In this paper, we analyze optimal control problems with control variables appearing linearly in the dynamics. We discuss different cost functionals involving the Lp-norm of the control. The case p = 0 represents the time-optimal control, the case p > 1 yields a standard smooth optimal control problem, whereas the case p = 1 leads to a nonsmooth cost functional. Several techniques are developed to deal with the nonsmooth case p = 1. We present a thorough theoretical discussion of the necessary conditions. Two types of numerical methods are developed: either a regularization technique is used or an augmentation approach is applied in which the number of control variables is doubled. We show the precise relations between the L1-minimal control and the bang-bang or singular controls in the augmented problem. Using second-order sufficient conditions (SSC) for bang-bang controls, we obtain SSC for L1-minimal controls. The different techniques and results are illustrated with an example of the optimal control for a free-flying robot which is taken from Sakawa.
