On an Extended Notion of Controllability

The notion of controllability has been already extended to the case when the trajectories approach the target asymptotically. Accordingly, we give a sufficient first order condition for the continuity of the Value Function V corresponding to an integral cost with a possibly zero nonnegative Lagrangian l. The fact that l is allowed to be zero outside the target makes the continuity of V possible even in situations where Petrov's inwards pointing field condition is not verified