Maximum-Hands-Off Control and L1 Optimality

TL;DR

This paper introduces maximum-hands-off control, a sparse control paradigm, and demonstrates its equivalence with L1-optimal control, providing methods for both sparse and continuous control solutions with practical examples.

Contribution

It establishes the theoretical link between maximum-hands-off control and L1 optimality, and proposes an L1/L2 approach for continuous control implementation.

Findings

L1-optimal control yields maximum-hands-off control solutions.

Maximum-hands-off control often exhibits a bang-off-bang structure.

L1/L2-optimal control produces continuous sparse controls.

Abstract

In this article, we propose a new paradigm of control, called a maximum-hands-off control. A hands-off control is defined as a control that has a much shorter support than the horizon length. The maximum-hands-off control is the minimum-support (or sparsest) control among all admissible controls. We first prove that a solution to an L1-optimal control problem gives a maximum-hands-off control, and vice versa. This result rationalizes the use of L1 optimality in computing a maximum-hands-off control. The solution has in general the "bang-off-bang" property, and hence the control may be discontinuous. We then propose an L1/L2-optimal control to obtain a continuous hands-off control. Examples are shown to illustrate the effectiveness of the proposed control method.