Boundary controllability of a cascade system coupling fourth- and second-order parabolic equations

A control system coupling fourth- and second-order parabolic equations is considered in this paper. The main topic is the study of the control properties of this system when we only control the second-order partial differential equation through a boundary condition. Depending on the choice of the diffusion coefficients, we obtain positive and negative results for approximate- and null-controllability. In particular, we prove that for any given positive time T0, we can find some diffusion coefficients such that the system is null-controllable in time T if T>T0 and is not null-controllable if T