This talk presents three complementary perspectives on the interplay between the theory of Hamilton--Jacobi (HJ) equations and modern computational approaches to optimal control and reinforcement learning. First, we examine the stability and convergence properties of value functions arising from Lipschitz-constrained control problems, and interpret them through the lens of viscosity solutions to Hamilton--Jacobi--Bellman (HJB) equations, providing a theoretical foundation for continuous-time reinforcement learning. Second, we explore the eradication time problem in controlled epidemic models, where the minimum-time solution emerges as the viscosity solution to a static HJB equation. This structure naturally lends itself to a physics-informed neural network (PINN) framework, enabling mesh-free approximation of both the value function and optimal bang-bang control. Lastly, we introduce a DeepONet-based policy iteration method that integrates operator learning with the Hamilton--Jacobi formulation to solve high-dimensional control problems efficiently, even in the absence of discretization. Through these case studies, we illustrate how HJ theory serves as a unifying backbone that connects control-theoretic objectives with modern machine learning methodologies.