Strict abnormal extremals in nonholonomic and kinematic control systems

TL;DR

This paper investigates the nature of abnormal extremals in nonholonomic and kinematic control systems, establishing conditions for their relation and illustrating their behavior through an example.

Contribution

It provides conditions linking optimal control problems for mechanical and kinematic systems and analyzes the properties of abnormal extremals in these contexts.

Findings

Conditions to relate mechanical and kinematic control problems.

Pontryagin's Maximum Principle applied to abnormal extremals.

Example illustrating abnormal solutions in nonholonomic systems.

Abstract

In optimal control problems, there exist different kinds of extremals, that is, curves candidates to be solution: abnormal, normal and strictly abnormal. The key point for this classification is how those extremals depend on the cost function. We focus on control systems such as nonholonomic control mechanical systems and the associated kinematic systems as long as they are equivalent. With all this in mind, first we study conditions to relate an optimal control problem for the mechanical system with another one for the associated kinematic system. Then, Pontryagin's Maximum Principle will be used to connect the abnormal extremals of both optimal control problems. An example is given to glimpse what the abnormal solutions for kinematic systems become when they are considered as extremals to the optimal control problem for the corresponding nonholonomic mechanical systems.