In this article, we investigate theoretical properties of the time-optimal hands-off control for linear time invariant systems. The purpose of the control is to maximize the time duration on which the control value is exactly zero (maximum hands-off control) and also to minimize the response time to achieve a given state transition (time-optimal control). For this, we introduce a cost function described by a linear combination of the L-0 measure and the response time of the control. Since the L-0 measure is non-convex and discontinuous, we adopt the L-1 relaxation technique for the analysis of the optimal control. By using this relaxation, we show the existence of the time-optimal hands-off control, and the equivalence between L-0 and L-1 optimal controls under the normality assumption. (C) 2018 Elsevier Ltd. All rights reserved.