Increasing Autonomy of Aerospace Systems via PINN-based Solutions of HJB Equation

Closed-loop optimal control is crucial for enhancing the autonomy of aerospace systems. However, its computation can be challenging, as it typically involves solving the Hamilton-Jacobi-Bellman (HJB) equation—a nonlinear partial differential equation (PDE) that poses significant numerical difficulties. This paper focuses on employing Bellman Neural Networks (BeNNs), a specialized framework within Physics-Informed Neural Networks (PINNs), to learn the solution of the HJB PDE and thereby ascertain the closed-loop optimal control. BeNNs leverage the constrained expressions from the Theory of Functional Connections and utilize shallow neural networks, trained via the Extreme Learning Machine (X-TFC) approach, to approximate the elusive solution of the HJB PDE. We achieve the solution to the nonlinear HJB by integrating the method of successive approximation with the solution of the linear Generalized HJB (GHJB) equation. The effectiveness of these frameworks is evaluated in the context of a missile pitch-plane autopilot optimal control problem. The results demonstrate that our framework can accurately compute the closed-loop optimal control within the specified domain, achieving low final errors relative to the reference states.