When are minimizing controls also minimizing relaxed controls?

Relaxation refers to the procedure of enlarging the domain of a variational problem or the search space for the solution of a set of equations, to guarantee the existence of solutions. In optimal control theory relaxation involves replacing the set of permissible velocities in the dynamic constraint by its convex hull. Usually the infimum cost is the same for the original optimal control problem and its relaxation. But it is possible that the relaxed infimum cost is strictly less than the infimum cost. It is important to identify such situations, because then we can no longer study the infimum cost by solving the relaxed problem and evaluating the cost of the relaxed minimizer. Following on from earlier work by Warga, we explore the relation between the existence of an infimum gap and abnormality of necessary conditions (i.e. they are valid with the cost multiplier set to zero). Two kinds of theorems are proved. One asserts that a local minimizer, which is not also a relaxed minimizer, satisfies an abnormal form of the Pontryagin Maximum Principle. The other asserts that a local relaxed minimizer that is not also a minimizer satisfies an abnormal form of the relaxed Pontryagin Maximum Principle.