Vanishing of the costate in pontryagin's maximum principle and singular time optimal controls

We provide examples of time and norm optimal controls that satisfy Pontryagin's maximum principle in an interval 0 less than or equal to t less than or equal to T but with a costate that vanishes in 0 less than or equal to t less than or equal to T - delta with delta < T. A refinement of this construction produces time optimal controls which do not satisfy the maximum principle, even in weak form. On the positive side, we show that when we drive to zero the costate is nonzero in the whole control interval.