Geodesic
Control of effects in the right-hand sides of a large ODE system of a block structure and optimization of sources in unseparated boundary conditions
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 24 (2021) no. 3, pp. 229-251 
K. R. Aida-zade; Y. R. Ashrafova. Control of effects in the right-hand sides of a large ODE system of a block structure and optimization of sources in unseparated boundary conditions. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 24 (2021) no. 3, pp. 229-251. http://geodesic.mathdoc.fr/item/SJVM_2021_24_3_a0/
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     author = {K. R. Aida-zade and Y. R. Ashrafova},
     title = {Control of effects in the right-hand sides of a large {ODE} system of a block structure and optimization of sources in unseparated boundary conditions},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {229--251},
     year = {2021},
     volume = {24},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2021_24_3_a0/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we investigate the problem of control of a complex object, described by a large ODE system of a block structure with unseparated boundary conditions between blocks. The controls in the right-hand sides of the equations and the values of the source parameters in the boundary conditions are to be optimized. We propose to apply the first order optimization methods for the numerical solution to the optimal control problem, using functional gradient formulas participating in the obtained necessary optimality conditions. Special schemes of the sweep method for the solution to the direct and conjugate boundary value problems, having a block structure, and unseparated non-local boundary conditions are offered. This method takes into account special features of ODE systems and boundary conditions, allows the transfer of boundary conditions for each block and each boundary condition in the block independent of each other. The obtained results of numerical experiments in solving the test problem and their analysis are given.

[1] Aschepkov L.T., “Optimalnoe upravlenie sistemoi s promezhutochnymi usloviyami”, Prikladnaya matematika i mekhanika, 45:2 (1981), 215–222 | MR | Zbl

[2] Vasilieva O.O., Mizukami K., “Optimality criterion for singular controllers: linear boundary conditions”, J. Math. Anal. and Appl., 213:2 (1997), 620–641 | DOI | MR | Zbl

[3] Vasileva O.O., Mizukami K., “Dinamicheskie protsessy, opisyvaemye kraevoi zadachei: neobkhodimye usloviya optimalnosti i metody resheniya”, Izv. RAN. Teoriya i sistemy upravleniya, 2000, no. 1, 95–100

[4] Abdullaev V.M., Aida-zade K.R., “O chislennom reshenii zadach optimalnogo upravleniya s nerazdelennymi mnogotochechnymi i integralnymi usloviyami”, Zhurn. vychisl. matem. i mat. fiziki, 52:12 (2012), 2163–2177 | Zbl

[5] Abdullaev V.M., Aida-zade K.R., “Podkhod k chislennomu resheniyu zadach optimalnogo upravleniya nagruzhennymi differentsialnymi uravneniyami s nelokalnymi usloviyami”, Zhurn. vychisl. matem. i mat. fiziki, 59:5 (2019), 739–751 | Zbl

[6] Sharifov Y.A., Mammadova N.B., “Optimal control problem described by impulsive differential equations with nonlocal boundary conditions”, Differ. Equations, 50:3 (2014), 403–411 | DOI | MR | Zbl

[7] Aida-zade K.R., Abdullaev V.M., “O chislennom reshenii nagruzhennykh sistem obyknovennykh differentsialnykh uravnenii s nerazdelennymi mnogotochechnymi i integralnymi usloviyami”, Sib. zhurn. vychisl. matematiki, 17:1 (2014), 1–16 | Zbl

[8] Aida-zade K.R., “K chislennomu resheniyu lineinykh differentsialnykh uravnenii s nelokalnymi nelineinymi usloviyami”, Zhurn. vychisl. matem. i mat. fiziki, 60:5 (2020), 828–836 | Zbl

[9] Mutallimov M.M., Amirova L.I., Aliev F.A., Faradjova Sh.A., Maharramov I.A., “Remarks to the paper: sweep algorithm for solving optimal control problem with multi-point boundary conditions”, TWMS J. Pure Appl. Math., 9:2 (2018), 243–246 | MR | Zbl

[10] Assanova A.T., “Solvability of a nonlocal problem for a hyperbolic equation with integral conditions”, Electronic J. of Differ. Equations, 2017:170 (2017), 1–12 | MR

[11] Aida-zade K.R., Ali-zade R.I., Novruzbekov I.G., Kalaushin M.A., “Dekompozitsionnyi metod analiza i sinteza ploskikh mekhanizmov”, Mekhanika mashin, 1980, no. 57, 26–30

[12] Geiser J., Decomposition Methods for Differential Equations: Theory and Applications, CRC Press, Taylor and Francis Group, 2009 | MR | Zbl

