An optimal control problem of the 3D viscous Camassa–Holm equations

Abstract

In this paper, we study an optimal control problem of the three-dimensional viscous Camassa–Holm equations in bounded domains with Dirichlet boundary conditions and a quadratic objective functional. We first prove the existence of an optimal solution. Then we establish the first-order necessary and second-order sufficient optimality conditions.

Keywords:

• viscous Camassa–Holm equations
• optimal control
• existence
• necessary optimality condition
• sufficient optimality condition

2010 Mathematics Subject Classifications:

• 49J20
• 49K20
• 76D55
• 35Q35

Acknowledgments

The authors would like to thank the reviewers for their helpful comments and suggestions, which improved the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) [grant number 101.02-2018.303].