In this paper, we investigate the existence of controls which allow to reach a given target through trajectories of a nonlinear control system in the case of a non exactly known initial state. For doing this, we use the key concept of weakly invariant tubes and we give a new compactness property for weakly invariant tubes with values in a prescribed collection of sets. We give some consequences of this property on the minimal time function to reach the target, and we prove a sufficient condition for the attainability of the target by weakly invariant tubes of the considered system. If the attainability property is not satisfied, we characterize a subcollection of initial sets from which the attainability property holds true, and we provide an algorithm to compute it.
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