Time-optimal control problem for a heat conduction equation with variable thermal conductivity

This paper investigates a time-optimal boundary control problem for the heat equation governed by a uniformly elliptic operator in a bounded multidimensional domain. Robin boundary conditions model passive heat exchange, while the control acts through a prescribed heat flux on a portion of the boundary. The admissible controls are assumed to be bounded measurable functions. The analysis relies on the spectral properties of the associated elliptic operator, including the discreteness of the spectrum and the positivity of the first eigenfunction. These properties are used to derive an optimal estimate for the minimal time required to achieve a prescribed average temperature in the domain.