## Abstract

This article proposes a new approach for computing a semiexplicit form of the solution to a class of traffic flow problems encoded by a Hamilton-Jacobi (HJ) partial differential equation (PDE), with time-switched Hamiltonian. Using a characterization of the problem derived from viability theory, we show that the solution associated with the problem can be formulated as a minimization problem involving the trajectory of an auxiliary dynamical system. A generalized Lax-Hopf formula for the switched Hamiltonian problem is derived, which enables us to compute the solution associated with affine initial or boundary conditions as a linear program involving the control function of the auxiliary dynamical system. This formulation allows us to compute the solution to the original problem exactly, unlike dynamic programming methods. In addition, this method allows one to very efficiently recompute the boundary conditions associated with an initial condition problem, allowing large-scale variable speed limit traffic control problems to be solved. Note to Practitioners - Most dynamic speed limit control techniques used to manage traffic flow on highways rely on discretizations of partial differential equations, which require one to compute the solution on a computational grid. This article focuses on an alternate solution method that does not require the solution to be found on all grid points, potentially saving computational time on large-scale problems.

Original language | English (US) |
---|---|

Pages (from-to) | 473-485 |

Number of pages | 13 |

Journal | IEEE Transactions on Automation Science and Engineering |

Volume | 19 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2022 |

### Bibliographical note

Funding Information:This work was supported by the Center for Advanced Multimodal Mobility Solutions and Education (CAMMSE) and the National Science Foundation under Grant 1636154 and Grant 1739964.

Publisher Copyright:

© 2004-2012 IEEE.

## Keywords

- Linear programming
- speed control
- switched fundamental diagrams