Hamilton-Jacobi-Bellman Control of Traffic Networks with Contextual Information

Transportation networks are increasingly exposed to contextual disturbances such as weather events, traffic incidents, and demand surges. Existing traffic management pipelines typically adopt a decoupled predict-then-optimize architecture, where contextual information is first used to forecast network parameters and the resulting predictions are subsequently incorporated into optimization models. While computationally convenient, this separation fails to integrate contextual signals directly into the continuous-time dynamics governing traffic flow. This paper develops a continuous-time optimal control framework in which contextual information dynamically modifies the congestion structure of the transportation network. Contextual variables are modeled as bounded signals that deform the network's macroscopic capacity landscape. Within this formulation, the traffic management problem is characterized by a contextual Hamilton-Jacobi-Bellman (HJB) equation that yields an optimal feedback policy mapping traffic states and contextual observations directly to operational control actions. We establish theoretical guarantees showing that the HJB value function acts as a control Lyapunov function for the closed-loop network dynamics. Under bounded contextual disturbances, the resulting policy guarantees bounded queue growth and asymptotic throughput recovery. To address the computational challenges of solving high-dimensional HJB equations, the value function is approximated using physics-informed neural networks (PINNs). Numerical experiments on a two-region urban network governed by macroscopic fundamental diagrams demonstrate that the contextual HJB controller mitigates congestion amplification and restores network throughput more rapidly than reactive control and predict-then-optimize baselines.