Characterization of maximum hands-off control

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 L0-(semi)norm, and also provides sufficient conditions for the optimality of such controls. Numerical example illustrates that adopting an L0 cost leads to a sparse control, whereas an L1-relaxation in singular problems leads to a non-sparse solution.