Geodesic
Optimal control of the Navier — Stokes system with a space variable in a network-like domain
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 19 (2023) no. 4, pp. 549-562
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The research of the problem of optimal control of the Navier — Stokes evolutionary differential system, considered in Sobolev spaces, the elements of which are functions with carriers in an $n$-dimensional network-like domain, is presented. Such domain consists of a finite number of subdomains, mutually adjacent to certain parts of the surfaces of their boundaries according to the graph type. For functions that are elements of these spaces, the conditions for the existence of traces on the surfaces of the joining are presented and the conditions of adjacency subdomains to which these functions satisfy are described. In applied questions of the analysis of the processes of transport of continuous media, the conditions of adjacency describe the regularities of the flow of fluid through the boundaries of the adjacent domains. The paper presents the results of following main research questions: 1) weak solvability of the initial boundary value problem for the Navier — Stokes system and obtaining the conditions for the existence of a weak solution to this problem; 2) the formation and solution of optimal control problems of various types of Navier — Stokes system. The fundamental approach to the analysis of the weak solvability of the initial boundary value problem is its reduction to the differential-difference problem (semi-digitization of the original system by a time variable) and subsequent use of a priori estimates for weak solutions of the obtained boundary value problems. The obtained a priori estimates are used to prove the theorem of the existence of a weak solution of the original differential system and indicate the way of the actual construction of this solution. A universal approach to solving the problems of optimal distributed and starting control of the Navier — Stokes evolutionary system is presented. The latter essentially expands the possibilities of analyzing non-stationary network-like processes of applied hydrodynamics (for example, processes of transporting various types of liquids through network or main line pipelines) and optimal control of these processes.
Keywords: differential-difference system, evolutionary Navier — Stokes system, network-like domain, solvability, optimal control. 
@article{VSPUI_2023_19_4_a10,
     author = {A. P. Zhabko and V. V. Provotorov and S. M. Sergeev},
     title = {Optimal control of the {Navier} {\textemdash} {Stokes} system with a space variable in a network-like domain},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {549--562},
     year = {2023},
     volume = {19},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VSPUI_2023_19_4_a10/}
}
TY  - JOUR
AU  - A. P. Zhabko
AU  - V. V. Provotorov
AU  - S. M. Sergeev
TI  - Optimal control of the Navier — Stokes system with a space variable in a network-like domain
JO  - Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
PY  - 2023
SP  - 549
EP  - 562
VL  - 19
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/VSPUI_2023_19_4_a10/
LA  - en
ID  - VSPUI_2023_19_4_a10
ER  - 
%0 Journal Article
%A A. P. Zhabko
%A V. V. Provotorov
%A S. M. Sergeev
%T Optimal control of the Navier — Stokes system with a space variable in a network-like domain
%J Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
%D 2023
%P 549-562
%V 19
%N 4
%U http://geodesic.mathdoc.fr/item/VSPUI_2023_19_4_a10/
%G en
%F VSPUI_2023_19_4_a10
A. P. Zhabko; V. V. Provotorov; S. M. Sergeev. Optimal control of the Navier — Stokes system with a space variable in a network-like domain. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 19 (2023) no. 4, pp. 549-562. http://geodesic.mathdoc.fr/item/VSPUI_2023_19_4_a10/

[1] Zhabko A. P., Provotorov V. V., Shindyapin A. I., “Optimal control of a differential-difference parabolic system with distributed parameters on the graph”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 17:4 (2021), 433–448 | DOI | MR

[2] Provotorov V. V., Sergeev S. M., Hoang V. N., “Point control of a differential-difference system with distributed parameters on a graph”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 17:3 (2021), 277–286 | DOI | MR

[3] Zhabko A. P., Provotorov V. V., Balaban O. R., “Stabilization of weak solutions of parabolic systems with distributed parameters on the graph”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 15:2 (2019), 187–198 | DOI | MR

[4] Zhabko A. P., Nurtazina K. B., Provotorov V. V., “Uniqueness solution to the inverse spectral problem with distributed parameters on the graph-star”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 16:2 (2020), 129–143 | DOI | MR

[5] Artemov M. A., Baranovskii E. S., Zhabko A. P., Provotorov V. V., “On a 3D model of non-isothermal flows in a pipeline network”, Journal of Physics. Conference Series, 1203 (2019), 012094 | DOI

[6] Baranovskii E. S., Provotorov V. V., Artemov M. A., Zhabko A. P., “Non-isothermal creeping flows in a pipeline network: existence results”, Symmetry, 13 (2021), 1300 | DOI

[7] Baranovskii E. S., “Steady flows of an Oldroyd fluid with threshold slip”, Communications on Pure and Applied Analysis, 18:2 (2019), 735–750 | DOI | MR | Zbl

[8] Artemov M. A., Baranovskii E. S., “Solvability of the Boussinesq approximation for water polymer solutions”, Mathematics, 7:7 (2019), 611 | DOI

[9] Lions J.-L., Some methods of solving non-linear boundary value problems, Mir Publ, M., 1972, 587 pp. (In Russian) | MR

[10] Volkova A. S., Provotorov V. V., “Generalized solutions and generalized eigenfunctions of boundary-value problems on a geometric graph”, Proceeding of Higher Educatianal Institutions, 58:3 (2014), 1–13 | MR | Zbl

[11] Lions J.-L., Madgenes E., Nonhomogeneous boundary problems and their applications, Mir Publ, M., 1971, 367 pp. (In Russian) | MR

[12] Ladyzhenskaya O. A., Boundary value problems of mathematical physics, Nauka Publ, M., 1973, 407 pp. (In Russian) | MR

[13] Kamachkin A. M., Potapov D. K., Yevstafyeva V. V., “Dynamics and synchronization in feedback cyclic structures with hysteresis oscillators”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 16:2 (2020), 186–199 (In Russian) | DOI | MR

[14] Zhabko A. P., Nurtazina K. B., Provotorov V. V., “About one approach to solving the inverse problem for parabolic equation”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 15:3 (2019), 322–335 | DOI | MR

[15] Fominyh A. V., Karelin V. V., Polyakova L. N., Myshkov S. K., Tregubov V. P., “The codifferential descent method in the problem of finding the global minimum of a piecewise affine objective functional in linear control systems”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 17:1 (2021), 47–58 (In Russian) | DOI | MR

[16] Aleksandrov A. Yu., Tikhonov A. A., “Stability analysis of mechanical systems with distributed delay via decomposition”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 17:1 (2021), 13–26 (In Russian) | DOI | MR

[17] Ekimov A. V., Zhabko A. P., Yakovlev P. V., “The stability of differential-difference equations with proportional time delay”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 16:3 (2020), 316–325 | DOI | MR