This study introduces a framework for computing optimal control inputs by integrating physics-informed deep learning with classical optimal control theory. The governing differential equations and terminal constraints are formulated via the Euler-Lagrange framework and incorporated into physics-informed neural networks, thereby enforcing adherence to the underlying dynamics and boundary conditions. In addition, a DeepONet-based architecture is utilized to estimate the terminal state corresponding to a given control input. The performance of the proposed framework is validated through its application to optimal impact time and impact angle guidance problems in missile engagement scenarios, considering cases both with and without aerodynamic drag. Numerical results demonstrate that the proposed method reliably identifies optimal solutions across a wide range of initial and terminal conditions, thereby offering insight into the structure of the optimal solution space and its potential usefulness for system design. When the training loss is adequately reduced, the solutions obtained are consistent with those computed using GPOPS, exhibiting only negligible discrepancies. Furthermore, the proposed methodology is readily extendable to a broader class of optimal control problems in aerospace engineering.
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