Localized Lane Guidance for a Bidirectional Pedestrian Cyclic Corridor: Equilibria and Generic Actuator Placement

This paper studies bidirectional pedestrian flow on a cyclic network of corridors. Each corridor consists of two lanes, while pedestrians may enter or leave the network at junctions. The dynamics are based on a 1-D, bidirectional, two-lane Lighthill-Whitham-Richards-type model, in which lane changes are described through explicit, density-dependent transfer rates that favor motion toward the less congested lane. We propose a localized lane-guidance control input that biases lane-changing behavior without creating pedestrians or reversing their walking direction. Such guidance may represent, for example, recommendations provided by staff or information screens. Under meaningful assumptions on the external inflows, we prove the existence and uniqueness of an admissible uniform equilibrium. We then linearize the system around this equilibrium and study the placement of localized guidance zones through a spectral detectability condition. We prove that, for any prescribed decay rate, two guidance patches located in a single, suitably chosen corridor are sufficient to satisfy the spectral detectability condition for all critical adjoint eigenmodes. This condition is necessary for the exponential stabilizability of the linearized system. Numerical experiments illustrate the spectral properties of the model and study the robustness of the modal observability condition with respect to the geometry and distribution of the guidance patches.