OnL1-minimization in optimal control and applications to robotics

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. Copyright © 2006 John Wiley & Sons, Ltd.