  The paper deals with the controllability properties of the Kawahara equation
posed on a periodic domain. We show that the equation is exactly controllable
by means of a control depending only on time and acting on the system through a
given shape function in space. Firstly, the exact controllability property is
established for the linearized system through a Fourier expansion of solutions
and the analysis of a biorthogonal sequence to a family of complex exponential
functions. Finally, the local controllability of the full system is derived by
combining the analysis of the linearized system, a fixed point argument and
some Bourgain smoothing properties of the Kawahara equation on a periodic
domain.
