Boundary Control of Traffic Density Waves Using a Markov-Switching Stochastic Parabolic PDE Model

Traffic flow control is essential for mitigating congestion and improving road efficiency. Traffic density waves characterize spatiotemporal variations in vehicle density and provide a useful tool for analyzing macroscopic traffic dynamics. In this article, boundary control of traffic density waves is investigated using a Markov-switching stochastic parabolic partial differential equation (MSSPDE) model. The model captures the macroscopic evolution of traffic density across multiple traffic conditions, including free flow, mild congestion, and severe congestion. Both partially unknown and fully known transition probabilities are considered. A boundary feedback controller is designed on the basis of the outlet boundary density and the spatial integral of traffic density. Stochastic stability analysis is performed for the closed-loop MSSPDE system. Sufficient conditions are derived for mean-square exponential stability (MSES) and robust MSES under parametric uncertainties. An H_∞ boundary control problem is also addressed to guarantee a prescribed disturbance attenuation level over a finite time horizon. Numerical simulations and quantitative comparisons demonstrate the effectiveness and practical relevance of the proposed control designs.