On the Pontryagin Maximum Principle under differential constraints of higher order

TL;DR

This paper extends the Pontryagin Maximum Principle to higher order differential constraints, providing a detailed proof and illustrating its applicability under weaker assumptions than traditional geometric frameworks.

Contribution

It derives a higher order PMP under less restrictive conditions, building on previous geometric results and demonstrating the approach's versatility for complex control systems.

Findings

Established a higher order PMP for differential constraints of arbitrary order.

Showed the approach's applicability under weaker regularity assumptions.

Suggested potential extensions to PDE control problems and mechanical systems.

Abstract

Exploiting our previous results on higher order controlled Lagrangians in [Nonlinear Anal. {\bf 207} (2021), 112263], we derive here an analogue of the classical first order Pontryagin Maximum Principle (PMP) for cost minimising problems subjected to 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 that 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, for further clarifying 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 studies on the dynamics of controlled mechanical systems.