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Stabilisation of stochastic single-file dynamics using port-Hamiltonian systems

Julia Ackermann, Matthias Ehrhardt, Thomas Kruse, Antoine Tordeux

Abstract

This study revisits a recently proposed symmetric port-Hamiltonian single-file model in one dimension. The uniform streaming solutions are stable in the deterministic model. However, the introduction of white noise into the dynamics causes the model to exhibit divergence. In response, we introduce a control term in a port-Hamiltonian framework. Our results show that this control term effectively stabilises the dynamics even in the presence of noise, providing valuable insights for the control of road traffic flows.

Stabilisation of stochastic single-file dynamics using port-Hamiltonian systems

Abstract

This study revisits a recently proposed symmetric port-Hamiltonian single-file model in one dimension. The uniform streaming solutions are stable in the deterministic model. However, the introduction of white noise into the dynamics causes the model to exhibit divergence. In response, we introduce a control term in a port-Hamiltonian framework. Our results show that this control term effectively stabilises the dynamics even in the presence of noise, providing valuable insights for the control of road traffic flows.
Paper Structure (9 sections, 4 theorems, 44 equations, 2 figures)

This paper contains 9 sections, 4 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Port-Hamiltonian single-file models
  3. Notations.
  4. Initial stochastic motion model
  5. Extended stochastic motion model
  6. Port-Hamiltonian formulation
  7. Long-time behaviour
  8. Simulation results
  9. Conclusion

Key Result

Proposition 3

Let $Z(t)=(Q(t),p(t))^\top\in\mathbb{R}^{2N}$, $t\in[0,\infty)$. Then the dynamics of the periodic system eq:modn2 are given by with $\mathbf{1}=(1,\ldots,1)\in\mathbb{R}^N$ and the Hamiltonian operator $H\colon \mathbb{R}^{2N}\to\mathbb{R}$, The matrix $J$ is skew-symmetric by $N\times N$ block, while $R$ is symmetric positive semi-definite.

Figures (2)

  • Figure 1: Illustration of the single-file motion system with periodic boundary conditions. Here, $q_n$ represents the curvilinear position, $Q_n = q_{n+1} - q_n$ is the distance to the right neighbour and $p_n$ denotes the velocity of the $n$-th vehicle.
  • Figure 2: Trajectories of 10 vehicles along a segment of length $L=501$ with periodic boundaries with the dynamics \ref{['eq:modn2']} with $\gamma=0$ (the ensemble's mean velocity diverges, upper panel) and with the pHS with $\gamma=0.1$ and $\gamma=1$ and the controlled input velocity $u=0$ (middle and lower panels).

Theorems & Definitions (14)

  • Remark 1
  • Remark 2
  • Proposition 3
  • proof
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Proposition 9
  • ...and 4 more