Geodesic
On approximate solvability set construction in a guidance problem for a time-invariant control system on a finite time interval
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 27 (2017) no. 2, pp. 210-221 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A time-invariant control system on a finite time interval in the finite-dimensional Euclidean space is considered. We discuss a problem of guidance with a compact target set for a control system on a given time interval. One way to solve the considered guidance problem is based on finding a solvability set in the phase space, namely, a set of all system positions from which, as from the initial ones, the guidance problem is solvable. The construction of the solvability set is an independent time-consuming problem which rarely has an exact solution. In this paper we discuss the approximate construction of a solvability set in the guidance problem for a time-invariant nonlinear control system. It is well-known that this problem is closely connected with the problem of constructing integral funnels and trajectory tubes of control systems. Integral funnels of control systems can be approximately constructed step-by-step as sets of corresponding attainability sets, therefore, attainability sets are considered to be the basic elements of the solving construction in this paper. Here, we propose a scheme of the solvability set approximate construction in a guidance problem for a time-invariant control system on a finite time interval. The basis of this scheme is reduction to the solvability sets approximate calculation of a finite number of simpler problems, namely, problems of guidance with the target set at fixed time moments from the given time interval. Wherein, the moments of time have to be chosen quite tightly in the mentioned time interval. As an example, we provide mathematical modeling of the guidance problem of the control system named “Translational Oscillator with Rotating Actuator” as well as the graphical support of the problem solution.
Keywords: control system, guidance problem, attainability set, solvability set, solvability set approximation. 
@article{VUU_2017_27_2_a4,
     author = {G. V. Parshikov},
     title = {On approximate solvability set construction in a guidance problem for a time-invariant control system on a finite time interval},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {210--221},
     year = {2017},
     volume = {27},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2017_27_2_a4/}
}
TY  - JOUR
AU  - G. V. Parshikov
TI  - On approximate solvability set construction in a guidance problem for a time-invariant control system on a finite time interval
JO  - Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
PY  - 2017
SP  - 210
EP  - 221
VL  - 27
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/VUU_2017_27_2_a4/
LA  - ru
ID  - VUU_2017_27_2_a4
ER  - 
%0 Journal Article
%A G. V. Parshikov
%T On approximate solvability set construction in a guidance problem for a time-invariant control system on a finite time interval
%J Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
%D 2017
%P 210-221
%V 27
%N 2
%U http://geodesic.mathdoc.fr/item/VUU_2017_27_2_a4/
%G ru
%F VUU_2017_27_2_a4
G. V. Parshikov. On approximate solvability set construction in a guidance problem for a time-invariant control system on a finite time interval. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 27 (2017) no. 2, pp. 210-221. http://geodesic.mathdoc.fr/item/VUU_2017_27_2_a4/

[1] Krasovskii N. N., Subbotin A. I., Positional differential games, Nauka, M., 1974, 456 pp.

[2] Kurzhanski A. B., Control and observation under uncertainty, Nauka, M., 1977, 392 pp.

[3] Krasovskii N. N., Control of dynamic system, Nauka, M., 1985, 520 pp.

[4] Kurzhanski A. B., Selected papers, Moscow State University, M., 2009, 756 pp.

[5] Ushakov V. N., Matviichuk A. R., Parshikov G. V., “A method for constructing a resolving control in an approach problem based on attraction to the feasibility set”, Proceedings of the Steklov Institute of Mathematics, 284, suppl. 1 (2014), 135–144 | DOI | MR

[6] Chernousko F. L., Dynamical systems phase state estimation: ellipsoid method, Nauka, M., 1988, 319 pp.

[7] Kurzhanski A., Valyi I., Ellipsoidal calculus for estimation and control, Birkhäuser, Boston, 1997, 321 pp. | MR | Zbl

[8] Lempio F., Veliov V., “Discrete approximation of differential inclusions”, Bayr. Math. Schriften., 54 (1998), 149–232 | MR | Zbl

[9] Guseinov Kh. G., Moiseev A. N., Ushakov V. N., “The approximation of reachable domains of control systems”, Journal of Applied Mathematics and Mechanics, 62:2 (1998), 169–175 | DOI | MR | Zbl

[10] Gusev M. I., “Estimates of reachable sets of multidimensional control systems with nonlinear interconnections”, Proceedings of the Steklov Institute of Mathematics, 269, suppl. 1 (2010), 134–146 | DOI | MR | Zbl

[11] Filippova T. F., “Construction of set-valued estimates of reachable sets for some nonlinear dynamical systems with impulsive control”, Proceedings of the Steklov Institute of Mathematics, 269, suppl. 1 (2010), 95–102 | DOI | MR | Zbl

[12] Matviychuk A. R., Ukhobotov V. I., Ushakov A. V., Ushakov V. N., “A guidance problem for a nonlinear control system on a finite time interval”, Prikl. Mat. Mekh., 81:2 (2017), 165–187 (in Russian)

[13] Khalil H. K., Nonlinear systems, Regular and Chaotic Dynamics, Izhevsk, 2009, 812 pp.