  In this paper, we study some controllability and observability properties for
a coupled system of time-discrete fourth- and second-order parabolic equations.
This system can be regarded as a simplification of the well-known stabilized
Kumamoto-Sivashinsky equation. Unlike the continuous case, we can prove only a
relaxed observability inequality which yields a $\phi(\triangle
t)$-controllability result. This result tells that we cannot reach exactly zero
but rather a small target whose size goes to 0 as the discretization parameter
$\triangle t$ goes to 0. The proof relies on a known Carleman estimate for
second-order time-discrete parabolic operators and a new Carleman estimate for
the time-discrete fourth-order equation.
