Inverse Design Method for the 2D Problems of Thermal Cloaking

We consider coefficient inverse problems for the 2D model of thermal scattering with variable coefficients. These problems arise when developing technologies of design of thermal cloaking devices. By the optimization method, the problems are reduced to respective control problems. The material parameters (radial and azimuthal heat conductivities) of the inhomogeneous anisotropic medium filling the thermal cloak and a heat flux on a part of the boundary play the role of controls. A unique solvability of direct thermal 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 for solving particular thermal cloaking problem is proposed.