A Hard-Constrained PMP-Based Warm-Start Framework for Nonlinear Optimal Control Using Physics-Informed Learning

Indirect methods based on Pontryagin’s Maximum Principle (PMP) offer theoretical rigor for nonlinear optimal control but suffer from extreme sensitivity to costate initialization. Physics-Informed Neural Networks (PINNs) provide a promising data-free approach to globally approximate trajectories and overcome this initialization barrier. However, they often lack strict numerical precision due to their reliance on soft penalty constraints. To bridge this gap, this paper proposes a hybrid framework that synergizes the global search capability of a structurally modified PINN with the rigorous precision of high-order Chebyshev–Gauss–Lobatto (CGL) spectral discretization. Within this framework, we first introduce a novel neural architecture that enforces the PMP stationarity condition as a hard constraint by analytically eliminating control inputs via costates, thereby reducing the optimization search space and ensuring strict optimality during training. The neural-generated trajectories subsequently provide a high-quality warm start for a CGL pseudospectral solver, transforming the problem into a single-shot convex quadratic programming formulation. Numerical experiments on the Van der Pol oscillator and elliptic PDE optimal control problems demonstrate that this strategy effectively mitigates the initialization sensitivity of indirect methods. The results show that the proposed method achieves superior accuracy and convergence stability compared to standalone PINN solvers, providing a robust initialization-free approach for complex nonlinear optimal control.