Optima and equilibria for a model of traffic flow.

*(English)*Zbl 1236.90024The authors considers the problem of the efficient control of road traffic flow. As a model for the traffic flow, the Lighthill-Whitham model is used, where the density of cars is described by a scalar conservation law and the speed of cars depends on the traffic density.

The paper starts with the study of the optimal control problem to find a control function, such that the overall traffic cost is minimized. The overall traffic cost is given as the integral of the individual cost of all drivers.

The authors prove the existence of an optimal control function that solves this problem. This control function is a departure rate that counts how many drivers enter the highway per unit time. To prove the existence, the authors use the Lax formulas for the representation of the unique entropy-admissible solution to a Cauchy problem that they obtain by switching the roles of the time and space variables. The authors show that this optimal control is continuous and generates a continuous state for which also the backwards dynamics is uniquely defined. They also provide an explicit description of the optimal control.

If every driver is free to choose the time to start to drive, a different traffic flow will be generated that corresponds to the Nash equilibrium solution where no driver can decrease her individual cost by changing her own departure time. The authors prove the existence of a Nash equilibrium. For a particular velocity function, they provide a method to explicitly compute this solution.

For the Nash equilibrium the overall traffic cost is in general higher than the optimal cost that can be obtained with the control constructed in the first part of the paper. To decrease the overall traffic cost, as an incentive to change the drivers behaviour time-dependent toll pricing can be used. The time-dependent toll price is introduced as an additional term in the individual cost function of the drivers. If the desired revenue is sufficiently large, the corresponding Nash equilibrium solution with the modified cost that includes the toll price is generated in such a way that the overall traffic cost that occurs if each driver behaves according to the corresponding Nash equilibrium solution is equal to the minimal cost for the optimal control problem from the first part of the paper. To obtain the corresponding optimal toll pricing strategy, the optimal control function obtained in the first part of the paper is used.

The paper starts with the study of the optimal control problem to find a control function, such that the overall traffic cost is minimized. The overall traffic cost is given as the integral of the individual cost of all drivers.

The authors prove the existence of an optimal control function that solves this problem. This control function is a departure rate that counts how many drivers enter the highway per unit time. To prove the existence, the authors use the Lax formulas for the representation of the unique entropy-admissible solution to a Cauchy problem that they obtain by switching the roles of the time and space variables. The authors show that this optimal control is continuous and generates a continuous state for which also the backwards dynamics is uniquely defined. They also provide an explicit description of the optimal control.

If every driver is free to choose the time to start to drive, a different traffic flow will be generated that corresponds to the Nash equilibrium solution where no driver can decrease her individual cost by changing her own departure time. The authors prove the existence of a Nash equilibrium. For a particular velocity function, they provide a method to explicitly compute this solution.

For the Nash equilibrium the overall traffic cost is in general higher than the optimal cost that can be obtained with the control constructed in the first part of the paper. To decrease the overall traffic cost, as an incentive to change the drivers behaviour time-dependent toll pricing can be used. The time-dependent toll price is introduced as an additional term in the individual cost function of the drivers. If the desired revenue is sufficiently large, the corresponding Nash equilibrium solution with the modified cost that includes the toll price is generated in such a way that the overall traffic cost that occurs if each driver behaves according to the corresponding Nash equilibrium solution is equal to the minimal cost for the optimal control problem from the first part of the paper. To obtain the corresponding optimal toll pricing strategy, the optimal control function obtained in the first part of the paper is used.

Reviewer: Martin Gugat (Erlangen)

##### MSC:

90B20 | Traffic problems in operations research |

91A10 | Noncooperative games |

35L65 | Hyperbolic conservation laws |

49K20 | Optimality conditions for problems involving partial differential equations |