Control Problem Concerned With the Process of Heating a Thin Plate
Previously, a mathematical model for the following problem was considered. On a part of the border of the right rectangle there is a heater with controlled temperature. It is required to find such a mode of its operation that the average temperature in some region reaches some given value. In this p...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
KamGU by Vitus Bering
2023-04-01
|
| Series: | Vestnik KRAUNC: Fiziko-Matematičeskie Nauki |
| Subjects: | |
| Online Access: | https://krasec.ru/dekhonov421023-eng/ |
| _version_ | 1797845257133490176 |
|---|---|
| author | Dekhkonov, F.N. |
| author_facet | Dekhkonov, F.N. |
| author_sort | Dekhkonov, F.N. |
| collection | DOAJ |
| description | Previously, a mathematical model for the following problem was considered. On a part of the border of the right rectangle there is a heater with controlled temperature. It is required to find such a mode of its operation that the average temperature in some region reaches some given value. In this paper, we consider a boundary control problem associated with a parabolic equation on a right rectangle. On the part of the border of the considered domain, the value of the solution with control parameter is given. Restrictions on the control are given in such a way that the average value of the solution in some part of the considered domain gets a given value. The auxiliary problem is solved by the method of separation of variables, while the problem in consideration is reduced to the Volterra integral equation. In addition, the definition of the generalized solution of the given initialboundary problem is given in the article and the existence of such a solution is proved. The solution of Volterra’s integral equation was found by the Laplace transform method and the existence theorem for admissible control functions was proved. It is also shown that the initial value of the admissible control function is equal to zero using the change of variable in the integral equation. The proof of this comes from the fact that the kernels of the integral equations are positive and finite, and the system has a single-valued solution. |
| first_indexed | 2024-04-09T17:35:50Z |
| format | Article |
| id | doaj.art-467453b71a7548a0b8b695634d3851ae |
| institution | Directory Open Access Journal |
| issn | 2079-6641 2079-665X |
| language | English |
| last_indexed | 2024-04-09T17:35:50Z |
| publishDate | 2023-04-01 |
| publisher | KamGU by Vitus Bering |
| record_format | Article |
| series | Vestnik KRAUNC: Fiziko-Matematičeskie Nauki |
| spelling | doaj.art-467453b71a7548a0b8b695634d3851ae2023-04-17T21:51:16ZengKamGU by Vitus BeringVestnik KRAUNC: Fiziko-Matematičeskie Nauki2079-66412079-665X2023-04-0120231697910.26117/2079-6641-2023-42-1-69-7910.26117/2079-6641-2023-42-1-69-79Control Problem Concerned With the Process of Heating a Thin PlateDekhkonov, F.N.0Namangan State UniversityPreviously, a mathematical model for the following problem was considered. On a part of the border of the right rectangle there is a heater with controlled temperature. It is required to find such a mode of its operation that the average temperature in some region reaches some given value. In this paper, we consider a boundary control problem associated with a parabolic equation on a right rectangle. On the part of the border of the considered domain, the value of the solution with control parameter is given. Restrictions on the control are given in such a way that the average value of the solution in some part of the considered domain gets a given value. The auxiliary problem is solved by the method of separation of variables, while the problem in consideration is reduced to the Volterra integral equation. In addition, the definition of the generalized solution of the given initialboundary problem is given in the article and the existence of such a solution is proved. The solution of Volterra’s integral equation was found by the Laplace transform method and the existence theorem for admissible control functions was proved. It is also shown that the initial value of the admissible control function is equal to zero using the change of variable in the integral equation. The proof of this comes from the fact that the kernels of the integral equations are positive and finite, and the system has a single-valued solution.https://krasec.ru/dekhonov421023-eng/parabolic equationsystem of integral equationsinitial-boundary problemadmissible controllaplace transformпараболическое уравнениесистема интегральных уравненийначально-краевая задачадопустимое управлениепреобразование лапласа |
| spellingShingle | Dekhkonov, F.N. Control Problem Concerned With the Process of Heating a Thin Plate Vestnik KRAUNC: Fiziko-Matematičeskie Nauki parabolic equation system of integral equations initial-boundary problem admissible control laplace transform параболическое уравнение система интегральных уравнений начально-краевая задача допустимое управление преобразование лапласа |
| title | Control Problem Concerned With the Process of Heating a Thin Plate |
| title_full | Control Problem Concerned With the Process of Heating a Thin Plate |
| title_fullStr | Control Problem Concerned With the Process of Heating a Thin Plate |
| title_full_unstemmed | Control Problem Concerned With the Process of Heating a Thin Plate |
| title_short | Control Problem Concerned With the Process of Heating a Thin Plate |
| title_sort | control problem concerned with the process of heating a thin plate |
| topic | parabolic equation system of integral equations initial-boundary problem admissible control laplace transform параболическое уравнение система интегральных уравнений начально-краевая задача допустимое управление преобразование лапласа |
| url | https://krasec.ru/dekhonov421023-eng/ |
| work_keys_str_mv | AT dekhkonovfn controlproblemconcernedwiththeprocessofheatingathinplate |