Method for solving bang-bang and singular optimal control problems using adaptive Radau collocation

A method is developed for solving bang-bang and singular optimal control problems using adaptive Legendre-Gauss-Radau collocation. The method is divided into several parts. First, a structure detection method is developed that 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, the objective function is augmented with a regularization term to avoid the singular arc. An iterative procedure is then developed for singular domains to obtain a control that lies in close proximity to the singular control. The method is demonstrated on four examples, three of which have either a bang-bang and/or singular optimal control while the fourth has a smooth and nonsingular optimal control. The results demonstrate that the method of this paper provides accurate solutions to problems whose solutions are either bang-bang or singular when compared against previously developed mesh refinement methods that are not tailored for solving nonsmooth and/or singular optimal control problems, and produces results that are equivalent to those obtained using previously developed mesh refinement methods for optimal control problems whose solutions are smooth.

Automated summary

A method is developed using adaptive Legendre-Gauss-Radau collocation to solve bang-bang and singular optimal control problems. The method includes steps such as structure detection, domain decomposition, and iterative procedure. The results demonstrate that this method provides accurate solutions to nonsmooth and/or singular optimal control problems, and equivalent results for optimal control problems with smooth solutions, compared to previously developed mesh refinement methods.