  We consider the dispersive Degasperis-Procesi equation $u_t-u_{x x
t}-\mathtt{c} u_{xxx}+4 \mathtt{c} u_x-u u_{xxx}-3 u_x u_{xx}+4 u u_x=0$ with
$\mathtt{c}\neq 0$. In \cite{Deg} the authors proved that this equation
possesses infinitely many conserved quantities. We prove that, in a
neighborhood of the origin, there are infinitely many of such constants of
motion which control the Sobolev norms and which are analytic in a neighborhood
of the origin of some $H^s$ Sobolev space, both on $\mathbb{R}$ and
$\mathbb{T}$. By the analysis of these conserved quantities we deduce a result
of global well-posedness for solutions with small initial data and we show
that, on the circle, the formal Birkhoff normal form of the Degasperis-Procesi
at any order is action-preserving.
