Geodesic
On the continuity of the optimal time as a function of the initial state for linear controlled objects with integral constraints on controls
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 2, pp. 130-137 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

A traditional object of study in the mathematical theory of optimal control is a controlled object with geometric constraints on the control vector $u$. At the same time, it turns out that sometimes it is more convenient to impose integral constraints on the control vector $u$. For example, in the theory of automatic design of optimal controllers, it is sometimes assumed that the control vector $u$ is not subject to any geometric constraints, but there is a requirement that the control $u(t)$ and its squared length $|u(t )|^2$ are Lebesgue summable on the corresponding interval. This circumstance, as well as the fact that the quality criterion has the form of a quadratic functional, makes it possible to construct an optimal control under rather broad assumptions. Quadratic integral constraints on controls can be interpreted as some energy constraints. Controlled objects under integral constraints on the controls are given quite a lot of attention in the mathematical literature on control theory. We mention the works of N.N. Krasovskii, E.B. Lee, L. Markus, A.B. Kurzhanski, M.I. Gusev, I.V. Zykov, and their students. The paper studies a linear time-optimal problem, in which the terminal set is the origin, under an integral constraint on the control. Sufficient conditions are obtained under which the optimal time as a function of the initial state $x_0$ is continuous.
Keywords: control, controlled object, integral constraint, time optimality. 
@article{TIMM_2024_30_2_a8,
     author = {M. S. Nikol'skii},
     title = {On the continuity of the optimal time as a function of the initial state for linear controlled objects with integral constraints on controls},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {130--137},
     year = {2024},
     volume = {30},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2024_30_2_a8/}
}
TY  - JOUR
AU  - M. S. Nikol'skii
TI  - On the continuity of the optimal time as a function of the initial state for linear controlled objects with integral constraints on controls
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2024
SP  - 130
EP  - 137
VL  - 30
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TIMM_2024_30_2_a8/
LA  - ru
ID  - TIMM_2024_30_2_a8
ER  - 
%0 Journal Article
%A M. S. Nikol'skii
%T On the continuity of the optimal time as a function of the initial state for linear controlled objects with integral constraints on controls
%J Trudy Instituta matematiki i mehaniki
%D 2024
%P 130-137
%V 30
%N 2
%U http://geodesic.mathdoc.fr/item/TIMM_2024_30_2_a8/
%G ru
%F TIMM_2024_30_2_a8
M. S. Nikol'skii. On the continuity of the optimal time as a function of the initial state for linear controlled objects with integral constraints on controls. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 2, pp. 130-137. http://geodesic.mathdoc.fr/item/TIMM_2024_30_2_a8/

[1] Krasovskii N.N., Teoriya upravleniya dvizheniem, Nauka, M., 1968, 476 pp.

[2] Li E.B., Markus L., Osnovy teorii optimalnogo upravleniya, Nauka, M., 1972, 576 pp.

[3] Kurzhanskii A.B., Upravlenie i nablyudenie v usloviyakh neopredelennosti, Nauka, M., 1977, 392 pp.

[4] Mezentsev A.V., Differentsialnye igry s integralnymi ogranicheniyami na upravlenie, Izd-vo Moskov. un-ta, 1988, 134 pp.

[5] Gusev M.I., Zykov I.V., “Ob ekstremalnykh svoistvakh granichnykh tochek mnozhestv dostizhimosti upravlyaemykh sistem pri integralnykh ogranicheniyakh”, Tr. In-ta matematiki i mekhaniki UrO RAN, 23:1 (2017), 103–115

[6] Zykov I.V., “O vneshnikh otsenkakh mnozhestv dostizhimosti upravlyaemykh sistem s integralnymi ogranicheniyami”, Izv. IMI UdGU, 53 (2019), 61–72 | DOI | Zbl

[7] Vasilev F.P., Metody optimizatsii, Faktorial press, M., 2002, 824 pp.

[8] Blagodatskikh V.I., Vvedenie v optimalnoe upravlenie. Lineinaya teoriya, Vysshaya shkola, M., 2001, 240 pp.

[9] Petrov N.N., Vvedenie v vypuklyi analiz, Izhevsk, 2009, 166 pp.

[10] Nikolskii M.S., “O nepreryvnosti vremeni optimalnogo bystrodeistviya kak funktsii nachalnogo sostoyaniya dlya lineinykh upravlyaemykh ob'ektov”, Vestn. Mosk. un-ta. Ser. 15. Vychisl. matematika i kibernetika, 2023, no. 2, 31–38 | DOI | Zbl