An optimal time control problem for the one-dimensional, linear heat equation, in the presence of a scaling parameter

In this paper, we study the optimal time problem for the one-dimensional, linear heat equation, in the presence of a scaling parameter. To begin with, we build an exact solution. The dependence of this solution as regards the scaling parameter naturally opens the way to study the existence and uniqueness of an optimal time control. If, moreover, one assumes the L∞ − null controllability, it enables to establish a bang-bang type property