Geodesic
On the properties of the set of trajectories of nonlinear control systems with integral constraints on the control functions
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 3, pp. 274-284 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The control systems described by nonlinear differential equations and integral constraints on the control functions are studied. Admissible control functions are chosen from a closed ball of the space $L_p,$ $p\in (1,\infty]$, with radius $r$ and centered at the origin. It is proved that the set of trajectories of the system is continuous at $p=\infty$ with respect to the Hausdorff pseudometric. It is shown that every trajectory is robust with respect to the fast and full consumption of the remaining control resource which implies that to achieve the desired result, it is advisable to spend the available control resource in small portions. This allows to prove that every trajectory can be approximated by the trajectory, generated by full consumption of the control resource.
Keywords: nonlinear control system; set of trajectories; integral constraint; geometric constraint; Hausdorff continuity; robustness. 
@article{TIMM_2022_28_3_a19,
     author = {N. Huseyin and A. Huseyin and Kh. G. Guseinov},
     title = {On the properties of the set of trajectories of nonlinear control systems with integral constraints on the control functions},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {274--284},
     year = {2022},
     volume = {28},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2022_28_3_a19/}
}
TY  - JOUR
AU  - N. Huseyin
AU  - A. Huseyin
AU  - Kh. G. Guseinov
TI  - On the properties of the set of trajectories of nonlinear control systems with integral constraints on the control functions
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2022
SP  - 274
EP  - 284
VL  - 28
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TIMM_2022_28_3_a19/
LA  - en
ID  - TIMM_2022_28_3_a19
ER  - 
%0 Journal Article
%A N. Huseyin
%A A. Huseyin
%A Kh. G. Guseinov
%T On the properties of the set of trajectories of nonlinear control systems with integral constraints on the control functions
%J Trudy Instituta matematiki i mehaniki
%D 2022
%P 274-284
%V 28
%N 3
%U http://geodesic.mathdoc.fr/item/TIMM_2022_28_3_a19/
%G en
%F TIMM_2022_28_3_a19
N. Huseyin; A. Huseyin; Kh. G. Guseinov. On the properties of the set of trajectories of nonlinear control systems with integral constraints on the control functions. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 3, pp. 274-284. http://geodesic.mathdoc.fr/item/TIMM_2022_28_3_a19/

[1] Buzikov M.E, Galyaev A.A., “Time-optimal interception of a moving target by a Dubins car”, Autom. Remote Control, 82:5 (2021), 745–758 | DOI | MR | Zbl

[2] Chentsov A.G., “An abstract problem on attainability: the “purely asymptotic” version”, Tr. Inst. Mat. Mekh. UrO RAN, 21:2 (2015), 289-305 (in Russian) | MR

[3] Chernous'ko F.L., Ovseevich A.L., “Properties of ellipsoids approximating attainable sets”, Dokl. Math., 67:1 (2003), 123–126 | Zbl

[4] Filippova T.F., “Control and estimation for a class of impulsive dynamical systems”, Ural Math. J., 5:2 (2019), 21–30 | DOI | MR | Zbl

[5] Kurzhanski A.B., Varaiya P., Dynamics and control of trajectory tubes. Theory and computation, Birkhäuser, Cham, 2014, 445 pp. | DOI | MR | Zbl

[6] Patsko V.S., Fedotov A.A., “The structure of the reachable set for a Dubins car with a strictly one-sided turn”, Tr. Inst. Mat. Mekh. UrO RAN, 25:3 (2019), 171–187 (in Russian) | DOI | MR

[7] Filippov A.F., Differential equations with discontinuous righthand sides, Springer, Dordrecht, 1988, 304 pp. | DOI | MR

[8] Panasyuk A.I., Panasyuk V.I., “An equation generated by a differential inclusion”, Math. Notes, 27:3 (1980), 213–218 | DOI | MR | Zbl

[9] Ershov A.A., Ushakov A.V., Ushakov V.N., “An approach problem for a control system with a compact set in the phase space in the presence of phase constraints”, Sb. Math., 210:8 (2019), 1092–1128 | DOI | MR | Zbl

[10] Krasovskii N.N., Subbotin A.I., Game-theoretical control problems, Springer, NY, 1988, 517 pp. ; Krasovskii N.N., Subbotin A.I., Pozitsionnye differentsial'nye igry, Original Russian text, Nauka Publ., Moscow, 1974, 456 pp. | MR | Zbl | MR

[11] Lukoyanov N.Yu., “A differential game with integral performance criterion”, Diff. Equat., 30:11 (1994), 1759–1766 | MR | Zbl

[12] Fominykh A.V., “On subdifferential and hypodifferential descent methods in a problem on constructing a program control with an integral constraint on the control”, Autom. Remote Control, 78:4 (2017), 608–617 | DOI | MR | Zbl

[13] Guseinov Kh.G., Nazlipinar A.S., “On the continuity property of $L_p$ balls and an application”, J. Math. Anal. Appl., 335:2 (2007), 1347–1359 | DOI | MR | Zbl

[14] Gusev M.I., Zykov I.V., “On extremal properties of the boundary points of reachable sets for control systems with integral constraints”, Proc. Steklov Inst. Math., 300:1 (2018), 114–125 | DOI | MR

[15] Gusev M.I., “An algorithm for computing boundary points of reachable sets of control systems under integral constraints”, Ural Math. J., 3:1 (2017), 44–51 | DOI | MR | Zbl

[16] Huseyin A., Huseyin N., “Precompactness of the set of trajectories of the controllable system described by a nonlinear Volterra integral equation”, Math. Model. Anal., 17:5 (2012), 686–695 | DOI | MR | Zbl

[17] Huseyin A., Huseyin N., Guseinov Kh.G., “Approximation of the integral funnel of a nonlinear control system with limited control resources”, Minimax Theory Appl., 5:2 (2020), 327–346 | MR | Zbl

[18] Huseyin N., Huseyin A., Guseinov Kh.G., “On the robustness property of a control system described by an Urysohn type integral equation”, Tr. Inst. Mat. Mekh. UrO RAN, 27:3 (2021), 263–270 | DOI | MR

[19] Huseyin A., Huseyin N., Guseinov K.G., “Continuity of $L_p$ balls and an application to input-output systems”, Mathematical Notes, 111:1–2 (2022), 58–70 | DOI | MR | Zbl

[20] Kostousova E.K., “On polyhedral estimation of reachable sets in the “extended” space for discrete-time systems with uncertain matrices and integral constraints”, Tr. Inst. Mat. Mekh. UrO RAN, 26:1 (2020), 141–155 (in Russian) | DOI | MR

[21] Krasovskii N.N., Teoriya upravleniya dvizheniem. Lineinye sistemy [Theory of motion control. Linear systems], Nauka Publ., Moscow, 1968, 475 pp.

[22] Subbotina N.N., Subbotin A.I., “Alternative for the encounter-evasion differential game with constraints on the momenta of the players controls”, J. Appl. Math. Mech., 39:3 (1975), 376–385 | DOI | MR | Zbl

[23] Wheeden R.L., Zygmund A., Measure and integral. An introduction to real analysis, M. Dekker Inc., NY, 1977, 274 pp. | MR | Zbl