On open-loop and feedback attainability of a closed set for nonlinear control systems

In this note, we investigate the-existence of controls which allow to reach a given closed set K through trajectories of a nonlinear control, system. In, the case where the set is sufficiently regular we give. a condition allowing to find a feedback control law which ensures the existence of trajectories to reach the set. We also consider the case where all the trajectories reach K. When K is not necessarily attainable but only viable, we build a set-valued feedback for which the set is invariant. Our approach concerns continuous dynamics possibly not C-1 so our methods do not come from geometric control theory. Furthermore, we do not require any regularity of the set K. in order to obtain our results, except when we want to establish the existence of a feedback control law to achieve our goals. 2002 Elsevier Science (USA). All rights reserved.