Geodesic
Control system depending on a parameter
Ural mathematical journal, Tome 7 (2021) no. 1, pp. 120-159 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A nonlinear control system depending on a parameter is considered in a finite-dimensional Euclidean space and on a finite time interval. The dependence on the parameter of the reachable sets and integral funnels of the corresponding differential inclusion system is studied. Under certain conditions on the control system, the degree of this dependence on the parameter is estimated. Problems of targeting integral funnels to a target set in the presence of an obstacle in strict and soft settings are considered. An algorithm for the numerical solution of this problem in the soft setting has been developed. An estimate of the error of the developed algorithm is obtained. An example of solving a specific problem for a control system in a two-dimensional phase space is given.
Keywords: control system, differential inclusion, reachable set, integral funnel, parameter dependence, approximation. 
@article{UMJ_2021_7_1_a10,
     author = {Vladimir N. Ushakov and Aleksandr A. Ershov and Andrey V. Ushakov and Oleg A. Kuvshinov},
     title = {Control system depending on a parameter},
     journal = {Ural mathematical journal},
     pages = {120--159},
     year = {2021},
     volume = {7},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UMJ_2021_7_1_a10/}
}
TY  - JOUR
AU  - Vladimir N. Ushakov
AU  - Aleksandr A. Ershov
AU  - Andrey V. Ushakov
AU  - Oleg A. Kuvshinov
TI  - Control system depending on a parameter
JO  - Ural mathematical journal
PY  - 2021
SP  - 120
EP  - 159
VL  - 7
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/UMJ_2021_7_1_a10/
LA  - en
ID  - UMJ_2021_7_1_a10
ER  - 
%0 Journal Article
%A Vladimir N. Ushakov
%A Aleksandr A. Ershov
%A Andrey V. Ushakov
%A Oleg A. Kuvshinov
%T Control system depending on a parameter
%J Ural mathematical journal
%D 2021
%P 120-159
%V 7
%N 1
%U http://geodesic.mathdoc.fr/item/UMJ_2021_7_1_a10/
%G en
%F UMJ_2021_7_1_a10
Vladimir N. Ushakov; Aleksandr A. Ershov; Andrey V. Ushakov; Oleg A. Kuvshinov. Control system depending on a parameter. Ural mathematical journal, Tome 7 (2021) no. 1, pp. 120-159. http://geodesic.mathdoc.fr/item/UMJ_2021_7_1_a10/

[1] Anan'evskii I. M., “Control of a nonlinear vibratory system of the fourth order with unknown parameters”, Autom. Remote Control, 62:3 (2001), 343–355 | DOI | MR

[2] Anan'evskii I. M., “Control synthesis for linear systems by methods of stability theory of motion”, Differential Equations, 39:1 (2003), 1–10 | DOI | MR

[3] Beznos A. V., Grishin A. A., Lensky A. V., Okhotsimsky D. E., Formalsky A. M., “Pendulum control using a flywheel”, Spetspraktikum po teoreticheskoi i prikladnoi mehanike [Special workshop on theoretical and applied mechanics], eds. V.V. Aleksandrov, Yu.V. Bolotov, MSU Press, Moscow, 2019, 170–195

[4] Bogachev V.I., Smoljanov O.G., Deistvitel'nyi i funktsional'nyi analiz: universitetskii kurs [Real and Functional Analysis: University Course], Research Center “Regular and Chaotic Dynamics”, Institute for Computer Research, Moscow-Izhevsk, 2009, 724 pp. (in Russian)

[5] Chernousko F. L., State Estimation for Dynamic Systems, Boca Raton, CRC Press, 1994, 320 pp.

[6] Chernousko F. L., Melikyan A. A., Igrovye zadachi upravlenija i poiska [Game Control and Search Problems], Nauka, Moscow, 1978, 270 pp. (in Russian) | MR

[7] Ershov A. A., Ushakov V. N., “An approach problem for a control system with an unknown parameter”, Sb. Math., 208:9 (2017), 1312–1352 | DOI | MR | Zbl

[8] Filippova T. F., “Construction of set-valued estimates of reachable sets for some nonlinear dynamical systems with impulsive control”, Proc. Steklov Inst. Math., 269, Suppl. 1 (2010), S95–S102 | DOI | MR

[9] Gusev M. I., “Estimates of reachable sets of multidimensional control systems with nonlinear interconnections”, Proc. Steklov Inst. Math, 269, Suppl. 1 (2010), S134–S146 | DOI | MR

[10] Krasovsky N. N., Upravlenie dinamicheskoi sistemoi: Zadacha o minimume garantirovannogo rezul'tata [Control of a Dynamical System: Problem on the Minimum of Guaranteed Result], Nauka, Moscow, 1985, 520 pp. (in Russian) | MR

[11] Krasovsky N. N., Subbotin A. I., Pozitsionnye differentsial'nye igry [Positional Differential Games], Fizmatlit, Moscow, 1974, 456 pp. (in Russian) | MR

[12] Kurzhansky A. B., Izbrannye Trudy [Selected Works], MSU Press, Moscow, 2009, 756 pp. (in Russian)

[13] Kurzhanski A. B., Valyi I., Ellipsoidal Calculus for Estimation and Control, Systems Control Found. Appl., Birkhäuser, Basel, 1997, 321 pp. | DOI | MR | Zbl

[14] Lee E. B., Markus L., Foundation of Optimal Control Theory, John Wiley Sons, New York–London–Sydney, 1967, 576 pp. | MR

[15] Leichtweiß K., Konvexe Mengen, Hochschultext, Springer-Verlag, Berlin, 1979, 330 pp. (in German) | MR

[16] Lempio F., Veliov V. M., “Discrete approximation of differential inclusions”, Bayreuth. Math. Schr., 54 (1998), 149–232 | MR | Zbl

[17] Nikol'skii M. S., “On the approximation of the reachable set of a differential inclusion”, Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet., 1987, no. 4, 31–34

[18] Nikol'skii M. S., “An inner estimate of the attainability set of Brockett's nonlinear integrator”, Differential Equations, 36:11 (2000), 1647–1651 | DOI | MR | Zbl

[19] Polyak B. T., Khlebnikov M. V., Shcherbakov, Upravleniye lineynymi sistemami pri vneshnih vozmushcheniyah: Tehnika lineynyh matrichnyh neravenstv [Control of linear systems under external disturbances: Technique of linear matrix inequalities], LENAND, Moscow, 2014, 560 pp. (in Russian) | DOI

[20] Ushakov V. N., Matviychuk A. R., Ushakov A. V., “Approximations of attainability sets and of integral funnels of differential inclusions”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2011, no. 4, 23–39 (in Russian) http://vst.ics.org.ru/journal/article/1816/ | DOI | Zbl

[21] Vdovin S. A., Taras'yev A. M., Ushakov V. N., “Construction of the attainability set of a Brockett integrator”, J. Appl. Math. Mech., 68:5 (2004), 631–646 | DOI | MR | Zbl