Theoretical analysis of the 2D thermal cloaking problem

Abstract

Coefficient inverse problems for the model of heat scattering with variable coefficients arising when developing technologies of design of thermal cloaking devices are considered. By the optimization method, these problems are reduced to respective control problems. The material parameters (radial and azimuthal conductivities) of the inhomogeneous anisotropic medium, filling the thermal cloak, play the role of control. The model of heat scattering acts as a functional restriction. A unique solvability of direct heat scattering problem in the Sobolev space is proved and the new estimates of solutions are established. Using these results, the solvability of control problem is proved and the optimality system is derived. Based on analysis of optimality system, the stability estimates of optimal solutions are established and efficient numerical algorithm of solving thermal cloaking problems is proposed.