Biharmonic equation in a highly oscillating domain and homogenization of an associated control problem

ABSTRACT

We consider a Dirichlet boundary control problem posed in an oscillating boundary domain governed by a biharmonic equation. Homogenization of a PDE with a non-homogeneous Dirichlet boundary condition on the oscillating boundary is one of the hardest problems. Here, we study the homogenization of the problem by converting it into an equivalent interior control problem. The convergence of the optimal solution is studied using periodic unfolding operator.

KEYWORDS:

• Homogenization
• optimal control
• oscillating boundary
• unfolding operator

2000 MSC CLASSIFICATIONS:

• 49J20
• 80M35
• 80M40
• 35B27

Acknowledgments

We acknowledge the referees for their careful and meticulous reading of the manuscript, and their comments immensely helped us in improving the presentation of the manuscript. The authors wish to thank Prof. A. K. Nandakumaran for stimulating discussions on the topic.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author would like to acknowledge the support of National Board for Higher Mathematics, Department of Atomic Energy, Government of India.