  In this paper, we consider a fourth order inner-controlled parabolic equation
on an open bounded subset of $R^d$, or a smooth compact manifold with boundary,
along with general boundary operators fulfilling the Lopatinskii-Sapiro
condition. We derive a spectral inequality for the solution of the parabolic
system that yields a null-controllability result. The spectral inequality is a
consequence of an interpolation inequality obtained via a Carleman inequality
for the bi-Laplace operator under the considered boundary conditions.
