Generalized Lie Algebroids - Examples by Distinguished Lie Algebroids with Applications to Optimal Control

TL;DR

This paper explores generalized Lie algebroids, demonstrating their distinction from traditional Lie algebroids, correcting a misconception about a key theorem, and applying the framework to an optimal control problem that standard Lie algebroids cannot solve.

Contribution

It establishes the generalized Lie algebroid as a significant example, corrects a previous theorem, and shows its applicability to complex optimal control problems.

Findings

Generalized Lie algebroids are a distinguished class within Lie algebroids.

Theorem 3.1 from prior work is invalid due to misconception.

Generalized Lie algebroids can solve certain optimal control problems beyond traditional Lie algebroid capabilities.

Abstract

We will prove that the generalized Lie algebroid is a distinguished example by Lie algebroid. The generality of it with respect to the Lie algebroid is similar with the generality of the pull-back vector bundle with respect to the vector bundle. Next, we will prove that the proof of Theorem 3.1 from [15] is a misconception and the mentioned Theorem has no validity. Finally, we anatomize an optimal control problem solvable in the generalized Lie algebroid framework whereas Lie algebroid instrumentation can not solve it.