DOI
10.34229/KCA2522-9664.24.3.11
UDC 519.622
1 Azerbaijan State Oil and Industry University; Institute of Control Systems of the National Academy of Sciences of Azerbaijan, Baku, Azerbaijan
vagif_ab@yahoo.com
|
2 Institute of Control Systems of the National Academy of Sciences of Azerbaijan, Baku, Azerbaijan
vugarhashimov@gmail.com
|
A PROBLEM OF OPTIMAL CONTROL OF LOADING POINTS AND THEIR
REACTION FUNCTIONS FOR A PARABOLIC EQUATION
Abstract. We consider the problem of optimal control of loading points and the corresponding reaction functions described by a loaded parabolic equation. Optimality conditions for control actions are obtained. The objective functional gradient formulas contained in these conditions are used in the algorithm for numerically solving the problem of optimization of loading points and reaction functions based on first-order optimization methods. The results of numerical experiments are provided.
Keywords: distributed-parameter system, loaded differential equation, necessary optimality condition, functional gradient.
full text
REFERENCES
- Nakhushev A.M. Loaded equations and their applications [in Russian]. Moscow: Nauka, 2012. 232 p.
- Dzhenaliev M.T. Optimal control of linear loaded parabolic equations. Differential equations. 1989. Vol. 25, N 4. P. 641–651.
- Dzhenaliev M.T. On the theory of linear boundary value problems for loaded differential equations [in Russian]. Almaty: ITPM Computer Center, 1995. 270p.
- Mardanov M.J., Sharifov Y.A., Zeynalli F.M. Existence and uniqueness of the solutions to impulsive nonlinear integro-differential equations with nonlocal boundary conditions. Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 2019. Vol. 45, N 2. P. 222–233. https://doi.org/ 10.29228/proc.6.
- Abdullayev V.M., Aida-zade K.R. Approach to the numerical solution of optimal control problems for loaded differential equations with nonlocal conditions. Comput. Math. Math. Phys. 2019. Vol. 59, N 5. P. 696–707. https://doi.org/10.1134/S0965542519050026.
- Abdullayev V.M. Identification of the functions of response to loading for stationary systems. Cybernetics and Systems Analysis. 2017. Vol. 53, N 3. P. 417–425. https://doi.org/10.1007/s10559-017-9942-6.
- Aida-zade K.R., Abdullayev V.M. Control synthesis for temperature maintaining process in a heat supply problem. Cybernetics and Systems Analysis. 2020. Vol. 56, N 3. P. 380–391. https://doi.org/10.1007/s10559-020-00254-5.
- Sergienko I.V., Deineka V.S. Optimal Control of Distributed Systems with Conjugation Conditions. New York: Kluwer Acad. Publ., 2005. 383 p.
- Ray W.H. Advanced Process Control. New York: McGraw-Hill, 1981. 376 p.
- Butkovsky A.G. Methods for controlling systems with distributed parameters [in Russian]. Moscow: Nauka, 1984. 568 p.
- Polyak B.T., Khlebnikov M.V., Rapoport L.B. Mathematical theory of automatic control [in Russian]. Moscow: LENAND, 2019. 504 p.
- Egorov A.I. Fundamentals of controltheory [in Russian]. Moscow: Fizmatlit, 2004. 504 p.
- Ayda-zade K.R., Bagirov A.G. On the problem of spacing of oil wells and control of their production rates. Autom. Remote Control. 2006. Vol. 67, N 1. P. 44–53. https://doi.org/10.1134/S0005117906010024.
- Afifi L., Lasri K., Joundi M., Amimi N. Feedback controls for exact remediability in disturbed dynamical systems. IMA J. of Mathematical Control and Information. 2018. Vol. 35, Iss. 2. P. 411–425. https://doi.org/10.1093/imamci/dnw054.
- Coron J.M., Wang Zh. Output feedback stabilization for a scalar conservation law with a nonlocal velocity. SIAM J. Math. Anal. 2013. Vol. 45, N 5. P. 2646–2665. https://doi.org/10.1137/120902203.
