Sharp Estimates of the One-Dimensional Boundary Control Cost for Parabolic Systems and Application to the N-Dimensional Boundary Null Controllability in Cylindrical Domains

In this paper we consider the boundary null controllability of a system of n parabolic equations on domains of the form Ω = (0, π)×Ω2 with Ω2 a smooth domain of RN−1, N > 1. When the control is exerted on {0} × ω2 with ω2 ⊂ Ω2, we obtain a necessary and sufficient condition that completely characterizes the null controllability. This result is obtained through the Lebeau– Robbiano strategy and requires an upper bound of the cost of the one-dimensional boundary null control on (0, π). The latter is obtained using the moment method and it is shown to be bounded by CeC/T when T goes to 0+.