Image assimilation methods are nowadays widely used to retrieve motion from image sequences with heuristics on the underlying dynamics. A mathematical model on the temporal evolution of the motion field has to be chosen, according to these heuristics, that approximately describes the evolution of the velocity at a pixel over the sequence. In order to quantify this approximation, we add an error term in the evolution equation of the motion field and design a weak formulation of 4D-Var image assimilation. The designed cost function simultaneously depends on the initial motion field and on the error value at each time step. The BFGS solver performs minimization to retrieve both motion field and errors. The method is evaluated and quantified on twin experiments, as no ground truth would be available for real data. The results demonstrate that the motion field is better estimated thanks to the error control.