Optimal Control of the Inverse Problem of the Burgers Equation for Representing the State of Sonic Vibration Velocity in Water
In this paper, we investigate the inverse of the set of unknown functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>v</mi><mo>,</mo><mi>g</m...
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2024-11-01
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| author | Jiale Qin Yiping Meng Shichao Yi |
| author_facet | Jiale Qin Yiping Meng Shichao Yi |
| author_sort | Jiale Qin |
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| description | In this paper, we investigate the inverse of the set of unknown functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>v</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> of the Burgers equation in the framework of optimal theory. Firstly, we prove the existence of functional minimizers in the optimal control problem and derive the necessary conditions for the optimal solution. Subsequently, the global uniqueness of the optimal solution and its stability are explored. After completing the ill-posed analysis of the Burgers equation, we can apply it to the problem of sonic vibration velocity in water. The desired result is obtained by inverse-performing an unknown initial state with known terminal vibration velocity. This is important for solving practical problems. |
| format | Article |
| id | doaj-art-2de38b5b52c84f688ae2bad7c68598a2 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-2de38b5b52c84f688ae2bad7c68598a22024-11-26T18:12:05ZengMDPI AGMathematics2227-73902024-11-011222362510.3390/math12223625Optimal Control of the Inverse Problem of the Burgers Equation for Representing the State of Sonic Vibration Velocity in WaterJiale Qin0Yiping Meng1Shichao Yi2School of Science, Jiangsu University of Science and Technology, Zhenjiang 212003, ChinaSchool of Science, Jiangsu University of Science and Technology, Zhenjiang 212003, ChinaSchool of Science, Jiangsu University of Science and Technology, Zhenjiang 212003, ChinaIn this paper, we investigate the inverse of the set of unknown functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>v</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> of the Burgers equation in the framework of optimal theory. Firstly, we prove the existence of functional minimizers in the optimal control problem and derive the necessary conditions for the optimal solution. Subsequently, the global uniqueness of the optimal solution and its stability are explored. After completing the ill-posed analysis of the Burgers equation, we can apply it to the problem of sonic vibration velocity in water. The desired result is obtained by inverse-performing an unknown initial state with known terminal vibration velocity. This is important for solving practical problems.https://www.mdpi.com/2227-7390/12/22/3625optimal controlnecessary conditionsstability and uniquenessill-posed analysis |
| spellingShingle | Jiale Qin Yiping Meng Shichao Yi Optimal Control of the Inverse Problem of the Burgers Equation for Representing the State of Sonic Vibration Velocity in Water Mathematics optimal control necessary conditions stability and uniqueness ill-posed analysis |
| title | Optimal Control of the Inverse Problem of the Burgers Equation for Representing the State of Sonic Vibration Velocity in Water |
| title_full | Optimal Control of the Inverse Problem of the Burgers Equation for Representing the State of Sonic Vibration Velocity in Water |
| title_fullStr | Optimal Control of the Inverse Problem of the Burgers Equation for Representing the State of Sonic Vibration Velocity in Water |
| title_full_unstemmed | Optimal Control of the Inverse Problem of the Burgers Equation for Representing the State of Sonic Vibration Velocity in Water |
| title_short | Optimal Control of the Inverse Problem of the Burgers Equation for Representing the State of Sonic Vibration Velocity in Water |
| title_sort | optimal control of the inverse problem of the burgers equation for representing the state of sonic vibration velocity in water |
| topic | optimal control necessary conditions stability and uniqueness ill-posed analysis |
| url | https://www.mdpi.com/2227-7390/12/22/3625 |
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