Analysis of one class of optimal control problems for distributed-parameter systems

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...

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Autores principales:	Kamil Mamtiyev, Tarana Aliyeva, Ulviyya Rzayeva
Formato:	article
Lenguaje:	EN RU UK
Publicado:	PC Technology Center 2021
Materias:	nonlinear boundary value problems functional convergence pontryagin’s maximum principle minimizing sequence Technology (General) T1-995 Industry HD2321-4730.9
Acceso en línea:	https://doaj.org/article/0d8d87b0fbf1473d8351ec47aa186a65
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spelling	oai:doaj.org-article:0d8d87b0fbf1473d8351ec47aa186a652021-11-04T14:06:45ZAnalysis of one class of optimal control problems for distributed-parameter systems1729-37741729-406110.15587/1729-4061.2021.241232https://doaj.org/article/0d8d87b0fbf1473d8351ec47aa186a652021-10-01T00:00:00Zhttp://journals.uran.ua/eejet/article/view/241232https://doaj.org/toc/1729-3774https://doaj.org/toc/1729-4061In 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 zeroKamil MamtiyevTarana AliyevaUlviyya RzayevaPC Technology Centerarticlenonlinear boundary value problemsfunctional convergencepontryagin’s maximum principleminimizing sequenceTechnology (General)T1-995IndustryHD2321-4730.9ENRUUKEastern-European Journal of Enterprise Technologies, Vol 5, Iss 4 (113), Pp 26-33 (2021)
institution	DOAJ
collection	DOAJ
language	EN RU UK
topic	nonlinear boundary value problems functional convergence pontryagin’s maximum principle minimizing sequence Technology (General) T1-995 Industry HD2321-4730.9
spellingShingle	nonlinear boundary value problems functional convergence pontryagin’s maximum principle minimizing sequence Technology (General) T1-995 Industry HD2321-4730.9 Kamil Mamtiyev Tarana Aliyeva Ulviyya Rzayeva Analysis of one class of optimal control problems for distributed-parameter systems
description	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
format	article
author	Kamil Mamtiyev Tarana Aliyeva Ulviyya Rzayeva
author_facet	Kamil Mamtiyev Tarana Aliyeva Ulviyya Rzayeva
author_sort	Kamil Mamtiyev
title	Analysis of one class of optimal control problems for distributed-parameter systems
title_short	Analysis of one class of optimal control problems for distributed-parameter systems
title_full	Analysis of one class of optimal control problems for distributed-parameter systems
title_fullStr	Analysis of one class of optimal control problems for distributed-parameter systems
title_full_unstemmed	Analysis of one class of optimal control problems for distributed-parameter systems
title_sort	analysis of one class of optimal control problems for distributed-parameter systems
publisher	PC Technology Center
publishDate	2021
url	https://doaj.org/article/0d8d87b0fbf1473d8351ec47aa186a65
work_keys_str_mv	AT kamilmamtiyev analysisofoneclassofoptimalcontrolproblemsfordistributedparametersystems AT taranaaliyeva analysisofoneclassofoptimalcontrolproblemsfordistributedparametersystems AT ulviyyarzayeva analysisofoneclassofoptimalcontrolproblemsfordistributedparametersystems
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Analysis of one class of optimal control problems for distributed-parameter systems

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