- Abdullayev V.M., Aida-zade K.R. Numerical solution of the problem of determining the number and locations of state observation points in feedback control of a heating process. Comput. Math. Math. Phys. 2018. Vol. 58, N 1. P. 78–89. https://doi.org/10.1134/S0965542518010025.
- Aida-zade K.R., Abdullayev V.M. Controlling the heating of a rod using the current and preceding time feedback. Autom. Remote Control. 2022. Vol. 83, N 1. P. 106–122. https://doi.org/10.1134/S0005117922010088.
- Aida-zade K.R., Hashimov V.A. Optimizing the arrangement of lumped sources and measurement points of plate heating. Cybernetics and Systems Analysis. 2019. Vol. 55, N 4. P. 605–615. https://doi.org/10.1007/s10559-019-00169-w .
- Aida-zade K.R., Bagirov A.H., Hashimov V.A. Feedback control of the power of moving sources in bar heating. Cybernetics and Systems Analysis. 2021. Vol. 57, N 4. P. 592–604. https://doi.org/10.1007/s10559-021-00384-4.
- Aida-zade K.R., Hashimov V.A. Feedback control of the plate heating process with optimization of the locations of sources and control. Autom. Remote Control. Vol. 81, N 4. P. 670–685. https://doi.org/10.1134/S0005117920040098.
- Asanova A.T., Kadirbaeva Zh.M., Bakirova А.A. On the unique solvability of a nonlocal boundary value problem for systems of loaded hyperbolic equations with impulsive actions. Ukrainian Mathematical Journal. 2018. Vol. 69, N 8. P. 1175–1195. https://doi.org/10.1007/ s11253-017-1424-5.
- Assanova A.T., Imanchiyev A.E., Kadirbayeva Zh.M. A nonlocal problem for loaded partial differential equations of fourth order. Vestnik Karagandinskogo Universiteta. Ser. Matematika. 2020. Vol. 97, N 1. P. 6–16. https://doi.org/10.31489/2020M1/6-16.
- Alikhanov A.A., Berezgov A.M., Shkhanukov-Lafishev M.X. Boundary value problems for certain classes of loaded differential equations and solving them by finite difference methods. Comp. Math. Math. Phys. 2008. Vol. 48, N 9. P. 1581–1590. https://doi.org/10.1134/ S096554250809008X .
- Abdullayev V.M., Aida-zade K.R. Finite-difference methods for solving loaded parabolic equations. Comp. Math. Math. Phys. 2016. Vol. 56, N 1. P. 93–105. https://doi.org/10.1134/S0965542516010036.
- Shkhanukov-Lafishev M.Kh. Locally one-dimensional scheme for a loaded heat equation with Robin boundary conditions. Comput. Math. Math. Phys. 2009. Vol. 49, N 7. P. 1167–1174. https://doi.org/10.1134/S0965542509070094.
- Abdullaev V.M., Aida-Zade K.R. Numerical solution of optimal control problems for loaded lumped parameter systems. Comput. Math. Math. Phys. 2006. Vol. 46, N 9. P. 1487–1502. https://doi.org/10.1134/S096554250609003X .
- Shor N.Z. Methods for minimizing nonsmooth functions and matrix optimization problems [in Russian]. Chisinau: Eureka, 2009. 272 p.
- Shor N.Z. Algorithms for sequential and non-smooth optimization [in Russian]. Chisinau: Eureka, 2012. 270 p.
- Stetsyuk P.I. Ellipsoid methods and r-algorithms [in Russian]. Chisinau: Eureka, 2014. 488 p.
- Pshenichnyi B.N., Danilin Yu.M. Numerical methods in extremal problems [in Russian]. Moscow: Nauka, 1975. 320 c.
- Samarsky A.A. Theory of difference schemes [in Russian]. Moscow: Nauka, 1983. 616 p.
- Walden J. On the approximation of singular source terms in differential equations. Numerical Methods for Partial Differential Equations. 1999. Vol. 15, N 4. P. 503–520. https://doi.org/10.1002/(SICI)1098-2426(199907)15:4<503::AID-NUM6>3.0.CO;2-Q .