Constrained Control of PDE Traffic Flow via Spatial Control Barrier Functions

In this paper, a constrained control approach to variable speed limit (VSL) control for macroscopic partial differential equations (PDE) traffic models is developed. Control Lyapunov function (CLF) theory for ordinary differential equations (ODE) is extended to account for spatially and temporally varying states and control inputs. The stabilizing CLF is then unified with safety constraints through the introduction of spatially varying control barrier functions (sCBF). These methods are applied to in-domain VSL control of the Lighthill-Whitham-Richards (LWR) model to regulate traffic density to a desired profile while ensuring the density remains below prescribed limits enforced by the sCBF. Results show that incorporating constrained control minimally affects the stabilizing control input while successfully maintaining the density with the defined safe set.