Traffic Control and Monitoring with a Macroscopic Model in the Presence of Strong Congestion Waves

This paper treats the problem of designing controllers and estimators for freeway management applications using macroscopic models and optimal control. The model is a nonlinear conservation law that may develop and propagate discontinuities known as shock waves and thus prevent the use of variational techniques requiring regularity. We show in this paper how a valid first variation of the model can be computed, give an explicit formula of its solution and use it to evaluate gradients of optimization problems using the adjoint method. The coordinated ramp metering problem and the state estimation problem are treated as illustrations.