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Volume 12
Issue 22
10.3390/math12223625
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Optimal Control of the Inverse Problem of the Burgers Equation for Representing the State of Sonic Vibration Velocity in Water

by
Jiale Qin
1,
Yiping Meng
1,* and
Shichao Yi
1,2
1
School of Science, Jiangsu University of Science and Technology, Zhenjiang 212003, China
2
Yangzijiang Shipbuilding Group, Taizhou 212299, China
*
Author to whom correspondence should be addressed.
Mathematics 2024, 12(22), 3625; https://doi.org/10.3390/math12223625
Submission received: 18 October 2024 / Revised: 15 November 2024 / Accepted: 16 November 2024 / Published: 20 November 2024
(This article belongs to the Section Engineering Mathematics)
Versions Notes

Abstract

In this paper, we investigate the inverse of the set of unknown functions (v,g) 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.
Keywords: optimal control; necessary conditions; stability and uniqueness; ill-posed analysis optimal control; necessary conditions; stability and uniqueness; ill-posed analysis

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MDPI and ACS Style

Qin, J.; Meng, Y.; Yi, S. Optimal Control of the Inverse Problem of the Burgers Equation for Representing the State of Sonic Vibration Velocity in Water. Mathematics 2024, 12, 3625. https://doi.org/10.3390/math12223625

AMA Style

Qin J, Meng Y, Yi S. Optimal Control of the Inverse Problem of the Burgers Equation for Representing the State of Sonic Vibration Velocity in Water. Mathematics. 2024; 12(22):3625. https://doi.org/10.3390/math12223625

Chicago/Turabian Style

Qin, Jiale, Yiping Meng, and Shichao Yi. 2024. "Optimal Control of the Inverse Problem of the Burgers Equation for Representing the State of Sonic Vibration Velocity in Water" Mathematics 12, no. 22: 3625. https://doi.org/10.3390/math12223625

APA Style

Qin, J., Meng, Y., & Yi, S. (2024). Optimal Control of the Inverse Problem of the Burgers Equation for Representing the State of Sonic Vibration Velocity in Water. Mathematics, 12(22), 3625. https://doi.org/10.3390/math12223625

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