  In recent years, controllability problems for dispersive systems have been
extensively studied. This work is dedicated to proving a new type of
controllability for a dispersive fifth order equation that models water waves,
what we will now call the overdetermination control problem. Precisely, we are
able to find a control acting at the boundary that guarantees that the
solutions of the problem under consideration satisfy an overdetermination
integral condition. In addition, when we make the control act internally in the
system, instead of the boundary, we are also able to prove that this condition
is satisfied. These problems give answers that were left open in [6] and
present a new way to prove boundary and internal controllability results for a
fifth order KdV type equation.
