Asian Research Journal of Mathematics

This study develops a mathematical model of a saturated section of the RN1 in Mbanza-Ngungu, applies the Hilbert Uniqueness Method (HUM) to regulate congestion, and validates the approach numerically using finite elements in MATLAB.

Based on 15 observation sessions, traffic is modelled with the macroscopic LWR approach through a linearized advection-diffusion equation with a source term. The point source of the car park, initially modelled by a Dirac mass, is regularized into a Gaussian , S_ε∈L² (Ω), thus making the model compatible with the functional framework of the HUM. The control, constructed via the adjoint operator on ω=[400,600]×[2,6]m, is applied to the equation discretized by P1 finite elements and an implicit Euler scheme (Δt = 0.1 s).

The final adjoint datum is obtained by inverting the regularized HUM operator with a conjugate gradient scheme (64 iterations, relative residual below 10⁻⁶). The simulations show that the localized control reduces the root-mean-square deviation from the target state from 21.3 to 2.5 veh·km⁻¹ in 60 seconds, i.e. an 88% reduction of the initial deviation, the residual deviation representing only 5.3% of the target density (2.5/47.0), and lowers the density peak from 94.2 to 78.7 veh·km⁻¹ compared with the uncontrolled evolution. It removes 10.6 vehicles from the segment, i.e. a mean extraction rate of 636 veh·h⁻¹. The adjoint solution ψ reveals a strong spatial sensitivity around the car park, confirming the relevance of the chosen control zone. The result obtained is one of approximate controllability, inherent to the parabolic nature of the linearized model, and the constructed control is subject to no sign or amplitude constraint: the extraction rate it requires exceeds the observed through-traffic flow (291 veh·h⁻¹), which currently limits its operational implementation.