Geodesic
External estimates of reachable sets for control systems with integral constraints
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 1, Tome 190 (2021), pp. 107-114.

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

In this paper, we consider the problem of constructing external estimates for reachable sets as level sets of some differentiable Lyapunov–Bellman function for a control system with an integral control constraint from $\mathbb{L}_p$, $p>1$. For an appropriate choice of this function, ellipsoidal and rectangular estimates can be obtained. Constructions proposed are based on integral estimates, maximal solutions, and the comparison principle for systems of differential inequalities. Using time as an argument of the Lyapunov–Bellman function, we obtain more accurate estimates. In the linear nonstationary case, such estimates may coincide with the reachable set. Also we present illustrative examples for nonlinear systems.
Keywords: reachable set, control system, integral constraint, integral inequality, comparison principle, external estimate. 
@article{INTO_2021_190_a9,
     author = {I. V. Zykov},
     title = {External estimates of reachable sets  for control systems with integral constraints},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {107--114},
     publisher = {mathdoc},
     volume = {190},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_190_a9/}
}
TY  - JOUR
AU  - I. V. Zykov
TI  - External estimates of reachable sets  for control systems with integral constraints
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2021
SP  - 107
EP  - 114
VL  - 190
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2021_190_a9/
LA  - ru
ID  - INTO_2021_190_a9
ER  - 
%0 Journal Article
%A I. V. Zykov
%T External estimates of reachable sets  for control systems with integral constraints
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2021
%P 107-114
%V 190
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2021_190_a9/
%G ru
%F INTO_2021_190_a9
I. V. Zykov. External estimates of reachable sets  for control systems with integral constraints. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 1, Tome 190 (2021), pp. 107-114. http://geodesic.mathdoc.fr/item/INTO_2021_190_a9/

[1] Gurman V. I., Printsip rasshireniya v zadachakh upravleniya, Fizmatlit, M., 1997

[2] Gusev M. I., “O vneshnikh otsenkakh mnozhestv dostizhimosti nelineinykh upravlyaemykh sistem”, Tr. In-ta mat. mekh. UrO RAN., 17:1 (2011), 60–69 | MR | Zbl

[3] Dykhta V. A., “Neravenstvo Lyapunova––Krotova i dostatochnye usloviya v optimalnom upravlenii”, Itogi nauki i tekhn. Ser. Sovrem. mat. priljozh. Temat. obzory., 110 (2006), 76–108

[4] Zykov I. V., “O zadache dostizhimosti dlya nelineinoi upravlyaemoi sistemy s integralnymi ogranicheniyami”, CEUR-WS Proc., 1894 (2017), 88–97

[5] Kurzhanskii A. B., “Printsip sravneniya dlya uravnenii tipa Gamiltona—Yakobi v teorii upravleniya”, Tr. In-ta mat. mekh. UrO RAN., 12:1 (2006), 173–183 | Zbl

[6] Martynyuk A. A., Lakshmikantam V., Lila S., Ustoichivost dvizheniya: metod integralnykh neravenstv, Naukova dumka, Kiev, 1989 | MR

[7] Nikolskii M. S., “Ob otsenivanii mnozhestva dostizhimosti dlya nekotorykh upravlyaemykh ob'ektov”, Mat. Mezhdunar. konf., posv. 110-letiyu so dnya rozhdeniya L. S. Pontryagina, M., 2018, 194–196

[8] Gusev M. I., Zykov I. V., “On extremal properties of boundary points of reachable sets for a system with integrally constrained control”, IFAC-PapersOnLine., 50 (2017), 4082–4087 | DOI

[9] Lee E. B., Marcus L., Foundations of Optimal Control Theory, Wiley, New York, 1967 | MR | Zbl