A. Kerimbekov. On the solvability of the problem of synthesizing distributed and boundary controls in the optimization of oscillation processes ... P. 128-140

We study the solvability of the problem of synthesis of distributed and boundary controls in the optimization of oscillation processes described by partial integro-differential equations with the Fredholm integral operator. Functions of external and boundary actions are nonlinear with respect to the controls. For the Bellman functional, an integro-differential equation of a specific form is obtained and the structure of its solution is found, which allows this equation to be represented as a system of two equations of a simpler form. An algorithm for constructing a solution to the problem of synthesizing distributed and boundary controls is described, and a procedure for finding the controls as a function (functional) of the state of the process is described.

Keywords: integro-differential equation, Fredholm operator, generalized solution, Bellman functional, FrДechet differential, optimal control synthesis

Received January 29, 2021

Revised March 22, 2021

Accepted April 2, 2021

Akylbek Kerimbekov, Dr. Phys.-Math. Sci., Prof., Kyrgyz-Russian Slavic university, Bishkek, 720022 Kyrgyzstan, e-mail: akl7@rambler.ru

REFERENCES

1.   Butkovskiy A.G. Distributed control systems. N Y: Elsevier, 1969, 446 p. ISBN: 0444000615 . Original Russian text published in Butkovskii A.G. Teoriya optimal’nogo upravleniya sistemami s raspredelennymi parametrami. Moscow: Nauka Publ., 1965, 476 p.

2.   Egorov A.I. Optimal’noe upravlenie teplovymi i diffuzionnymi protsessami [Optimal control of thermal and diffusion processes]. Moscow: Nauka Publ., 1978, 464 p.

3.   Egorov A.I., Znamenskaya L.N. Vvedenie v teoriyu upravleniya sistemami s raspredelennymi parametrami [Introduction to the control theory of systems with distributed parameters]. SPb.: Lan’, 2017, 292 p. ISBN: 978-5-8114-2554-9 .

4.   Egorov A.I. Optimal stabilization of the distributed parameter systems. In: Marchuk G.I. (eds), Optimization Techniques IFIP Technical Conference, (Novosibirsk, July 1–7, 1974), Lecture Notes in Computer Science, vol 27, Berlin; Heidelberg: Springer, 1975, pp. 167–172 .
doi: 10.1007/3-540-07165-2_22 

5.   Shenfel’d G.B. Optimal control synthesis for an elastic structure. In: Optimizatsiya protsessov v sistemakh s rapredelennymi parametrami. Frunze: Ilim Pibl., 1975, pp. 23–26. (in Russian)

6.   Rakhimov M. O sinteze optimal’nogo upravleniya uprugimi kolebaniyami [On the synthesis of optimal control of elastic vibrations], Candidate Sci. (Phys.-Math.) Dissertation, Ashkhabad, 1979, 128 p.

7.   Rakhimov M. Primenenie metodov dinamicheskogo programmirovaniya i spektral’nogo razlozheniya k zadacham optimal’nogo upravleniya sistemami s raspredelennymi parametrami [Application of dynamic programming and spectral decomposition methods to problems of optimal control of systems with distributed parameters], Abstract of Doctor Sci. (Phys.-Math.) Dissertation: 01.01.02, Mos. Gos. Univ. Moscow, 1989, 32 p.

8.   Kerimbekov A. Nelineinoe optimal’noe upravlenie lineinymi sistemami s raspredelennymi parametrami [Nonlinear optimal control of linear systems with distributed parameters], Doctor Sci. (Phys.-Math.) Dissertation, Bishkek: Ilim Publ, 2003, 224 p.

9.   Volterra V. Theory of functionals and of integral and integro-differential equations. N Y: Dover, 1959, 226 p. ISBN: 0486442845 . Translated to Russian under the title Funktsional’naya teoriya, integral’nye i integrodifferentsial’nye uravneniya, Moscow: Nauka Publ., 1984, 456 p.

10.   Vladimirov V.S. Mathematical problems of the uniform-speed transport theory of particles. Trudy Mat. Inst. Steklov, 1961, vol. 61, pp. 3–158 (in Russian).

