Controllability and stabilization of gravity-capillary surface water waves in a basin

Jing Cui; Guangyue Gao; Shu-Ming Sun

doi:10.3934/cpaa.2021158

Article Contents

2022, Volume 21, Issue 6: 2035-2063. Doi: 10.3934/cpaa.2021158

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Controllability and stabilization of gravity-capillary surface water waves in a basin

1.
Huawei Beijing Research Center, 156 Beiqing Rd, Haidian District, Beijing, 100039, China
2.
Beijing National Day School, No. 66 Yuquan Lu, Haidian District, Beijing, 100039, China
3.
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA

^* Corresponding author
^* Corresponding author

It is dedicated to Professor Goong Chen on the occasion of his 70th birthday

Received: December 31, 2020

Revised: June 30, 2021

Published: June 2022

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Abstract

The paper concerns the controllability and stabilization of surface water waves in a two-dimensional rectangular basin under the forces of gravity and surface tension. The surface waves are generated by a wave-maker placed at the left side-boundary and it is physical relevant to see whether the surface waves are controllable or can be stabilized using appropriate motion of the wave-maker. Due to the surface tension, an edge condition must be imposed at the contact point between the free surface and a solid boundary. Two types of wave-makers are considered: "flexible" or "rigid". It is shown that the surface waves are approximately controllable, but not exactly controllable, for both "flexible" and "rigid" wave-makers. In addition, under a static feedback to control a "rigid" wave-maker, the strong stability of feedback control system is obtained.

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Mathematics Subject Classification: Primary: 76B15, 76B45; Secondary: 93C20.

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