10th U6+ Consortium of African Universities International Conference

Rapid urbanization and increasing vehicle populations have intensified traffic congestion, leading to substantial economic losses, environmental degradation, and reduced mobility. Understanding and mitigating these challenges require mathematically rigorous models capable of capturing the nonlinear interactions that govern traffic dynamics. Although nonlinear differential equation models have successfully described phenomena such as shock waves, stop-and-go oscillations, and phase transitions, their analytical complexity and limited integration with advanced control strategies continue to constrain their practical application. This study proposes an integrated mathematical framework that combines optimal control theory and perturbation techniques to investigate unidirectional traffic flow governed by nonlinear differential equations. The research will formulate a nonlinear traffic flow model that accurately represents the evolution of traffic density and velocity, derive optimal control strategies to minimize congestion while enhancing traffic stability and efficiency, and employ perturbation methods to obtain reliable analytical approximations where exact solutions are intractable. Qualitative and stability analyses will further be undertaken to examine the dynamical behaviour of the controlled system and evaluate the effectiveness of the proposed control interventions. The anticipated outcome is a robust analytical framework that advances the application of nonlinear differential equations in transportation systems while demonstrating the complementary strengths of optimal control and perturbation techniques. The study is expected to contribute to the fields of applied mathematics and intelligent transportation by providing theoretical insights and practical strategies for improving traffic flow and supporting evidence-based traffic management.