Null-controllability for a fourth order parabolic equation under general boundary conditions

In this paper, we consider a fourth-order inner-controlled parabolic equation on an open bounded subset of $ {\mathbb{R}}^d $, or a smooth compact manifold with boundary, along with general boundary operators fulfilling the Lopatinskiĭ-Šapiro 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.