On the Pontryagin Maximum Principle under differential constraints of higher order
Volume 130 / 2023
Abstract
Exploiting our previous results on higher order controlled Lagrangians [Nonlinear Anal. 207 (2021), art. 112263], we derive an analogue of the classical first order Pontryagin Maximum Principle (PMP) for cost minimising problems with higher order differential constraints $\frac {d^k x^j}{dt^k} = f^j\big (t, x(t), \frac {d x}{dt}(t), \ldots , \frac {d^{k-1} x}{dt^{k-1}}(t), u(t)\big )$, $t \in [0,T]$, where $u(t)$ is a control curve in a compact set $K \subset \mathbb {R}^m$. This result and its proof can be considered as a detailed illustration of one of the claims of the previous paper, namely that the results of that paper, originally established in a smooth differential-geometric framework, yield directly properties holding under much weaker and more common assumptions. In addition, to further clarify our motivations, in the last section we display a couple of quick indications on how the two-step approach of this paper (i.e., a preliminary easy-to-get differential geometric discussion followed by a refining analysis to weaken the regularity assumptions) might be fruitfully exploited also in the context of control problems governed by partial differential equations or in the dynamics of controlled mechanical systems.