11.   Richtmyer R.D. Principles of advanced mathematical physics, vol. 1. N Y: Springer-Verlag, 1978, 424 p. doi: 10.1007/978-3-642-46378-5 . Translated to Russian under the title Printsipy sovremennoi matematicheskoi fiziki, Moscow: Mir Publ., 1982, 488 p.

12.    Sachs E.W., Strauss A.K. Efficient solution of a partial integro-differential equation in finance. Applied Numerical Mathematics, 2008, vol. 58, no. 11, pp. 1687–1703. doi: 10.1016/j.apnum.2007.11.002 

13.   Thorwe J., Bhaleker S. Solving partial integro-differential equations using Laplace transform method. Am. J. Comput. Appl. Math., 2012, vol. 2, no. 3, pp. 101–104. doi: 10.5923/j.ajcam.20120203.06 

14.   Kerimbekov A.K., Abdyldaeva E.F. Optimal distributed control for the processes of oscillation described by Fredholm integro-differential equations. Eurasian Math. J., 2015, vol. 6, no. 2, pp. 18–40.

15.   Kerimbekov A.K., Abdyldaeva E.F. On the property of equal ratios in the problem of boundary vector control of elastic vibrations described by Fredholm integro-differential equations. Trudy Inst. Mat. i Mekh. UrO RAN, 2016, vol. 22, no. 2, pp. 163–176 (in Russian).
doi: 10.21538/0134-4889-2016-22-2-163-176 

16.   Kerimbekov A., Abdyldaeva E. On the solvability of a nonlinear tracking problem under boundary control for the elastic oscillations described by Fredholm integro-differential equations. In: L. Bociu, J.-A. D$\acute{\mathrm{e}}$sid$\acute{\mathrm{e}}$ri, A. Habbal (eds), System Modeling and Optimization (CSMO 2015), IFIP Advances in Information and Communication Technology, vol. 494, Cham: Springer, 2016, pp. 312–321. doi: 10.1007/978-3-319-55795-3_29 

17.   Kerimbekov A., Abdyldaeva E., Duyshenalieva U. Generalized solution of a boundary value problem under point exposure of external forces. International J. Pure Appl. Math., 2017, vol. 113, no. 4, pp. 87–101. doi: 10.12732/ijpam.v113i4.9 

18.   Kerimbekov A., Abdyldaeva E. The optimal vector control for the elastic oscillations described by Fredholm integral-differential equations. In: J. Delgado, M. Ruzhansky (eds.), Analysis and Partial Differential Equations: Perspectives from Developing Countries, Springer Proceedings in Mathematics & Statistics, vol. 275, Cham: Springer, 2019, pp. 14–30. doi: 10.1007/978-3-030-05657-5_3 

19.   Kerimbekov A., Tairova O.K. On the solvability of synthesis problem for optimal point control of oscillatory processes. IFAC-PapersOnLine, 2018, vol. 51, no. 32, pp. 754–758. doi: 10.1016/j.ifacol.2018.11.455 

20.   Bellman R. The theory of dynamic programming. Bull. Amer. Math. Soc., 1954, vol. 60, no. 6, pp. 503–515. doi: 10.1090/S0002-9904-1954-09848-8 

21.   Plotnikov V.I. An energy inequality and the overdeterminacy property of a system of eigenfunctions. Math. USSR-Izv., 1968, vol. 2, no. 4, pp. 695–707. doi: 10.1070/IM1968v002n04ABEH000656 

22.   Krasnov M. Integral’nye uravneniya [Integral equations]. Moscow: Nauka Publ., 1975, 304 p.

23.   Lusternik L.A. Sobolev V.J. Elements of functional analysis. International monographs on advanced mathematics and physics. Delhi: Hindustan Publishing Corp., 1974, 360 p. ISBN: 0470556501 . Original Russian text published in Lyusternik L.A., Sobolev V.I. Elementy funktsional’nogo analiza, Moscow: Nauka Publ., 1965, 520 p.

24.   Smirnov V.I. A course of higher mathematics, vol. 1. Oxford: Pergamon Press, 1964, 546 p. ISBN: 9781483152592 . Original Russian text published in Smirnov V.I. Kurs vysshei matematiki. T. 1. Moscow: Nauka Publ., 1974, 480 p.

Cite this article as: A. Kerimbekov. On the solvability of the problem of synthesizing distributed and boundary controls in the optimization of oscillation processes, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 2, pp. 128–140.