2020_16_3_a7
A. V. Ekimov; A. P. Zhabko; P. V. Yakovlev. The stability of differential-difference equations with proportional time delay. I. Linear controlled system. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 16 (2020) no. 3, pp. 316-325. http://geodesic.mathdoc.fr/item/VSPUI_2020_16_3_a7/

[1] R. Bellman, K. L. Cooke, Differential-difference equations, Academic Press, London, United Kingdom, 1963, 461 pp. | MR | MR | Zbl

[2] A. P. Zhabko, O. N. Chizhova, “Stability analysis of homogeneous differential-difference equation with linear delay”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Sciences. Control Processes, 2015, no. 3, 105–115 (In Russian)

[3] A. V. Ekimov, O. N. Chizhova, U. P. Zaranik, “Stability of homogeneous nonstationary systems of differential-difference equations with linearly time delay”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Sciences. Control Processes, 15:4 (2019), 415–424 (In Russian) | MR

[4] A. A. Laktionov, A. P. Zhabko, “Method of differential transformations for differential systems with linear time-delay”, Proceedings of the First IFAC Conference LTDS-98 (France, 1998), 201–205

[5] B. G. Grebenshhikov, “Methods for investigating the stability of systems with linear delay”, Siberian Mathematical Journal, 42:1 (2001), 41–51 (In Russian) | MR

[6] B. G. Grebenshhikov, A. B. Lozhnikov, “On stabilization of one system with aftereffect”, Automation and Remote Control, 2011, no. 1, 13–26 (In Russian)

[7] A. P. Zhabko, O. N. Chizhova, “On stabilization of systems of linear equations with linear increasing time-delay by observations”, Proceedings of RuPAC2016 (St. Petersburg, Russia, 2016), 261–263

[8] A. P. Zhabko, O. G. Tikhomirov, O. N. Chizhova, “On stabilization of a class of systems with time proportional delay”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Sciences. Control Processes, 14:2 (2018), 165–172 (In Russian) | MR

[9] V. I. Zubov, Lectures on control theory, Nauka, M., 1975, 496 pp. (In Russian)

[10] A. Zhabko, O. Chizhova, U. Zaranik, “Stability analysis of the linear time delay systems with linearly increasing delay”, Cybernetics and Physics, 5:2 (2016), 67–72 | MR