On the solvability of control synthesis problems for nonlinear oscillatory optimization processes described by integro-differential equations

A. K. Kerimbekov ; E. F. Abdyldaeva ; A. A. Anarbekova

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 213 (2022), pp. 63-71

Résumé

The solvability of synthesis problems for distributed and boundary controls in minimizing problems for piecewise linear functionals for oscillatory processes described by partial integro-differential equations with Fredholm integral operators are examined. For the Bellman functional, a specific integro-differential equation is obtained. An algorithm for constructing a solution of the control synthesis problem of distributed and boundary controls is described. A procedure for determining controls as functions (functionals) of the state of the controlled process is constructed.

Détail

Keywords: integro-differential equation, Fredholm operator, generalized solution, Bellman functional, optimal control synthesis.
Mots-clés : Fréchet differential

Bibliographie

[1] Arguchintsev A. V., Optimalnoe upravlenie giperbolicheskimi sistemami, Fizmatlit, M., 2007

[2] Butkovskii A. G., Teoriya optimalnogo upravleniya sistemami s raspredelennymi parametrami, Nauka, M., 1965

[3] Egorov A. I., Optimalnoe upravlenie teplovymi i diffuzionnymi protsessami, Nauka, M., 1978

[4] Egorov A. I., Znamenskaya L. N., Vvedenie v teoriyu upravleniya sistemami s raspredelennymi parametrami, Lan, SPb., 2017

[5] Kerimbekov A., Nelineinoe optimalnoe upravlenie lineinymi sistemami s raspredelennymi parametrami, Ilim, Bishkek, 2003

[6] Kerimbekov A., “Sintez raspredelennogo optimalnogo upravleniya v zadache slezheniya pri optimizatsii teplovykh protsessov, opisyvaemykh integro-differentsialnymi uravneniyami”, Itogi nauki i tekhn. Ser. Sovr. mat. prilozh. Temat. obz., 183 (2020), 85–97 | MR

[7] Kerimbekov A., “O razreshimosti zadachi sinteza raspredelennogo i granichnogo upravlenii pri optimizatsii kolebatelnykh protsessov”, Tr. In-ta mat. mekh. UrO RAN., 27:2 (2021), 128–140 | MR

[8] Kerimbekov A., Abdyldaeva E. F, “O ravnykh otnosheniyakh v zadache granichnogo vektornogo upravleniya uprugimi kolebaniyami, opisyvaemymi fredgolmovymi integro-differentsialnymi uravneniyami”, Tr. In-ta mat. mekh. UrO RAN., 22:2 (2016), 163–176 | MR

[9] Kerimbekov A., Nametkulova R. Zh., Kadirimbetova A. K., “Usloviya optimalnosti v zadache upravleniya teplovymi protsessami s integro-differentsialnym uravneniem”, Izv. Irkut. gos. un-ta. Ser. Mat., 15 (2016), 50–61 | Zbl

[10] Kerimbekov A., Nametkulova R. Zh., Kadirimbetova A. K., “Priblizhennoe reshenie zadachi raspredelennogo i granichnogo upravleniya teplovym protsessom”, Izv. Irkut. gos. un-ta. Ser. Mat., 16 (2016), 71–78

[11] Lions Zh. L., Optimalnoe upravlenie sistemami, opisyvaemymi uravneniyami v chastnykh proizvodnykh, Fizmatlit, M., 1972

[12] Sirazetdinov T. K., Optimizatsiya sistem s raspredelennymi parametrami, Nauka, M., 1977 | MR

[13] Arguchintsev A. V., Kedrina M. S., “Determination of functional parameters in boundary conditions of linear hyperbolic systems by optimal control methods”, J. Phys. Conf. Ser., 1847 (2021), 012014 | DOI

[14] Arguchintsev A., Poplevko P., “An optimal control problem by parabolic equation with boundary smooth control and an integral constraint”, Num. Alg. Contr. Optim., 8:2 (2018), 193–202 | DOI | MR | Zbl

[15] Arguchintsev A., Poplevko P., “An optimal control problem by a hybrid system of hyperbolic and ordinary differential equations”, Games., 12:1 (2021), 23 | DOI | MR | Zbl

[16] Arguchintsev A., Poplevko P., “An optimal control problem by a hyperbolic system with boundary delay”, Izv. Irkut. gos. un-ta. Ser. Mat., 35 (2021), 3–17 | DOI | MR | Zbl

[17] Egorov A. I., “Optimal stabilization of systems with distributed parameters”, Optim. Tech. IFIP Tech. Conf., 27 (1975), 167–172 | Zbl

[18] Kerimbekov A., Abdyldaeva E. F., “Optimal distributed control for the processes of oscillation described by Fredholm integro-differential equations”, Eurasian Math., 26 (2015), 28–40 | MR

[19] Kerimbekov A., Abdyldaeva E. F., “On the solvability of a nonlinear tracking problem under boundary control for the elastic oscillations described by Fredholm integro-differential equations”, 27th IFIP Conf. on System Modeling and Optimization, Springer, Sophia Antipolis, France, 2016, 312–321 | DOI | MR

[20] Kerimbekov A., Abdyldaeva E. F., “The optimal vector control for the elastic oscillations described by Fredholm integro-differential equations”, Analysis and Partial Differential Equations: Perspectives from Developing Countries, Springer, 2019, 14–30 | MR | Zbl

[21] Kerimbekov A., Abdyldaeva E. F., Duyshenalieva U. E., “Generalized solution of a boundary value problem under point exposure of external forces”, Int. J. Pure Appl. Math., 113:4 (2017), 87–101 | DOI

[22] Kerimbekov A., Seidakmat E., “On solvability of tracking problem under nonlinear boundary control”, 11th ISAAC Congr. “Analysis, Probability, Applications, and Computation”, Springer, 2019, 312–321 | MR

[23] Kerimbekov A., Tairova O. K., “On the solvability of synthesis problem for optimal point control of oscillatory processes”, IFAC-PapersOnLine., 51 (2018), 754–758 | DOI

[24] Sachs E. W., Strauss A. K., “Efficient solution of a partial integro-differntial equation in finance”, Appl. Numer. Math., 58:11 (2008), 1687–1703 | DOI | MR | Zbl

[25] Thorwe J., Bhaleker S., “Solving partial integro-differential equations using Laplace transform method”, Am. J. Comput. Appl. Math., 2:3 (2012), 101–104 | DOI