[13] Samarskii A.A., Nikolaev E.S., Metody resheniya setochnykh uravnenii, Nauka, M., 1978 | MR

[14] Voevodin A.F. Metod progonki dlya raznostnykh uravnenii, opredelennykh na komplekse, Zhurn. vychisl. matem. i mat. fiziki, 13:2 (1973), 494–497 | Zbl

[15] Voevodin A.F., Shugrin S.M., Metody resheniya odnomernykh evolyutsionnykh sistem, Nauka, Sibirskoe izd-vo, Novosibirsk, 1993 | MR

[16] Abramov A.A., “O perenose granichnykh uslovii dlya sistem lineinykh obyknovennykh differentsialnykh uravnenii (variant metoda progonki)”, Zhurn. vychisl. matem. i mat. fiziki, 1:3 (1961), 542–545 | Zbl

[17] Abramov A.A., Burago N.G., Dyshko A.L. i dr., “Paket prikladnykh programm dlya resheniya lineinykh dvukhtochechnykh kraevykh zadach”, Soobscheniya po programmnomu obespecheniyu EVM, VTs AN SSSR, M., 1982

[18] Aida-zade K.R., Ashrafova E.R., “Raschet sostoyaniya sistemy diskretnykh lineinykh protsessov, svyazannykh nerazdelennymi kraevymi usloviyami”, Sib. zhurn. industr. matematiki, 19:4 (2016), 3–14 | Zbl

[19] Aida-zade K.R., Ashrafova Y.R., “Solving systems of differential equations of block structure with nonseparated boundary conditions”, J. of Applied and Industrial Mathematics, 9:1 (2015), 1–10 | DOI | MR

[20] Ashrafova Y.R., “On one method of block transfer of conditions for a system of three-step discrete processes connected only by boundary conditions”, Informatics and Control Problems, 39:2 (2019), 48–56

[21] Abdullaev V.M., “Reshenie differentsialnykh uravnenii s nerazdelennymi mnogotochechnymi i integralnymi usloviyami”, Sib. zhurn. industr. matematiki, 15:3 (2012), 3–15 | MR | Zbl

[22] Aida-zade K.R., Ashrafova Y.R., “Numerical solution to an inverse problem on a determination of places and capacities of sources in the hyperbolic systems”, J. of Industrial and Management Optimization, 16:6 (2020), 3011–3033 | DOI | MR

[23] Aida-zade K.R., Ashrafova E.R., “Chislennoe reshenie zadachi opredeleniya mest i ob'emov utechek pri neustanovivshemsya dvizhenii zhidkosti v truboprovodnoi seti slozhnoi struktury”, Zhurn. vychisl. matem. i mat. fiziki, 57:12 (2017), 36–52

[24] Kabanikhin S.I., Shishlenin M.A., “Vosstanovlenie koeffitsienta diffuzii, zavisyaschego ot vremeni, po nelokalnym dannym”, Sib. zhurn. vychisl. matematiki, 21:1 (2018), 55–63 | MR | Zbl

[25] Penenko A.V., “Soglasovannye chislennye skhemy dlya resheniya nelineinykh obratnykh zadach identifikatsii istochnikov gradientnymi algoritmami i metodami Nyutona-Kantorovicha”, Sib. zhurn. vychisl. matematiki, 21:1 (2018), 99–116 | Zbl

[26] Karchevskii A.L., Dedok V.A., “Vosstanovlenie koeffitsienta dielektricheskoi pronitsaemosti po modulyu rasseyannogo elektricheskogo polya”, Sib. zhurn. industr. matematiki, 21:3 (2018), 50–59 | Zbl

[27] Karchevskii A.L., “Korrektnaya skhema deistvii pri chislennom reshenii obratnoi zadachi optimizatsionnym metodom”, Sib. zhurn. vychisl. matematiki, 11:2 (2008), 39–149

[28] Shishlenin M.A., “Matrichnyi metod v zadachakh opredeleniya istochnikov kolebanii”, Sib. elektron. matem. izvestiya, 11 (2014), 161–171 | MR

[29] Rothe E., “Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben”, Math. Annalen, 102:1 (1930), 650–670 | DOI | MR | Zbl

[30] Vasilev F.P., Metody optimizatsii, Faktorial Press, M., 2002

[31] Polyak B.T., Vvedenie v optimizatsiyu, Nauka, M., 1983 | MR

[32] Pontryagin L.S., Obyknovennye differentsialnye uravneniya, Nauka, M., 1975

[33] Egorov A.I., Obyknovennye differentsialnye uravneniya s prilozheniyami, 2-e izd., ispr., FIZMATLIT, M., 2005