  The local and global control results for a general higher-order KdV-type
operator posed on the unit circle are presented. Using spectral analysis, we
are able to prove local results, that is, the equation is locally controllable
and exponentially stable. To extend the local results to the global one we
captured the smoothing properties of the Bourgain spaces, the so-called
propagation of singularities, which are proved with a new perspective. These
propagation, together with the Strichartz estimates, are the key to extending
the local control properties to the global one, precisely, higher-order
KdV-type equations are globally controllable and exponentially stabilizable in
the Sobolev space $H^{s}(\mathbb{T})$ for any $s \geq 0$. Our results recover
previous results in the literature for the KdV and Kawahara equations and
extend, for a general higher-order operator of KdV-type, the Strichartz
estimates as well as the propagation results, which are the main novelties of
this work.
