In this paper, event-based boundary controls to stabi- lize a linear hyperbolic system of **conservation** **laws** have been designed. The analysis of global exponential stabil- ity is based on Lyapunov techniques. Moreover, we have proved that under the two event-based stabilization ap- proaches, the solution to the closed-loop system exists and is unique. This paper might be considered as the first contribution to event-based control of Hyperbolic PDEs, and complements the work of [13] and [31] on sampled data control of parabolic PDEs and on event-based con- trol of parabolic PDEs, respectively.

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Robust feedforward boundary control of hyperbolic **conservation** **laws** Xavier Litrico, Vincent Fromion and Gérard Scorletti
Abstract— The paper proposes a feedforward boundary con-
trol to reject measured disturbances for systems modelled by hyperbolic partial differential equations obtained from conser- vation **laws**. The controller design is based on frequency domain methods. Perfect rejection of measured perturbations at one boundary is obtained by controlling the other boundary. This result is then extended to design robust open-loop controller when the model of the system is not perfectly known, e.g. in high frequencies. Frequency domain comparisons and time-domain simulations illustrates the good performance of the feedforward boundary controller.

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assumption on the data, hence there is nothing to be generalized at this point.
In order to deduce the uniqueness and the continuous dependence on the data for (EvPb) with L ∞ data, we
use the property of finite speed of propagation. Indeed, let u be a G-entropy solution of (EvPb) with some L ∞
datum. Firstly, applying the result of [22] (for **conservation** **laws** in Ω l and Ω r ) we readily see that the solution is uniquely defined by the datum outside the triangle T := {(t, x) | t ∈ (0, T ], |x| ≤ Lt} where L = L 0 + 1

1 Introduction
The main difficulty in the numerical solution of systems of **conservation** **laws** is the complexity of con- structing the Riemann solvers. One way to overcome this difficulty is to consider centered schemes as in [15, 18, 22, 23, 3]. However, in general these schemes are more diffusive than Godunov type methods based on exact or approximate Riemann solvers when this alternative is available. Therefore in this paper we will consider Godunov type methods. Most often the numerical solution requires the calculation of eigenvalues or eigenvectors of the Jacobian matrix of the system. This is even more complicated when the system is non-strictly hyperbolic, i.e. eigenvectors are not linearly independent. In this paper we present a new approach which do not require such eigenvalue and eigenvector calculations.

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Fractional spaces and **conservation** **laws** Pierre CASTELLI, Pierre-Emmanuel JABIN, St´ephane JUNCA
Abstract In 1994, Lions, Perthame and Tadmor conjectured the maximal smoothing effect for multidimensional scalar **conservation** **laws** in Sobolev spaces. For strictly smooth convex flux and the one-dimensional case we detail the proof of this conjec- ture in the framework of Sobolev fractional spaces W s,1 , and in fractional BV spaces: BV s . The BV s smoothing effect is more precise and optimal. It implies the optimal

Stability estimates for non-local scalar **conservation** **laws**
Felisia Angela Chiarello 1 Paola Goatin 1 Elena Rossi 1
Abstract
We prove the stability of entropy weak solutions of a class of scalar **conservation** **laws** with non-local flux arising in traffic modelling. We obtain an estimate of the dependence of the solution with respect to the kernel function, the speed and the initial datum. Stability is obtained from the entropy condition through doubling of variable technique. We finally provide some numerical simulations illustrating the dependencies above for some cost functionals derived from traffic flow applications.

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to the Riemann Problem (2.4) computed at time, say, t = 1.
As it is usual when dealing with nonclassical scalar **conservation** **laws**, see [ 21 , Chapter II], we need first to introduce the auxiliary function ψ, see Figure 2, left. Let ψ(R) = R and, for ρ 6= R, let ψ(ρ) be such that the straight line through ρ, q(ρ) and ψ(ρ), q ◦ ψ(ρ) is tangent to the graph of q at ψ(ρ), q ◦ ψ(ρ). Besides, for ¯ ρ ∈ [0, R T [ the line through ¯ ρ, q(¯ ρ)

