Strict sense minimizers which are relaxed extended minimizers in general optimal control problems

We provide a sufficient condition to avoid Lavrentiev phenomena, namely the occurrence of a gap between the infimum cost of a general optimal control problem subject to endpoint and state constraints and an extension of it. This condition is expressed in terms of a nonsmooth constrained version of the Pontryagin Maximum Principle. In particular, we prove that, in the absence of abnormal feasible extremals (as customary, abnormal means that the cost multiplier is zero), any process which is locally optimal in the set of original (strict sense) processes must also be locally optimal in a larger set of relaxed extended processes. Our relaxed extended problem includes as special cases both the classical extension by convex relaxation of the velocities set and the impulsive extension of control-polynomial systems with unbounded controls.