Bellman Neural Networks for the Class of Optimal Control Problems With Integral Quadratic Cost

This article introduces Bellman neural networks (BeNNs) and employs them to learn the optimal control actions for the class of optimal control problems (OCPs) with integral quadratic cost. BeNNs represent a particular family of physics-informed neural networks specifically designed and trained to tackle OCPs via applying the Bellman principle of optimality (BPO). The BPO provides necessary and sufficient optimality conditions, which result in a nonlinear partial differential equation known as the Hamilton–Jacobi–Bellman (HJB) equation. BeNNs learn the optimal control actions from the unknown solution of the arising HJB equation (i.e., the value function), where the unknown solution is modeled using a neural network. In addition, this article shows how to estimate the upper bounds on the generalization error of BeNNs while learning the solutions for the OCP class under consideration. The generalization error estimate is provided in terms of the choice and number of the training points as well as the training error. Numerical studies show that BeNNs can be successfully applied to learn the feedback control actions for the class of OCPs considered and, after the training is completed, deployed to control the system in a closed-loop fashion.