L2 Norm-Based Control Regularization for Solving Optimal Control Problems

Solutions to practical optimal control problems (OCPs) may consist of control profiles that switch between control limits or assume values interior to their admissible set, either due to activation of inequality state path constraints or existence of singular control arcs. Abrupt switches in the control (i.e., bang-bang control) jeopardizes the numerical solution of OCPs unless care is taken to isolate precise time transition points where sharp switches occur (excluding the chattering phenomenon). We propose a novel control regularization method, called Bang-Bang Singular Regularization (BBSR), based on L2 norm-based regularization. We present an analysis on the L2 norm-based regularization at two levels: 1) its connection to trigonometric regularization and 2) its ability to approximate regular and singular control arcs. The utility of the method is demonstrated in solving three classes of trajectory optimization problems: 1) space minimum-fuel low-thrust trajectories with bang-bang thrust profiles, 2) the Goddard rocket problem with its known bang-singular-bang control structure, and 3) minimum-time spacecraft reorientation with both bang-bang and second-order singular arcs. The results demonstrate the utility of the BBSR method in approximating extremal control profiles that may consist of pure regular and/or mixed regular and singular control arcs.