In this paper, we analyse the optimality of broken Pontryagin extremal for an n-dimensional affine control system with a control parameter, taking values in a k-dimensional closed ball. We prove the optimality of broken normal extremals in many cases that include (but not are exhausted by) the cases of the involutive driftless part of the system for any n > k and of the contact driftless part for n = 3 and k = 2.
