Geodesic
Solution of a local boundary problem for a non-linear non-stationary system in the class of discrete controls
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 18 (2022) no. 1, pp. 18-36 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This article proposes an algorithm of construction for the discrete controlling function which is restricted by a norm and provides transition for the wide class of the systems of non-stationary nonlinear ordinary differential equations from the initial state to the setting final state. A constructive sufficient condition that provides this transition is obtained. Efficiency of the method is demonstrated by the solution of the robot-manipulator control problem and its numerical modeling.
Keywords: discrete control, non-linear non-stationary system, stabilization, boundary conditions. 
@article{VSPUI_2022_18_1_a1,
     author = {A. N. Kvitko and N. N. Litvinov},
     title = {Solution of a local boundary problem for a non-linear non-stationary system in the class of discrete controls},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {18--36},
     year = {2022},
     volume = {18},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSPUI_2022_18_1_a1/}
}
TY  - JOUR
AU  - A. N. Kvitko
AU  - N. N. Litvinov
TI  - Solution of a local boundary problem for a non-linear non-stationary system in the class of discrete controls
JO  - Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
PY  - 2022
SP  - 18
EP  - 36
VL  - 18
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/VSPUI_2022_18_1_a1/
LA  - ru
ID  - VSPUI_2022_18_1_a1
ER  - 
%0 Journal Article
%A A. N. Kvitko
%A N. N. Litvinov
%T Solution of a local boundary problem for a non-linear non-stationary system in the class of discrete controls
%J Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
%D 2022
%P 18-36
%V 18
%N 1
%U http://geodesic.mathdoc.fr/item/VSPUI_2022_18_1_a1/
%G ru
%F VSPUI_2022_18_1_a1
A. N. Kvitko; N. N. Litvinov. Solution of a local boundary problem for a non-linear non-stationary system in the class of discrete controls. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 18 (2022) no. 1, pp. 18-36. http://geodesic.mathdoc.fr/item/VSPUI_2022_18_1_a1/

[1] Kalman R. E., Falb P. L., Arbib M. A., Topics in mathematical system theory, Mir Publ, M., 1971, 399 pp. (In Russian) | MR

[2] Petrov N. N., “Local controllability of autonomous systems”, Differential Equations, 4:7 (1968), 1218–1232 (In Russian) | Zbl

[3] Petrov N. N., “A solution of a certain problem in control theory”, Differential Equations, 5:5 (1969), 962–963 (In Russian) | Zbl

[4] Vereshchagin I. F., Methods for investigating flight regimes of changing mass apparatus, v. 2, Perm' State University Press, Perm', 1972, 294 pp. (In Russian)

[5] Zubov V. I., Lectures in control theory, Nauka Publ, M., 1975, 496 pp. (In Russian)

[6] Furi M., Nistri P., Pera M. P., Zezza P., “Linear controllability by piecewise constant controls with assigned switching times”, J. Optim. Theory Appl., 45:2 (1985), 219–229 | DOI | MR | Zbl

[7] Seilova R. D., Amanov T. D., “Construction of piece wise constant controls for linear impulsive systems”, Proceedings of International Symposium “Reliability and quality”, 2005, 4–5

[8] Kvitko A. N., Maksina A. M., Chistyakov S. V., “On a method for solving a local boundary problem for a nonlinear stationary system with perturbations in the class of piecewise constant controls”, Intern. J. Robust Nonlinear Control, 29 (2019), 4515–4536 | DOI | MR | Zbl

[9] Baier R., Gerdts M., “A computational method for non-convex reachable sets using optimal control”, European Control Conference (ECC) (Budapest, Hungary, 2009), 97–102

[10] Plotnikov A. V., Arsiry A. V., Komleva T. A., “Piecewise constant controller linear fuzzy systems”, Intern. J. Ind. Math., 4:2 (2012), 77–85

[11] Kvitko A. N., Yakusheva D. B., “Synthesis of discrete stabilization for a non-linear stationary control system under incomplete information”, Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 45:2 (2012), 65–72 (In Russian) | MR | Zbl

