### Abstract

In the steady-state assignment model where each link has a non-decreasing cost flow curve we have monotonicity not just at the link level but also at the route level. In our dynamical system we assume that the users swap to cheaper routes. Monotonicity of the route cost function is enough to guarantee that the given function V, detailed later, is in fact a Lyapunov function and hence that the system converges to equilibrium.

In the dynamic assignment model, the route cost function is not a monotone function of route flow, as was shown in [1]. Therefore convergence does not immediately follow, as it does in the steady-state case. This paper essentially shows that the dynamic counterpart of the steady-state Lyapunov function is in fact not a Lyapunov function. This does not at all imply non-convergence of the dynamical system simulating a swap to cheaper routes, but it does raise the question of convergence. Obviously, if another function could be found that satisfies the criteria of being a Lyapunov function this would be sufficient for convergence.

Original language | English |
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Title of host publication | 2001 IEEE Intelligent Transportation Systems - Proceedings |

Place of Publication | New York |

Publisher | IEEE Press |

Pages | 569-572 |

Number of pages | 4 |

ISBN (Print) | 0-7803-7194-1 |

DOIs | |

Publication status | Published - 2001 |

Event | IEEE Intelligent Transportation Systems Conference (ITSC'01) - Oakland, Canada Duration: 25 Aug 2001 → 29 Aug 2001 |

### Conference

Conference | IEEE Intelligent Transportation Systems Conference (ITSC'01) |
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Country | Canada |

Period | 25/08/01 → 29/08/01 |

### Cite this

*2001 IEEE Intelligent Transportation Systems - Proceedings*(pp. 569-572). New York: IEEE Press. https://doi.org/10.1109/ITSC.2001.948722

**Non-convergence in dynamic assignment networks?** / Mounce, R.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*2001 IEEE Intelligent Transportation Systems - Proceedings.*IEEE Press, New York, pp. 569-572, IEEE Intelligent Transportation Systems Conference (ITSC'01), Canada, 25/08/01. https://doi.org/10.1109/ITSC.2001.948722

}

TY - GEN

T1 - Non-convergence in dynamic assignment networks?

AU - Mounce, R

PY - 2001

Y1 - 2001

N2 - In the steady-state assignment model where each link has a non-decreasing cost flow curve we have monotonicity not just at the link level but also at the route level. In our dynamical system we assume that the users swap to cheaper routes. Monotonicity of the route cost function is enough to guarantee that the given function V, detailed later, is in fact a Lyapunov function and hence that the system converges to equilibrium.In the dynamic assignment model, the route cost function is not a monotone function of route flow, as was shown in [1]. Therefore convergence does not immediately follow, as it does in the steady-state case. This paper essentially shows that the dynamic counterpart of the steady-state Lyapunov function is in fact not a Lyapunov function. This does not at all imply non-convergence of the dynamical system simulating a swap to cheaper routes, but it does raise the question of convergence. Obviously, if another function could be found that satisfies the criteria of being a Lyapunov function this would be sufficient for convergence.

AB - In the steady-state assignment model where each link has a non-decreasing cost flow curve we have monotonicity not just at the link level but also at the route level. In our dynamical system we assume that the users swap to cheaper routes. Monotonicity of the route cost function is enough to guarantee that the given function V, detailed later, is in fact a Lyapunov function and hence that the system converges to equilibrium.In the dynamic assignment model, the route cost function is not a monotone function of route flow, as was shown in [1]. Therefore convergence does not immediately follow, as it does in the steady-state case. This paper essentially shows that the dynamic counterpart of the steady-state Lyapunov function is in fact not a Lyapunov function. This does not at all imply non-convergence of the dynamical system simulating a swap to cheaper routes, but it does raise the question of convergence. Obviously, if another function could be found that satisfies the criteria of being a Lyapunov function this would be sufficient for convergence.

U2 - 10.1109/ITSC.2001.948722

DO - 10.1109/ITSC.2001.948722

M3 - Conference contribution

SN - 0-7803-7194-1

SP - 569

EP - 572

BT - 2001 IEEE Intelligent Transportation Systems - Proceedings

PB - IEEE Press

CY - New York

ER -