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We consider a partial differential equation - ordinary differential equation system to describe the dynamics of traffic flow with autonomous vehicles. In the model, the bulk flow of human drivers is represented by a scalar conservation law, while each autonomous vehicle is described by an ordinary differential equation. The coupled PDE-ODE model is introduced, and existence of solutions for this model is shown, along with a proposed algorithm to construct approximate solutions. Next, we propose a control strategy for the speeds of the autonomous vehicles to minimize the average fuel consumption of the entire traffic flow. Existence of solutions for the optimal control problem is proved, and we numerically show that a reduction in average fuel consumption is possible with an AV acting as a moving bottleneck.

We address a problem of stabilization of a scalar conservation laws with fixed flux pointwiseconstraints. The PDE models the impact of two toll gates on the traffic flow and the feedbacklaws is designed to limit the number of cars passing through the two toll gates. The existenceof solutions is proved using a Schauder’s fixed point theorem, together with a coupled PDEODE. To prove stabilization results, we study the structure of the solutions using an extensionof backward generalized characteristics.

In this paper, we study the problem of initial data identification for the one-dimensional Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the time-irreversibility of the Burgers equation, some target functions are unattainable from solutions of this equation, making the inverse problem under consideration ill-posed. To get around this issue, we introduce an non-smooth optimization problem which consists in minimizing the difference between the predictions of the Burgers equation and the observations of the system at a final time in L^2(R) norm. The two main contributions of this work are as follows. -We fully characterize the set of minimizers of the aforementioned non-smooth optimization problem. -A wave-front tracking method is implemented to construct numerically all of them. One of minimizers is the backward entropy solution, constructed using a backward-forward method.

Autonomous vehicles (AV) offer new avenues for transportation applications and call for a new understanding of traffic dynamics when both regular cars and AV coexist. Many works looked at this question with a microscopic approach where both the AVs and the regular cars are modelled as ODEs. Here, present a second order model describing the interaction between a macroscopic traffic flow described by PDEs and autonomous vehicles (AV) described by ODEs. This model is inspired by recent development on moving bottlenecks for scalar conservation laws and is intrinsically multiscale. We give an analysis of this model and we show the existence of weak solutions with bounded variations.

In this paper, we study the problem of identification for the one-dimensional Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the property of non-backward uniqueness of Burgers equation, there may exist multiple initial data leading to the same given target. In recent works, a fully characterization of the set of initial data leading to a given target is given using the classical Lax-Hopf formula. In this note, an alternative proof based only on generalized backward characteristics is given. This leads to the hope of investigate systems of conservation laws in one dimension where the classical Lax-Hopf formula doesn't hold anymore. Moreover, numerical illustrations are presented using as a target, a function optimized for minimum pressure rise in the context of sonic-boom minimization problems. All of initial data leading to this given target are constructed using a wave-front tracking algorithm.

International audience; This paper introduces a new sliding mode methodology for a system of two hyperbolic equations. The main difference with the existing literature is the definition of the sliding variable, given here by the gradient of a Lyapunov functional. We state and prove an existence theorem and a global asymptotic stability result. The efficiency of our feedback-law is illustrated by some numerical simulations relying on implicit schemes.