On stability of one-dimensional Hughes' dynamics with affine costs

TL;DR

This paper analyzes the stability of the one-dimensional Hughes model with affine costs, focusing on entropy solutions with a discontinuous flux and a density-dependent turning curve $\xi(t)$. By restricting to well-separated solutions, it proves uniqueness and $L^1$-continuous dependence on both the initial data and the cost parameter $\alpha$, leveraging an open-end formulation and BV-regularity arguments; the turning curve acquires $W^{1,\infty}$ regularity under affine costs, enabling a decoupled fixed-point analysis. The study complements existence results for affine costs by establishing stability and continuous dependence, highlighting the role of BV regularity and the necessity of the well-separated hypothesis for uniqueness. Numerical experiments using a deterministic microscopic approximation demonstrate that the evacuation time can be highly sensitive to $\alpha$ when turning-curve interactions are present, even though the underlying density may remain stable in $L^1$, underscoring a practical instability in a key engineering metric. Overall, the work clarifies stability regimes for the Hughes model and informs both analytical techniques for discontinuous-flux problems and interpretable numerical diagnostics in crowd evacuation scenarios.

Abstract

We investigate stability issues for the one-dimensional variant of the celebrated Hughes model for pedestrian evacuation. The cost function is assumed to be affine, which is a setting where existence of solutions with BV loc in space regularity, away from the so-called turning curve, was recently established. We provide a uniqueness result for solutions having the special property that agents never cross the turning curve (which implies that they are BV globally). In the same setting, continuous dependence of solutions on the cost parameter is highlighted. On the other hand, numerical simulations using the manyparticle approximation of the model, with more general initial conditions that allow the support of the solutions to intersect the turning curve, demonstrate the strong sensitivity of the evacuation time to the same cost parameter; this instability arises from interactions between agents and the turning curve.

Figures (4)

• Figure 1: $T_{\rm mic}$ as function of $\alpha$ with initial datum \ref{['num_in_con']}. Left: $\alpha \in [0,20]$, step size $\Delta \alpha=0.1$. Right: $\alpha \in [10,14]$, step size $\Delta \alpha=0.05$.
• Figure 2: $T_{\rm mic}$ as function of $\delta$ with $\Delta \delta = 0.01$ and initial datum \ref{['num_in_con_2']}. Left: $\delta \in [0,1]$. Right: $\delta \in [0,0.1]$. Top: $\alpha=1$. Bottom: $\alpha=12.7$.
• Figure 3: $T_{\rm mic}$ as function of $\delta$ with $\Delta \delta = 0.01$, initial datum \ref{['num_in_con_4']} and $\alpha=12.7$.
• Figure 4: Explanation of the discontinuities at $\delta = 0.1$ and $\delta=0.26$ of $T_{\rm mic}$ plotted in Figure \ref{['fig:epsilon']}. We select values of $\delta$ around $\delta=0.1$ (top) and $\delta=0.26$ (bottom). For $\delta=0.08$ no intersection between particles path (blue lines in the online version) and the turning curve (magenta line in the online version) occurs, whereas intersections are observed for $\delta=0.12$. A similar explanation applies to the case of $\delta=0.26$.

Theorems & Definitions (14)

• Definition 2.1: Open-end formulation
• Theorem 2.2: Existence for affine costs
• Definition 3.1: Well-separated solution
• Theorem 3.2: Stability of well-separated solutions
• Lemma 3.3
• Proposition 3.4
• Proposition 3.5
• Proposition 3.6
• Corollary 3.7
• proof