  In this paper we prove a uniform controllability result for a fourth order
parabolic partial differential equation which includes a transport term, when
the coefficients of higher order terms vanish. We prove the null
controllability of the system with a single boundary control, and also it is
obtained that the cost of the control decreases to zero, under the hypothesis
of the control time being large enough. Moreover, we prove that, if the control
time is small enough, the cost of controllability increases to infinity.