5 Conclusion
In this work, we have derived new symmetric hyperbolic systems of **conservation** **laws** to model viscoelastic flows with Upper-Convected Maxwell fluids, either 3D compressible or 2D incompressible with hydrostatic pressure and a free surface. The systems yield the first well-posedness results for causal multi-dimensional viscoelastic motions satisfying the locality principle (i.e. information propagates at finite-speed) as small-time smooth solutions to Cauchy initial-value problems. The systems also suggest a promising route to unify models for solid and fluid motions. Like K-BKZ theory for viscoelastic fluids with fading memory, they extend standard symmetric-hyperbolic systems (polyconvex elastodynamics and Saint-Venant shallow-water systems). However, they are formulated differently, with the help of an additional material metric variable. Now, using the same methodology, other viscoelastic models with a K-BKZ integro-differential for- mulation could in fact be similarly formulated as systems of **conservation** **laws**. Moreover, varying the relaxation limit of the additional material metric variable should yield (symmetric-hyperbolic formulations of) many possible flow models in between elastic solids and fluids, like elasto-plastic models. New rheological extensions of the polyconvex elastodynamics and Saint-Venant shallow-water systems will be studied in future works.

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illustrating the behaviour of the solutions of a non-local traffic flow model, when the size and the position of the kernel support or the velocity function vary. In particular, we analyze the impact on two cost functionals, measuring traffic congestion.
• In Chapter 3, we consider a system of M non-local **conservation** **laws** in one space dimension, given by a non-local multi-class model obtained as a generalization of the M -populations model for traffic flow described in [ 8 ]. This is a multi-class version of the one dimensional scalar **conservation** law with non-local flux proposed in [ 11 ]. We allow different anisotropic kernels for each equation of the system. The model takes into account the distribution of heterogeneous drivers and vehicles characterized by their maximal speeds and look-ahead visibility in a traffic stream. The main result of this chapter is the existence of weak solutions locally in time. We remark that, since the convolution kernels are not smooth on R, the results in [ 2 ] cannot be applied due to the lack of L ∞ -bounds on their derivatives. We do not address the question of uniqueness

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18 Adimurthi & G. D. Veerappa Gowda & J. Jaffr´e
family includes the Force and proposes an alternative for the Musta schemes. For all these schemes we proved convergence of the approximate solution to the entropy solution of the continuous problem. We also gave hints on how to extend them to systems and high resolution schemes. In forthcoming papers we will extend these schemes to higher resolution schemes and to the discontinuous flux case. We will also give an example of application to a 2 × 2 system of **conservation** **laws** representing a problem of polymer flooding.

Over the last decades, traffic congestion, car accidents and pollution became daily issues. To understand and overcome road traffic problems, scientists from different research fields are creating advanced mathematical models. Mathematical models help to understand road traffic phenomena, develop optimal road network with efficient movement of traffic and minimal traffic congestion. This thesis is devoted to macroscopic traffic flow modelling, which describes traffic flow by variables av- eraged over multiple vehicles: density, velocity and flow. Macroscopic models naturally lead to **conservation** **laws**, which are hyperbolic partial differential equa- tions. In recent years, this class of equations is more widely considered, but few theoretical results are available. This is caused by two main difficulties. The for- mer is the non-linear hyperbolic nature of equations, which leads to consider weak solutions, instabilities and diffusivity of numerical schemes. The latter is the non- uniqueness of weak solutions and the need to introduce exotic functionals to select a unique physically reasonable solution.

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Since C. DeLellis and M. Westdickenberg [Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire 20 (2003), 1075–1085] have proved the existence of such solutions that do not belong to B q,loc s,p if either s > 1/ max(p, 3) or s = 1/3 and 1 ≤ q < p < 3 or s = 1/p with p ≥ 3 and q < ∞, this regularizing effect is optimal. The proof is based on the kinetic formulation of scalar **conservation** **laws** and on an interaction estimate in physical space.

Besides, Colombo and Rosini [12] introduced a model for pedestrian flow accounting for panic appearance and consisting in a scalar **conservation** law in one space-dimension displaying nonclassical shocks. Such a simplified model can be used for example to describe the motion of a crowd along a corridor or a bridge. Moreover, in [13] the authors show that the flux constraint represented by the presence of a door may cause the onset of panic states from a normal situation. In this model, the flux function is not concave (nor convex) and therefore it does not match the available results about **conservation** **laws** with constrained flux. A rigorous analysis of this pedestrian flow model thus needs the extension of the above cited results to general fluxes.

