A variational-geometric approach for the optimal control of nonholonomic systems

TL;DR

This paper introduces a geometric framework using Lie algebroids and Hamiltonian systems to derive necessary conditions for optimal control of nonholonomic systems, unifying and extending previous approaches.

Contribution

It provides a novel geometric interpretation and a unifying formalism for analyzing optimal control problems with nonholonomic constraints.

Findings

Derived necessary conditions as solutions to constrained variational problems.

Unified the analysis of normal extremals using Lie algebroid theory.

Extended the framework to include previously unconsidered cases.

Abstract

Necessary conditions for existence of normal extremals in optimal control of systems subject to nonholonomic constraints are derived as solutions of a constrained second order variational problems. In this work, a geometric interpretation of the derivation is studied from the theory of Lie algebroids. We employ such a framework to describe the problem into a unifying formalism for normal extremals in optimal control of nonholonomic systems and including situations that have not been considered before in the literature from this perspective. We show that necessary conditions for existence of extremals in the optimal control problem can be also determined by a Hamiltonian system on the cotangent bundle of a skew-symmetric algebroid.