Homogenization of a boundary optimal control problem governed by Stokes equations

Abstract

This article considers an optimal control problem for the stationary Stokes system in a three-dimensional domain with a highly oscillating boundary. The controls are acting on the state through the Neumann data on the oscillating part of the boundary with appropriate scaling parameters ${ϵ}^{\alpha }$ with $\alpha \ge 1$. The periodic unfolding operators are used to characterize the optimal controls. Using the unfolding operators, we analyse the asymptotic behaviour of the optimal control problem under consideration. For $\alpha =1$, the limit optimal control problem has both boundary and interior controls. For $\alpha >1$, the limit optimal control problem has only boundary controls.

Keywords:

• Rough boundary
• optimal control
• unfolding operator
• homogenization

AMS Subject Classifications:

• 35B27
• 35B40
• 49J20

Acknowledgments

The authors are profoundly grateful to the reviewer for reviewing the manuscripts very carefully and make many constructive suggestions to improve the article. The first author acknowledges the support of Science & Engineering Research Board (SERB) (SRG/2019/000997), Government of India.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Science and Engineering Research Board (SERB) [SRG/2019/000997].