Maximum hands-off control and L1 optimality

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 L
¹
-optimal control problem gives a maximum hands-off control, and vice versa. This result rationalizes the use of L
¹
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 L
¹
/L
²
-optimal control to obtain a continuous hands-off control. Examples are shown to illustrate the effectiveness of the proposed control method.