In this paper, a new framework for the tracking of closed curves and their associated motion fields is described. The proposed method enables a continuous tracking along an image sequence of both a deformable curve and its velocity field. Such an approach is formalized through the minimization of a global spatio-temporal continuous cost functional, w.r.t a set of variables representing the curve and its related motion field. The resulting minimization process relies on optimal control approach and consists in a forward integration of an evolution law followed by a backward integration of an adjoint evolution model. This latter pde includes a term related to the discrepancy between the current estimation of the state variable and discrete noisy measurements of the system. The closed curves are represented through implicit surface modeling, whereas the motion is described either by a vector field or through vorticity and divergence maps depending on the kind of targeted applications. The efficiency of the approach is demonstrated on two types of image sequences showing deformable objects and fluid motions.