The Pontryagin maximum principle: the constancy of the Hamiltonian

The maximum principle was proved by Pontryagin using the assumption that the controls involved were measurable and bounded functions of time. However in many applications the optimal control is piecewise continuous and bounded. Thus it is natural to try to consturct a proof assuming that the admissible controls lie in the class of piecewise continuous bounded functions of t. This turns out to involve some subtle difficulties, and several of the proofs currently in the literature are defective. Here we address one of these difficulties: that of showing that, for an optimal control, the Hamiltonian is constant for all values of t. This result is a crucial one for anyone wishing to solve problems of optimal control. It is by no means a simple corollary of the maximum property. What a number of books actually establish is that the Hamiltonian is piecewise constant, taking a fixed value between the discontinuities of the optimal control. We supply a continuity argument which resolves this difficulty.