[12] Peregudova O. A., Filatkina E. V., “On stabilization of cascade-type non-linear systems with piecewise constant control”, Review Appl. Ind. Math., 21:1 (2014), 80–82 (In Russian) | MR

[13] Popkov A. S., “Construction of reachability and controllability sets in a special linear control problem”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Sciences. Control Processes, 17:3 (2021), 294–308 (In Russian) | DOI | MR

[14] Gryn L., “Discrete feedback stabilization of semilinear control systems”, Control, Optim. Calc. Var., v. 1, 1996, 207–224 | MR

[15] Gabdrakhimov A. F., “On the stabilization of linear stationary control systems with incomplete feedback”, Vestnik of Udmurtsky University. Mathematics. Mechanics. Computer Sciences, 2008, no. 2, 30–31 (In Russian)

[16] Lizina E. A., Shchennicov V. N., Shchennicova E. V., “Stabilization of continuous-discrete system with periodic matrix of coefficients”, Proceedings of Higher Educational Investigations. Povolzhsk Region. Physics and Mathematics Sciences. Physics, 2013, no. 1(25), 181–195 (In Russian)

[17] Lapin S. V., “Piecewise-constant stabilization of systems that are linear with respect to control”, Autom. Remote Control, 1992, no. 6, 37–45 (In Russian) | Zbl

[18] Ailon A., Segev R., “Driving a linear constant system by a piecewise constant control”, Intern. J. Control, 47 (1988), 815–825 | DOI | MR | Zbl

[19] Shushlyapin E. A., “On the equivalency of piecewise-constant control with a known number of switchings and arbitrary amplitude bounded control in a terminal problem for a linear non-stationary system”, J. Sov. Math., 65:2 (1993), 1550–1554 | DOI | MR

[20] Bulgakov A. I., Zhukovskii S. E., “The “bang-bang” principle for second order linear differential equations”, Vestnik of Tambov State University, 6:2 (2001), 150–154 (In Russian)

[21] Alzabut J. O., “Piecewise constant control of boundary value problem for linear impulsive differential systems”, Math. Methods Eng., 2007, 123–129 | DOI | Zbl

[22] Maksimov V. P., Chadov A. L., “On a class of controls for a functional-differential continuous-discrete system”, Proceedings of Higher Educational Investigations. Mathematics, 2012, no. 9, 72–76 (In Russian) | Zbl

[23] Oaks O. J., Cook G., “Piecewise linear control of nonlinear systems”, IEEE Transactions on Industrial Electronics and Control Instrumentation, 23:1 (1976), 56–63 | DOI

[24] Sachkov Y. L., Ardentov A. A., Mashtakov A. P., “Constructive solution to control problem via nilpotent approximation method”, Proceedings of Program Systems Institute Scientific Conference (Pereslavl'-Zalesskij, Russia, 2009), v. 2, 2009, 5–23 (In Russian)

[25] Yurkevich V. D., Design of two-time-scale non-linear time-varying control systems, Nauka Publ, St Petersburg, 2000, 287 pp. (In Russian)

[26] Kuznetsov A. V., “Local optimality conditions for non-linear controlled systems in the class of piecewise-constant control laws”, Vestnik of Rjazan' State Radiotechnical University, 38:4 (2011), 125–128 (In Russian)

[27] Smirnov E. Y., Stabilization of programmed motion, Saint Petersburg University Press, St Petersburg, 1997, 301 pp. (In Russian)

[28] Barbashin E. A., Introduction to stability theory, Nauka Publ, M., 1967, 224 pp. (In Russian)

[29] Afanas'ev V. N., Kolmanovskii V. B., Nosov V. R., Mathematical theory of control systems design, Vysshaya shkola Publ, M., 2003, 614 pp. (In Russian)

[30] Kvitko A. N., “Solution of the local boundary value problem for a nonlinear non-stationary system in the class of synthesising controls with account of perturbations”, Intern. J. Control, 93:8 (2020), 1931–1941 | DOI | MR | Zbl