# Optimal control of a thermistor problem with vanishing conductivity

## TL;DR

This paper studies the optimal control of a steady thermistor problem with vanishing conductivity, establishing existence results and deriving optimality conditions in arbitrary dimensions.

## Contribution

It introduces a novel analysis of thermistor control with conductivity that can vanish, extending results to arbitrary dimensions.

## Key findings

- Existence of steady state proven
- Optimal control existence established
- Optimality system derived

## Abstract

An optimal control of a steady state thermistor problem is considered, where the convective boundary coefficient is taken as the control variable. A distinctive feature of this paper is that the problem is considered in arbitrary dimensions, and the electrical conductivity is allowed to vanish above a threshold temperature value. The existence of a steady state is proved, an objective functional is introduced, the existence of the optimal control is proved, and the optimality system is derived.