  A class of models of long waves in dispersive media with coupled quadratic
nonlinearities on a periodic domain $\mathbb{T}$ are studied. We used two
distributed controls, supported in $\omega\subset\mathbb{T}$ and assumed to be
generated by a linear feedback law conserving the "mass" (or "volume"), to
prove global control results. The first result, using spectral analysis,
guarantees that the system in consideration is locally controllable in
$H^s(\mathbb{T})$, for $s\geq0$. After that, by certain properties of Bourgain
spaces we show a property of global exponential stability. This property
together with the local exact controllability ensures for the first time in the
literature that long waves in nonlinear dispersive media are globally exactly
controllable in large time. Precisely, our analysis relies strongly on the
bilinear estimates using the Fourier restriction spaces in two different
dispersions that will guarantee a global control result for coupled systems of
the Korteweg-de Vries type. This result, of independent interest in the area of
control of coupled dispersive systems, provides a necessary first step for the
study of global control properties to the coupled dispersive systems in
periodic domains.
