Upper bound analysis of smoothing parameter in the smoothing method for minimum-fuel low-thrust trajectory optimization

The optimal control law for minimum-fuel low-thrust trajectories exhibits Bang-Bang characteristics, leading to high numerical sensitivity and convergence difficulties in trajectory optimization problems. The smoothing method introduces a smoothing parameter to approximate the Bang-Bang control with a continuous optimal control law, enhancing the convergence probability of the trajectory optimization problem. By gradually reducing the smoothing parameter from a larger positive value to zero through numerical continuation, the minimum-fuel solution to the original problem can be obtained. However, in some cases, if the starting value of the smoothing parameter exceeds a certain upper bound, the corresponding problem cannot achieve convergence, hindering the subsequent numerical continuation process. To address this issue, this work employs the indirect method to model the minimum-fuel low-thrust trajectory optimization problem as a two-point boundary value problem. We introduce four common smoothing forms for Bang-Bang control and conduct an in-depth analysis of the upper bound of the smoothing parameter. By studying the impact of the smoothing parameter on the control law, the relationship between the upper bound of the smoothing parameter and the minimum thrust magnitude required to reach the given terminal state is established. Based on the minimum thrust magnitude, criteria for determining the existence of an upper bound is provided, and formulas for calculating the upper bounds in four smoothing forms are derived, providing a reference for selecting the starting value of the smoothing parameter.