Physics-Informed Pontryagin Neural Networks for Path-Constrained Optimal Control Problems

Solving constrained optimal control problems (OCPs) is essential to ensure safety in real world scenarios. Recent machine learning techniques have shown promise in addressing OCPs. This paper introduces a novel methodology for solving OCPs with path constraints using Physics-Informed Neural Networks (PINNs). Specifically, Pontryagin Neural Networks (PoNNs), which solve the boundary value problem arising from the indirect method and Pontryagin Minimum Principle (PMP), are extended to handle path constraints. In this new formulation, pathconstraints are incorporated into the Hamiltonian through additional Lagrange multipliers, which are treated as optimization variables. The complementary slackness conditions are enforced by ensuring the zero value of the Fischer-Burmeister function within the loss functions to be minimized. This approach adds minimal complexity to the original PoNN framework, as it avoids the need for continuation methods, penalty functions, or additional differential equations, which are often required in traditional methods to solve path-constrained OCPs via the indirect method. Numerical results for a benchmark OCP and a fixed-time energy-optimal rendezvous with various path constraints demonstrate the effectiveness of the proposed method in solving path-constrained OCPs.