Hamilton Jacobi Bellman

The Hamilton-Jacobi-Bellman (HJB) equation is a fundamental tool in optimal control theory, used to find optimal control strategies for dynamic systems by solving for an optimal value function. Current research focuses on developing efficient numerical methods to solve HJB equations, particularly for high-dimensional systems, employing techniques like deep learning (including neural SDEs and physics-informed neural networks), kernel methods, and tensor train approximations to mitigate the "curse of dimensionality." These advancements are significantly impacting fields like reinforcement learning, economics, and robotics by enabling the design of more effective and scalable control algorithms for complex, real-world applications.