  Maximum hands-off control is a control that has the minimum L0 norm among all
feasible controls. It is known that the maximum hands-off (or L0-optimal)
control problem is equivalent to the L1-optimal control under the assumption of
normality. In this article, we analyze the maximum hands-off control for linear
time-invariant systems without the normality assumption. For this purpose, we
introduce the Lp-optimal control with 0<p<1, which is a natural relaxation of
the L0 problem. By using this, we investigate the existence and the
bang-off-bang property (i.e. the control takes values of 1, 0 and -1) of the
maximum hands-off control. We then describe a general relation between the
maximum hands-off control and the L1-optimal control. We also prove the
continuity and convexity property of the value function, which plays an
important role to prove the stability when the (finite-horizon) control is
extended to model predictive control.
