---
abstract: 'This paper presents analyses for the maximum hands-off control using the geometric methods developed for the theory of turnpike in optimal control. First, a sufficient condition is proved for the existence of the maximum hands-off control for linear time-invariant systems with arbitrarily fixed initial and terminal points using the relation with $L^1$ optimal control. Next, a sufficient condition is derived for the maximum hands-off control to have the turnpike property, which may be useful for approximate design of the control.'
author:
- 'Noboru Sakamoto${}^{1,\ast}$ and Masaaki Nagahara${}^{2,\S}$[^1][^2][^3][^4]'
bibliography:
- 'biblio/Main.bib'
- 'biblio/OptimalControl.bib'
title: '**The turnpike property in the maximum hands-off control** '
---

Conclusions
===========

In this paper, we considered the maximum hands-off control problem, which attracts much attention from the viewpoints of solving environmental problems [@Nagahara:16:ieeetac], using the geometric analysis method developed for the theory of turnpike in optimal control. Using the equivalence of maximum hands-off control and $L^1$ optimal control under certain hypotheses, the existence of the maximum hands-off control for linear time-invariant systems is proved. Using the invariant manifold theory, it has been shown that the turnpike phenomenon appears in the maximum hands-off control under the conditions of normality and spectral conditions. The result may be useful from the fact that the occurrence of turnpike often leads to simplification of optimal control design.

[^1]: 1: Faculty of Science and Engineering, Nanzan University, Yamazato-cho 18, Showa-ku, Nagoya, 464-8673, Japan

[^2]: 2:Institute of Environmental Science and Technology, The University of Kitakyushu, Hibikino 1-1, Wakamatsu-ku, Kitakyushu, Fukuoka, 808-0135, Japan

[^3]: $\ast$: Supported, in part, by JSPS KAKENHI Grant Numbers JP26289128, JP19K04446 and by Nanzan University Pache Research Subsidy I-A-2 for 2020 academic year. [E-mail: noboru.sakamoto@nanzan-u.ac.jp]{}

[^4]: §: This work has been partially funded by JSPS KAKENHI Grant Number 19H02301. [E-mail:nagahara@ieee.org]{}
