An Indirect Approach to the Existence of Quasi-Optimal Controls in Coefficients for Multi-Dimensional Thermistor Problem

The paper studies a problem of an optimal control in coefficients for the system of two coupled elliptic equations also known as thermistor problem which provides a simultaneous description of the electric field $u=u(x)$ and temperature $ heta(x)$. The coefficient $b$ of operator $mathrm{div},left(b(x), abla , heta(x) ight)$ is used as the control in $W^{1,q}(Omega)$ with $q>N$. The optimal control problem is to minimize the discrepancy between a given distribution $ heta_din L^1(Omega)$ and the temperature of thermistor $ hetain W^{1,gamma}_0(Omega)$ by choosing an appropriate anisotropic heat conductivity $b(x)$. Basing on the perturbation theory of extremal problems and the concept of fictitious controls, we propose an extquotedblleft approximation approach extquotedblright and discuss the existence of the so-called quasi-optimal and optimal solutions to the given problem.