Epsilon-Trig Regularization Method for Bang-Bang Optimal Control Problems

Bang-bang control problems have numerical issues due to discontinuities in the control structure and require smoothing when using optimal control theory that relies on derivatives. Traditional smooth regularization introduces a small error into the original problem using error controls and an error parameter to enable the construction of accurate smoothed solutions. When path constraints are introduced into the problem, the traditional smooth regularization fails to bound the error controls involved. It also introduces a dimensional inconsistency related to the error parameter. Moreover, the traditional approach solves for the error controls separately, which makes the problem formulation complicated for a large number of error controls. The proposed Epsilon-Trig regularization method was developed to address these issues by using trigonometric functions to impose implicit bounds on the controls. The system of state equations is modified such that the smoothed control is expressed in sine form, and only one of the state equations contains an error control in cosine form. Since the Epsilon-Trig method has an error parameter only in one state equation, there is no dimensional inconsistency. Moreover, the Epsilon-Trig method only requires the solution to one control, which greatly simplifies the problem formulation. Its simplicity and improved capability over the traditional smooth regularization method for a wide variety of problems including the Goddard rocket problem have been discussed in this study.