Group adaptive direct optimization for switching time in control parameterization

Traditional control parameterization methods are effective for solving general constrained optimal control problems, but the predefined control switching times limit their flexibility. The sequential time-scaling transformation technique addresses this by introducing a new control function to indirectly optimize switching times. However, as the number of controls increases, this method becomes increasingly complex and significantly alters the problem structure. To overcome these challenges, we propose a group-adaptive direct optimization method for switching times. Our results show that the previously perceived shortcomings of direct optimization can be effectively mitigated, leading to higher computational efficiency compared to the sequential time-scaling transformation. We derive gradient calculation formulas using the variational and co-state methods and validate the effectiveness of our approach through numerical examples.