Structure Identification Method for Nonsmooth and Singular Optimal Control Problems

This work introduces a method developed for identifying the structure of nonsmooth and singular optimal control problems using jump function approximations and the Hamiltonian of the system. The structure identification procedure is paired with a regularization method to solve the problem using a multiple-domain Legendre-Gauss-Radau (LGR) collocation. The method is divided into several parts. First, the structure detection method described identifies switch times in the control and analyzes the corresponding switching function for segments where the solution is either bang-bang or singular. Second, after the structure has been detected, the domain is decomposed into multiple domains such that the multiple-domain formulation includes additional decision variables that represent the switch times in the optimal control. In domains classified as bang-bang, the control is set to either its upper or lower limit. In domains identified as singular, a regularization procedure is employed. The method is demonstrated on an example with a finite and infinite-order singular arc. The results demonstrate that the method of this paper can accurately identify the control structure of a complex optimal control problem. Furthermore, when the structure identification method is paired with a strategy for solving singular optimal control problems accurate solutions are obtained. The results are compared against a previously developed mesh refinement method that is not tailored for solving nonsmooth and/or singular optimal control problems.