LyZNet with Control: Physics-Informed Neural Network Control of Nonlinear Systems with Formal Guarantees

Optimal control for high-dimensional nonlinear systems remains a fundamental challenge. One bottleneck is that classical approaches for solving the Hamilton-Jacobi-Bellman (HJB) equation suffer from the curse of dimensionality. Recently, physics -informed neural networks have demonstrated potential in overcoming the curse of dimensionality in solving certain classes of PDEs, including special cases of HJB equations. However, one perceived limitation of neural networks is their lack of formal guarantees in the solutions they provide. To address this issue, we have built LyZNet, a Python tool that combines physics-informed learning with formal verification. The previous version of the tool demonstrated the capability for stability analysis and region of attraction estimates. In this paper, we present the tool for solving optimal control problems. We expand the functionalities of the tool to support the formulation and solving of optimal control problems for control-affine systems via physics-informed neural network policy iteration (PINN-PI). We outline the methodology that enables the learning and verification of PINN for optimal stabilization tasks. We demonstrate with a classical control example that the learned optimal controller indeed has significantly improved performance and verifiable regions of attraction. Copyright (c) 2024 The Authors.