Geodesic
Asymptotic Control Theory for a Closed String. II
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 194-214.

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

We develop an asymptotic control theory for one of the simplest distributed (infinite-dimensional) oscillating systems, namely, for a closed string under a bounded load applied to a single distinguished point. We give a precise description of the classes of string states that admit complete damping, and find an asymptotically exact value of the required time. By using approximate reachable sets instead of the exact ones, we design a feedback control, which turns out to be asymptotically optimal. The main results are exact algebraic formulas for the asymptotic shape of the reachable sets, for the asymptotically optimal time of motion, and for the asymptotically optimal control thus constructed.
Keywords: maximum principle, reachable sets, linear system, string control. 
@article{TM_2023_321_a12,
     author = {Lev V. Lokutsievskiy and Alexander I. Ovseevich},
     title = {Asymptotic {Control} {Theory} for a {Closed} {String.} {II}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {194--214},
     publisher = {mathdoc},
     volume = {321},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2023_321_a12/}
}
TY  - JOUR
AU  - Lev V. Lokutsievskiy
AU  - Alexander I. Ovseevich
TI  - Asymptotic Control Theory for a Closed String. II
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2023
SP  - 194
EP  - 214
VL  - 321
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2023_321_a12/
LA  - ru
ID  - TM_2023_321_a12
ER  - 
%0 Journal Article
%A Lev V. Lokutsievskiy
%A Alexander I. Ovseevich
%T Asymptotic Control Theory for a Closed String. II
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2023
%P 194-214
%V 321
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2023_321_a12/
%G ru
%F TM_2023_321_a12
Lev V. Lokutsievskiy; Alexander I. Ovseevich. Asymptotic Control Theory for a Closed String. II. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 194-214. http://geodesic.mathdoc.fr/item/TM_2023_321_a12/

[1] Fedorov A., Ovseevich A., “Asymptotic control theory for a system of linear oscillators”, Moscow Math. J., 16:3 (2016), 561–598 | DOI | MR | Zbl

[2] Fedorov A.K., Ovseevich A.I., “Asymptotic control theory for a closed string”, Russ. J. Math. Phys., 25:2 (2018), 200–219 | DOI | MR | Zbl

[3] E. V. Goncharova and A. I. Ovseevich, “Comparative analysis of the asymptotic dynamics of reachable sets to linear systems”, J. Comput. Syst. Sci. Int., 46:4 (2007), 505–513 | DOI | MR | Zbl

[4] A. Ya. Helemskii, Lectures and Exercises on Functional Analysis, Transl. Math. Monogr., 233, Am. Math. Soc., Providence, RI, 2006 | MR | Zbl

[5] G. G. Magaril-Il'yaev and V. M. Tikhomirov, Convex Analysis: Theory and Applications, Transl. Math. Monogr., 222, Am. Math. Soc., Providence, RI, 2003 | MR | Zbl

[6] Yosida K., Hewitt E., “Finitely additive measures”, Trans. Amer. Math. Soc., 72 (1952), 46–66 | DOI | MR | Zbl