Geodesic
Stabilization of the second-order linear time-invariant control systems by a~delayed feedback
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2010), pp. 87-90.

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

We consider a static stabilization problem for a two-dimensional linear time-invariant control system with a delayed feedback. We obtain the necessary and sufficient conditions for the stabilizability of the system under consideration. The theorems proved in this paper show that such a delayed feedback approach is efficient in stabilizing the second-order linear systems.
Keywords: linear time-invariant system, delayed feedback, stationary stabilization, asymptotic stability. 
@article{IVM_2010_12_a8,
     author = {M. M. Shumafov},
     title = {Stabilization of the second-order linear time-invariant control systems by a~delayed feedback},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {87--90},
     publisher = {mathdoc},
     number = {12},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2010_12_a8/}
}
TY  - JOUR
AU  - M. M. Shumafov
TI  - Stabilization of the second-order linear time-invariant control systems by a~delayed feedback
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2010
SP  - 87
EP  - 90
IS  - 12
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2010_12_a8/
LA  - ru
ID  - IVM_2010_12_a8
ER  - 
%0 Journal Article
%A M. M. Shumafov
%T Stabilization of the second-order linear time-invariant control systems by a~delayed feedback
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2010
%P 87-90
%N 12
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2010_12_a8/
%G ru
%F IVM_2010_12_a8
M. M. Shumafov. Stabilization of the second-order linear time-invariant control systems by a~delayed feedback. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2010), pp. 87-90. http://geodesic.mathdoc.fr/item/IVM_2010_12_a8/

[1] Bernstein D. S., “Some open problems in matrix theory arising in linear systems and control”, Linear Algebra Appl., 162–164 (1992), 409–432 | DOI | MR | Zbl

[2] Blondel V., Gevers M., Lindquist A., “Survey on the state of systems and control”, Eur. J. Control, 1:1 (1995), 5–23 | Zbl

[3] Syrmos V. L., Abdallah C. T., Dorato P., Grigoriadis K., “Static output feedback –a survey”, Automatica, 33:2 (1997), 125–137 | DOI | MR | Zbl

[4] Blondel V., Sontag E., Vidyasagar M., Willems J., Open problems in mathematical systems and control theory, Springer, London, 1999 | MR | Zbl

[5] Polyak B. T., Scherbakov P. S., “Trudnye zadachi lineinoi teorii upravleniya. Nekotorye podkhody k resheniyu”, Avtomatika i telemekhanika, 2005, no. 5, 7–46 | MR | Zbl

[6] Leonov G. A., Shumafov M. M., Metody stabilizatsii lineinykh upravlyaemykh sistem, Izd-vo S.-Peterb. un-ta, SPb., 2005

[7] Leonov G. A., “Stabilizatsionnaya problema Broketta”, Avtomatika i telemekhanika, 2001, no. 5, 190–193 | MR | Zbl

[8] Leonov G. A., “Problema Broketta v teorii ustoichivosti lineinykh differentsialnykh uravnenii”, Algebra i analiz, 13:4 (2001), 134–155 | MR | Zbl

[9] Moreau L., Aeyels D., “Periodic output feedback stabilization of single-input single-output continuous-time systems with odd relative degree”, Syst. Control Lett., 51:5 (2004), 395–406 | DOI | MR | Zbl

[10] Pyragas K., “Continuous control of chaos by self-controlling feedback”, Phys. Lett. A, 170:6 (1992), 421–428 | DOI

[11] Pyragas K., “Control of chaos via extended delay feedback”, Phys. Lett. A, 206:5–6 (1995), 323–330 | DOI | MR | Zbl

[12] Pyragas V., Pyragas K., “Delayed feedback control of the Lorenz system: an analytical treatment at a subcritical Hopf bifurcation”, Phys. Rev. E, 73 (2006), 036215 | DOI | MR

[13] Pyragas K., “Delayed feedback control of chaos”, Philos. Trans. R. Soc. Lond. Ser. A. Math. Phys. Eng. Sci., 364 (1846), 2309–2334 | DOI | MR

[14] Tamaševičius A., Mykolaitis G., Pyragas V., Pyragas K., “Delayed feedback control of periodic orbits without torsion in nonautonomous systems: theory and experiment”, Phys. Rev. E, 76:2 (2007), 026203, 6 pp. | DOI

[15] Andreev Yu. N., Upravlenie konechnomernymi lineinymi ob'ektami, Nauka, M., 1976 | MR

[16] Pontryagin L. S., “O nulyakh nekotorykh elementarnykh transtsendentnykh funktsii”, Izv. AN SSSR. Ser. Matem., 6:3 (1942), 115–134 | MR | Zbl

[17] Chebotarev N. G., Meiman N. N., “Problema Rausa–Gurvitsa dlya polinomov i tselykh funktsii”, Tr. Matem. in-ta im. V. A. Steklova, 26, Izd-vo AN SSSR, M., 1949, 3–331 | MR | Zbl

[18] Elsgolts L. E., Norkin S. B., Vvedenie v teoriyu differentsialnykh uravnenii s otklonyayuschimsya argumentom, Nauka, M., 1971 | MR

[19] Neimark Yu. I., Dinamicheskie sistemy i upravlyaemye protsessy, Nauka, M., 1978 | MR

[20] Kiryanen A. I., Ustoichivost sistem s posledeistviem i ikh prilozheniya, Izd-vo S.-Peterb. un-ta, SPb., 1994 | MR