In this paper (vector valued) bandlimited functions have been used as the class of admissible controls. Essential properties of the class of admissible controls are developed. Properties of attainable sets and the set of trajectories of the dynamical system under consideration are presented. These results are proved without the convexity condition usually imposed on the ‘ velocity field ’ of the dynamical system. In the final section several interesting optimal control problems are solved using the compactness properties of attainable sots and admissible trajectories.