Stability of a Traffic Flow Model with Nonconvex Relaxation

Li, Tong; Liu, Hailiang · 2005 · OpenAlex-citations

DOI: 10.4310/cms.2005.v3.n2.a1

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Summary

This paper investigates the nonlinear stability of traveling wave solutions for a quasi-linear relaxation model characterized by a nonconvex equilibrium flux. The research is motivated by the Payne-Whitham (PW) dynamic continuum traffic flow model, which incorporates a nonconcave fundamental diagram. Such nonconcave diagrams are observed in real-world traffic and are necessary for modeling complex phenomena like stop-and-go waves. The study aims to establish the asymptotic stability of traveling waves under the sub-characteristic condition, a requirement for linear stability, using a weighted energy method adapted from previous work on semi-linear systems. The authors analyze the system $u_t + v_x = 0$ and $v_t + g(u)_x = -(1/\epsilon)(v - f(u))$, where $f$ is the nonconvex equilibrium flux. They demonstrate the existence of unique traveling wave solutions connecting adjacent equilibria, satisfying the Rankine-Hugoniot and entropy conditions. The core of the analysis involves reformulating the problem in terms of perturbations to these traveling waves and applying a weighted $L^2$ energy method. A specific weight function is chosen based on the decay rates of the traveling wave at infinity to handle the nonlinearity of $g$ and the nonconvexity of $f$. The proof relies on establishing a priori estimates for the perturbations, ensuring global existence and convergence to the traveling wave profile. The main results distinguish between two cases based on the decay behavior of the traveling wave. In the non-degenerate case, where the wave exhibits exponential decay at infinity, the traveling wave is asymptotically stable against small initial perturbations in the Sobolev space $H^2$. In the degenerate case, where the wave exhibits algebraic decay, stability is established against small perturbations in the weighted space $H^2 \cap L^2_{\langle x \rangle_+}$. In both scenarios, the solution converges to the traveling wave as time approaches infinity, provided the initial disturbances are sufficiently small. These findings provide a rigorous mathematical foundation for the stability of traffic flow models with nonconcave fundamental diagrams. By proving stability under the sub-characteristic condition, the paper validates the use of the PW model for analyzing stable traffic regimes. The results also imply that convergence rates can be further explored if initial perturbations possess additional spatial decay. This work extends previous stability analyses, which were largely limited to convex fluxes, to the more complex and realistic nonconvex setting relevant to traffic engineering.

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