Note on the cost of the approximate controllability for the heat equation with potential

We prove the approximate controllability for the heat equation with potential with a cost of order ec/ε when the target is in H1 0 (Ω) with a precision in L2(Ω) norm. Also a quantification estimate of the unique continuation for initial data in L2(Ω) of the heat equation with potential is established.  2004 Elsevier Inc. All rights reserved.