Analysis of one class of optimal control problems for distributed-parameter systems
DOI:
https://doi.org/10.15587/1729-4061.2021.241232Keywords:
nonlinear boundary value problems, functional convergence, Pontryagin’s maximum principle, minimizing sequenceAbstract
In the paper, the method of straight lines approximately solves one class of optimal control problems for systems, the behavior of which is described by a nonlinear equation of parabolic type and a set of ordinary differential equations. Control is carried out using distributed and lumped parameters. Distributed control is included in the partial differential equation, and lumped controls are contained both in the boundary conditions and in the right-hand side of the ordinary differential equation. The convergence of the solutions of the approximating boundary value problem to the solution of the original one is proved when the step of the grid of straight lines tends to zero, and on the basis of this fact, the convergence of the approximate solution of the approximating optimal problem with respect to the functional is established.
A constructive scheme for constructing an optimal control by a minimizing sequence of controls is proposed. The control of the process in the approximate solution of a class of optimization problems is carried out on the basis of the Pontryagin maximum principle using the method of straight lines. For the numerical solution of the problem, a gradient projection scheme with a special choice of step is used, this gives a converging sequence in the control space. The numerical solution of one variational problem of the mentioned type related to a one-dimensional heat conduction equation with boundary conditions of the second kind is presented. An inequality-type constraint is imposed on the control function entering the right-hand side of the ordinary differential equation. The numerical results obtained on the basis of the compiled computer program are presented in the form of tables and figures.
The described numerical method gives a sufficiently accurate solution in a short time and does not show a tendency to «dispersion». With an increase in the number of iterations, the value of the functional monotonically tends to zero
References
- Butkovsky, A. G. (1965). Theory of optimal control of systems with distributed parameters. Moscow: Science, 474.
- Butkovsky, A. G., Egorov, A. I., Lurie, K. A. (1968). Optimal Control of Distributed Systems (A Survey of Soviet Publications). SIAM Journal on Control, 6 (3), 437–476. doi: https://doi.org/10.1137/0306029
- Egorov, A. I., Znamenskaya, L. N. (2005). Control of vibrations of coupled objects with distributed and lumped parameters. Computational Mathematics and Mathematical Physics, 45 (10), 1701–1718. Available at: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=zvmmf&paperid=578&option_lang=eng
- Egorov, A. I., Znamenskaya, L. N. (2006). Controllability of vibrations of a system of objects with distributed and lumped parameters. Computational Mathematics and Mathematical Physics, 46 (6), 955–970. doi: https://doi.org/10.1134/s0965542506060054
- Egorov, A. I., Znamenskaya, L. N. (2009). Controllability of vibrations of a net of coupled objects with distributed and lumped parameters. Computational Mathematics and Mathematical Physics, 49 (5), 786–796. doi: https://doi.org/10.1134/s0965542509050054
- Panferov, V. I., Anisimova, E. Y. (2009). On optimal control over heating of buildings as a distributed-parameter process. Bulletin of South Ural State University, 3, 24–28. Available at: https://dspace.susu.ru/xmlui/bitstream/handle/0001.74/708/5.pdf?sequence=1&isAllowed=y
- Teimurov, R. A. (2013). The problem of optimal control for moving sources for systems with distributed parameters. Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 1 (21), 24–33. Available at: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=vtgu&paperid=290&option_lang=eng
- Andreev, Yu. N., Orkin, V. M. (1969). Concerning approximate solution of problem of distributed system optimal control. Automation and Remote Control, 30 (5), 681–690. Available at: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=at&paperid=10161&option_lang=eng
- Kamil, M., Tarana, A., Ulviyya, R. (2020). Solution of One Problem on Optimum Gas Well Operation Control. Economic computation and economic cybernetics studies and research, 54 (4/2020), 249–264. doi: https://doi.org/10.24818/18423264/54.4.20.16
- Sakawa, Y. (1964). Solution of an optimal control problem in a distributed-parameter system. IEEE Transactions on Automatic Control, 9 (4), 420–426. doi: https://doi.org/10.1109/tac.1964.1105753
- Leonchuk, M. P. (1964). Numerical solution of problems of optimal processes with distributed parameters. USSR Computational Mathematics and Mathematical Physics, 4 (6), 189–198. doi: https://doi.org/10.1016/0041-5553(64)90091-6
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2021 Kamil Mamtiyev, Tarana Aliyeva, Ulviyye Rzayeva
This work is licensed under a Creative Commons Attribution 4.0 International License.
The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.
A license agreement is a document in which the author warrants that he/she owns all copyright for the work (manuscript, article, etc.).
The authors, signing the License Agreement with TECHNOLOGY CENTER PC, have all rights to the further use of their work, provided that they link to our edition in which the work was published.
According to the terms of the License Agreement, the Publisher TECHNOLOGY CENTER PC does not take away your copyrights and receives permission from the authors to use and dissemination of the publication through the world's scientific resources (own electronic resources, scientometric databases, repositories, libraries, etc.).
In the absence of a signed License Agreement or in the absence of this agreement of identifiers allowing to identify the identity of the author, the editors have no right to work with the manuscript.
It is important to remember that there is another type of agreement between authors and publishers – when copyright is transferred from the authors to the publisher. In this case, the authors lose ownership of their work and may not use it in any way.
