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
C. D'Apice;U. De Maio;
2018-01-01
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
