Namkyeong Cho

Namkyeong Cho, Yeoneung Kim

We address the crucial yet underexplored stability properties of the Hamilton--Jacobi--Bellman (HJB) equation in model-free reinforcement learning contexts, specifically for Lipschitz continuous optimal control problems. We bridge the gap between Lipschitz continuous optimal control problems and classical optimal control problems in the viscosity solutions framework, offering new insights into the stability of the value function of Lipschitz continuous optimal control problems. By introducing structural assumptions on the dynamics and reward functions, we further study the rate of convergence of value functions. Moreover, we introduce a generalized framework for Lipschitz continuous control problems that incorporates the original problem and leverage it to propose a new HJB-based reinforcement learning algorithm. The stability properties and performance of the proposed method are tested with well-known benchmark examples in comparison with existing approaches.

Yeongjong Kim, Yeoneung Kim, Minseok Kim et al.

We propose a physics-informed neural network policy iteration (PINN-PI) framework for solving stochastic optimal control problems governed by second-order Hamilton--Jacobi--Bellman (HJB) equations. At each iteration, a neural network is trained to approximate the value function by minimizing the residual of a linear PDE induced by a fixed policy. This linear structure enables systematic $L^2$ error control at each policy evaluation step, and allows us to derive explicit Lipschitz-type bounds that quantify how value gradient errors propagate to the policy updates. This interpretability provides a theoretical basis for evaluating policy quality during training. Our method extends recent deterministic PINN-based approaches to stochastic settings, inheriting the global exponential convergence guarantees of classical policy iteration under mild conditions. We demonstrate the effectiveness of our method on several benchmark problems, including stochastic cartpole, pendulum problems and high-dimensional linear quadratic regulation (LQR) problems in up to 10D.