  Maximum hands-off control aims to maximize the length of time over which zero
actuator values are applied to a system when executing specified control tasks.
To tackle such problems, recent literature has investigated optimal control
problems which penalize the size of the support of the control function and
thereby lead to desired sparsity properties. This article gives the exact set
of necessary conditions for a maximum hands-off optimal control problem using
an $L_0$-(semi)norm, and also provides sufficient conditions for the optimality
of such controls. Numerical example illustrates that adopting an $L_0$ cost
leads to a sparse control, whereas an $L_1$-relaxation in singular problems
leads to a non-sparse solution.
