We consider 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 $\theta(x)$. The coefficients of operator $\mathrm{div}\,\left(A(x)\, \nabla \, \theta(x)\right)$ are used as the controls in $L^\infty(\Omega)$. The optimal control problem is to minimize the discrepancy between a given distribution $\theta_d\in L^r(\Omega)$ and the temperature of thermistor $\theta\in W^{1,\gamma}_0(\Omega)$ by choosing an appropriate anisotropic heat conductivity matrix $B$. Basing on the perturbation theory of extremal problems and the concept of fictitious controls, we propose an \textquotedblleft approximation approach\textquotedblright\ and discuss the existence of the so-called quasi-optimal and optimal solutions to the given problem.

### Thermistor Problem: Multi-Dimensional Modelling, Optimization, and Approximation

#### Abstract

We consider 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 $\theta(x)$. The coefficients of operator $\mathrm{div}\,\left(A(x)\, \nabla \, \theta(x)\right)$ are used as the controls in $L^\infty(\Omega)$. The optimal control problem is to minimize the discrepancy between a given distribution $\theta_d\in L^r(\Omega)$ and the temperature of thermistor $\theta\in W^{1,\gamma}_0(\Omega)$ by choosing an appropriate anisotropic heat conductivity matrix $B$. Basing on the perturbation theory of extremal problems and the concept of fictitious controls, we propose an \textquotedblleft approximation approach\textquotedblright\ and discuss the existence of the so-called quasi-optimal and optimal solutions to the given problem.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11386/4761344
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