The cost of the control in the case of a minimal time of control: The example of the one-dimensional heat equation

In this article, we consider the controllability of the one-dimensional heat equation with Dirichlet boundary conditions, internal control depending only on the time variable and an imposed profile depending on the space variable. It is well-known that in this context, there might exist a positive minimal time of null-controllability T0, depending on the behavior of the Fourier coefficients of the profile. We prove two different results. The first one, which is surprising, is that the cost of the controllability in time T>T0 close to T0 may explode in an arbitrary way. On the other hand, we prove as a second result that for a large class of profiles, the cost of controllability at time T>T0 is bounded from above by exp⁡(C(T0)/(T−T0)) for some constant C(T0)>0 depending on T0. The main method used here is the moment method.