# Optimal control on distributions

## TL;DR

This paper explores optimal control problems on nonholonomic manifolds, analyzing various formulations and proving that such systems can be controlled with bang-bang controls in single or multiple time settings.

## Contribution

It introduces a comprehensive analysis of optimal control on nonholonomic manifolds, including infinitesimal deformations, adjointness, and control strategies for complex functionals.

## Key findings

- Nonholonomic systems can be controlled by bang-bang controls in single or bi-temporal settings.
- Analysis of infinitesimal deformations and adjointness in control problems.
- Extension to multitime optimal control problems with various functional types.

## Abstract

This paper studies (single-time and multitime) optimal control problems on a nonholonomic manifold (described either by the kernel of a Gibbs-Pfaff form or by the span of appropriate vector fields). For both descriptions we analyse: infinitesimal deformations and adjointness, single-time optimal control problems, multitime optimal control problem of maximizing a multiple integral functional, multitime optimal control problem of maximizing a curvilinear integral functional, Curvilinear functionals depending on curves, optimization of mechanical work on Riemannian manifolds. Also we prove that a nonholonomic system can be always controlled by uni-temporal or bi-temporal bang-bang controls.