Physics-Informed Deep Learning Approach to Solve Optimal Control Problem

The proposed method aims to construct optimal control solutions using physics-informed deep learning and optimal control theory. The methodology involves deriving differential equations and final constraints from the Euler-Lagrange equations. Physics-informed neural networks are then designed to match these differential equations and conditions. Additionally, a physics-informed deepONet is introduced to predict the final state when the control input is provided. The effectiveness of the proposed method is demonstrated through the application of the optimal impact time guidance problem. The method can find optimal solutions even with varying initial and final conditions. The obtained results are compared with those from a parametric optimization solver, and despite not using any optimal control solution data, the difference between them is relatively small. An important characteristic of the proposed method is that the neural networks are time-dependent and pre-trained, meaning that the solution is a continuous function of time and conditions and can be applied in real-time.