A simple 'finite approximations' proof of the Pontryagin maximum principle under reduced differentiability hypotheses

Traditional proofs of the Pontryagin maximum principle (PMP) require the continuous differentiability of the dynamics with respect to the state variable on a neighbourhood of the minimizing state trajectory, when arbitrary values of the control variable are inserted into the dynamic equations. Recently, Sussmann has drawn attention to the fact that the PMP remains valid when the dynamics are differentiable with respect to the state variable, merely when the minimizing control is inserted into the dynamic equations. This weakening of earlier hypotheses has been referred to as the Lojasiewicz refinement. Besides, it suffices to hypothesize that the dynamics are differentiable with respect to the state variable merely along the minimizing state trajectory. We show that these extensions of early versions of the PMP can be simply proved by finite approximations, application of a Lagrange multiplier rule in finite dimensions and passage to the limit. Furthermore, our analysis requires that the minimizer in question is merely a Pontryagin local minimizer, a weaker notion of 'local minimizer' than has previously been considered, in connection with these extensions.