Optimal Control of Nonlocal Thermistor Equations

TL;DR

This paper investigates the optimal control of nonlocal thermistor equations, establishing existence, uniqueness, and deriving the optimality system, complemented by numerical simulations to demonstrate the results.

Contribution

It provides the first comprehensive analysis of optimal control for nonlocal thermistor problems, including existence, uniqueness, and numerical methods.

Findings

Existence of an optimal control is proved.

The optimality system and control characterization are derived.

Numerical simulations validate the theoretical results.

Abstract

We are concerned with the optimal control problem of the well known nonlocal thermistor problem, i.e., in studying the heat transfer in the resistor device whose electrical conductivity is strongly dependent on the temperature. Existence of an optimal control is proved. The optimality system consisting of the state system coupled with adjoint equations is derived, together with a characterization of the optimal control. Uniqueness of solution to the optimality system, and therefore the uniqueness of the optimal control, is established. The last part is devoted to numerical simulations.