Numerical solution of delay fractional optimal control problems with free terminal time

This paper considers a class of delay fractional optimal control problems with free terminal time. The fractional derivatives in this class of problems are described in the Caputo sense and they can be of different orders. We first show that for this class of problems, the well-known time-scaling transformation for mapping the free time horizon into a fixed time interval yields a new fractional-order system with variable time-delay. Then, we propose an explicit numerical scheme for solving the resulting fractional time-delay system, which gives rise to a discrete-time optimal control problem. Furthermore, we derive gradient formulas of the cost and constraint functions with respect to decision variables. On this basis, a gradient-based optimization approach is developed to solve the resulting discrete-time optimal control problem. Finally, an example problems is solved to demonstrate the effectiveness of our proposed solution approach.