Deterministic and Fractional-Order Analysis of Urban Traffic Congestion With Optimal Control

Urban traffic congestion is still a huge problem because of the convoluted way that infrastructure, changing flow conditions, and driver behavior all work together. This research is aimed at developing a mathematical model that effectively studies and manages congestion through memory-dependent dynamics. The authors propose a fractional-order compartmental model that classifies drivers into four distinct categories: susceptible, irregular, jammed, and regulated. The Atangana–Baleanu fractional derivative is used in the model to account for nonlinear effects and long-term dependencies. The traffic congestion influence number (TCIN) is a new way to measure the likelihood that traffic will get stuck. Analytical studies show the existence of equilibrium states, positivity, and boundedness. Stability analyses show that congestion decreases as the congestion number falls below 1. Bifurcation analysis is used to identify important changes, such as transcritical, forward, reverse, and Hopf bifurcations. These bifurcations can reveal fast, oscillatory patterns of congestion. Using numerical simulations, the memory dependency in congestion dynamics is captured, and insights into the evolution of traffic congestion patterns are gained using the fractional-order framework. Based on the modeled conditions, adaptive regulation and signal optimization appear capable of alleviating congestion. The proposed framework provides a theoretical framework for traffic-management strategies and estimates their possible impact on urban congestion dynamics. The study primarily involves a theoretical and computational analysis, using analytical results to complement numerical simulations to examine congestion dynamics and congestion control mechanisms.