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Green Waves I: A PDE Approach to Homogenization of Lines of Traffic Lights (Preliminary version)

  • Ali Assi
  • Nicolas Forcadel
  • Regis Monneau
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Abstract

The paper is concerned with the homogenization of first-order, time-dependent Hamilton- Jacobi equations on a one-dimensional spatial domain with infinitely many junctions located at integer points. The Hamiltonian may be non-convex and exhibits discontinuities both in time and space: the set of time discontinuities may be dense in time, while the spatial discontinuities occur at infinitely many junctions that are discrete in space. This configu- ration arises naturally for example in the modeling of a infinite number of trafic lights on a single road and creates significant analytical challenges, in particular due to the absence of a standard viscosity solution theory adapted to this setting. To overcome this difficulty, we introduce a new notion of viscosity solution, called Lipschitz lattice viscosity solution, for which we prove a comparison principle. The Lipschitz regularity plays a crucial role in this definition and we then have to prove, under suitable structural assumptions, global Lipschitz estimates for the solutions.

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Cite

BibTeXRIS
BibTeX
@misc{assi2026green,
  title = {{Green Waves I: A PDE Approach to Homogenization of Lines of Traffic Lights (Preliminary version)}},
  author = {Ali Assi and Nicolas Forcadel and Regis Monneau},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  url = {https://hal.science/hal-05837643v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Green Waves I: A PDE Approach to Homogenization of Lines of Traffic Lights (Preliminary version)
AU  - Ali Assi
AU  - Nicolas Forcadel
AU  - Regis Monneau
PY  - 2026
DA  - 2026/10/23
PB  - HAL
UR  - https://hal.science/hal-05837643v1
AB  - The paper is concerned with the homogenization of first-order, time-dependent Hamilton- Jacobi equations on a one-dimensional spatial domain with infinitely many junctions located at integer points. The Hamiltonian may be non-convex and exhibits discontinuities both in time and space: the set of time discontinuities may be dense in time, while the spatial discontinuities occur at infinitely many junctions that are discrete in space. This configu- ration arises naturally for example in the modeling of a infinite number of trafic lights on a single road and creates significant analytical challenges, in particular due to the absence of a standard viscosity solution theory adapted to this setting. To overcome this difficulty, we introduce a new notion of viscosity solution, called Lipschitz lattice viscosity solution, for which we prove a comparison principle. The Lipschitz regularity plays a crucial role in this definition and we then have to prove, under suitable structural assumptions, global Lipschitz estimates for the solutions.
ER  -