Optimal control of vibrations of a dynamic Gao beam in contact with a reactive foundation

ABSTRACT

In this paper, we investigate the optimal control of vibrations of a nonlinear viscoelastic beam, which is acted upon by a horizontal traction, that may come in contact with a reactive foundation underneath it. By the Dubovitskii and Milyutin functional analytical approach, we derive the Pontryagin maximum principle of the system governed by the Gao beam equation. And the first-order necessary optimality condition is presented for the optimal control problem in fixed final horizon case. Finally, we also sketch the numerical solution based on the obtained theoretical results.

Keywords:

• Nonlinear viscoelastic beam
• Gao beam
• optimal control
• maximum principle
• necessary optimality condition

AMS subject classifications:

• 35L75
• 49B22
• 49K20
• 74K10
• 74M05

Acknowledgements

The author would like to thank the editor and the anonymous referees for their very careful reading and constructive suggestions that improve substantially the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported in part by the National Natural Science Foundation of China [grant number 11471036].

Notes on contributors

Bing Sun

Bing Sun received his Ph.D. degree in operational research and control theory from the Academy of Mathematics and Systems Science, the Chinese Academy of Sciences in 2005. From 2005 to 2006, he was a Postdoctoral Fellow at the School of Computational and Applied Mathematics, University of the Witwatersrand, South Africa. Since 2008, he has been with the Beijing Institute of Technology, first as an Assistant Professor (2008–2010), an Associate Professor (2010–2015) and subsequently a Professor (since 2015) in mathematical system theory. He is the author or coauthor almost 30 international peer/refereed journal papers. His research interests include the optimal control theory of systems described by partial differential equations and numerical solutions of optimal feedback control for both finite- and infinite-dimensional systems. Dr Sun received the Training Program Foundation for the Beijing Municipal Excellent Talents in 2012.