Cloaking via impedance boundary condition for the 2-D Helmholtz equation

Abstract

We consider control problems for the 2-D Helmholtz equation in an unbounded domain with partially coated boundary. Dirichlet boundary condition is given on one part of the boundary and the impedance boundary condition is imposed on another its part. The role of control in control problem under study is played by boundary impedance. Quadratic tracking–type functionals for the field play the role of cost functionals. Solvability of control problems is proved. The uniqueness and stability of optimal solutions with respect to certain perturbations of both cost functional and incident field are established.

Keywords:

• Helmholtz equation
• mixed boundary value problem
• impedance
• control problem
• boundary control
• solvability
• stability estimates

AMS Subject Classifications:

• 35R30
• 35Q60
• 35J15
• 78A25

Acknowledgments

This research was suppotted in parts by grants from the Russian Foundation for Basic Research project no. 10-01-00219-a), the Far East Branch of the Russian Academy of Sciences (project no. 12-I-P17-03) and the Ministry of Education and Science of Russia (project no. 14.A18.21.0353).