Pontryagin Maximum Principle - a generalization

TL;DR

This paper extends the Pontryagin maximum principle to almost Lie algebroids, providing a unified geometric framework that generalizes classical optimal control results and includes problem reduction schemes.

Contribution

It introduces a generalization of PMP to AL algebroids, integrating homotopy concepts and unifying reduced and unreduced control problem formulations.

Findings

Extended PMP to AL algebroids

Unified reduced and unreduced problem formalism

Framework based on homotopy and AL algebroid geometry

Abstract

The fundamental theorem of the theory of optimal control, the Pontryagin maximum principle (PMP), is extended to the setting of almost Lie (AL) algebroids, geometrical objects generalizing Lie algebroids. This formulation of the PMP yields, in particular, a scheme comprising reductions of optimal control problems similar to the reduction for the rigid body in analytical mechanics. On the other hand, in the presented approach the reduced and unreduced PMPs are parts of the same universal formalism. The framework is based on a very general concept of homotopy of measurable paths and the geometry of AL algebroids.