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Exploiting our previous results on higher order controlled Lagrangians in[Nonlinear Anal. {\bf 207} (2021), 112263], we derive here an analogue of theclassical first order Pontryagin Maximum Principle (PMP) for cost minimisingproblems subjected to higher order differential constraints $\frac{d^kx^j}{dt^k} = f^j\big(t, x(t), \frac{d x}{dt}(t), \ldots, \frac{d^{k-1}x}{dt^{k-1}}(t), u(t)\big)$, $t \in [0,T]$, where $u(t)$ is a control curve ina compact set $K \subset \mathbb R^m$. This result and its proof can beconsidered as a detailed illustration of one of the claims of that previouspaper, namely that the results of that paper, originally established in asmooth differential geometric framework, yield directly properties holdingunder much weaker and more common assumptions. In addition, for furtherclarifying our motivations, in the last section we display a couple of quickindications on how the two-step approach of this paper (i.e., a preliminaryeasy-to-get differential geometric discussion followed by a refining analysisto weaken the regularity assumptions) might be fruitfully exploited also in thecontext of control problems governed by partial differential equations or instudies on the dynamics of controlled mechanical systems.

On the Pontryagin Maximum Principle under differential constraints of higher order