Time-Optimal Control of the Heat Conduction Equation

Abstract

We study a time-optimal control problem for the heat conduction equation with a general elliptic operator and Robin boundary condition. The goal is to drive the weighted average temperature to a prescribed value in minimal time using an admissible control subject to a uniform bound. Using spectral decomposition and the Green function, the problem is reduced to a Volterra integral equation. Analyzing its kernel via the operator's eigenpairs, we derive the maximal attainable average temperature. For any target below this threshold, we prove the existence of a unique minimal time and establish its asymptotic expansion, revealing logarithmic growth governed by the principal eigenvalue.