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[3] P. Castelli, S. Junca. Smoothing eect in BV − Φ for scalar **conservation** **laws**. preprint (2015), hal-01133725, 28 pp.
[4] G.-Q. Chen, S. Junca, M. Rascle. Validity of nonlinear geometric optics for entropy solutions of multidimensional scalar **conservation** **laws**, J. Dierential. Equations, 222, (2006), 439-475. [5] K. S. Cheng. The space BV is not enough for hyperbolic **conservation** **laws**. J. Math. Anal. App.,

∂ t U 2 + ∂ x U 2 ϕ(U ) = 0 (3.6)
is a 2 × 2 system of **conservation** **laws** in one space dimension in the unknown U = (U 1 , U 2 ). System (3.6) has been extended and studied from many different points of
view. For instance, the existence of solutions in the case of several space dimensions was obtained in [2, Theorem 2.6]. The convergence of approximate solutions was proved in [16] in the case of one dimensional symmetric systems.

• vehicles must travel with postive speed,
• vehicles can only have a zero velocity when they encounter maximum density ρ max ,
• right before a red light, the density must be equal to the maximum density.
Solving **conservation** **laws** on networks is not trivial, because of the difficulty of modeling the dynamics at the junctions, and more precisely the fact that the boundary conditions on incoming and outgoing links at a given junction are coupled and may depend on the solu- tions on adjactent links. In order to allocate flow at junctions one has to define additional constraints. The LWR model was first applied to unidirectional networks in [91]. The authors note that drivers will generally try to avoid congestion by possibly changing paths. That is why they propose an "entropy condition" which maximizes the flux of vehicles at the level of the junction. Their work was extended to general networks in [44]. We also refer the reader to [32, 57, 67, 68, 69, 70, 87, 89]. A road network can be represented by a directed graph G = {L, J } (Figure 1.9a), where L is a set of oriented links representing the roads and J is a set of nodes representing the intersections. Each link l ∈ L can be spa- tially represented by an open set I l . We can then define the density of vehicles ρ l = ρ l (t, x)

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Hyperbolic **Conservation** **Laws**
A. Mazzia ∗
Abstract. In this paper we consider numerical approximations of hyperbolic **conservation** **laws** in the one-dimensional scalar case, by studying Godunov and van Leer’s methods. Before to present the numerical treatment of hyperbolic **conservation** **laws**, a theoretical introduction is given together with the definition of the Riemann problem. Next the numerical schemes are discussed. We also present numerical experiments for the linear advection equation and Burgers’ equation. The first equation is used for modeling discontinuities in fluid dynamics; the second one is used for modeling shocks and rarefaction waves. In this way we can compare the different behavior of both schemes.

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[13] T.-P. Liu & T. Yang, A new entropy functional for a scalar **conservation** law, Commun. Pure Appl. Math. 52 no. 11 (1999), 1427–1442.
[14] D. Serre, Systems of **conservation** **laws** – 1. Hyperbolicity, entropies, shock waves – 2. Geometric structures, oscillations, and initial-boundary value problems, Cam- bridge Univ. Press, 1999 & 2000.

pairs of the ABI equations. An explicit formula of the entropy solution of the BI equations is also given.
Finally, we remark that the equivalence of the two coordinate systems in several space dimensions is discussed in [16, 17, 45]. See also [20] for a numerical investigation in two dimensional equations of gas dynamics and the references therein. However, due to the essential difference between the systems of **conservation** **laws** in one dimension and several dimensions, it is not so obvious to obtain the results of this paper in several dimensions. A further study on the problem will be given in a forthcoming work.

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FOR UNCERTAIN SCALAR **CONSERVATION** **LAWS**
J. TRYOEN † ‡ , O. LE MAˆITRE ‡ , AND A. ERN †
Abstract. This paper deals with the design of adaptive anisotropic discretization schemes for **conservation** **laws** with stochastic parameters. A Finite Volume scheme is used for the deter- ministic discretization, while a piecewise polynomial representation is used at the stochastic level. The methodology is designed in the context of intrusive Galerkin projection methods with Roe-type solver. The adaptation aims at selecting the stochastic resolution level based on the local smoothness of the solution in the stochastic domain. In addition, the stochastic features of the solution greatly vary in the space and time so that the constructed stochastic approximation space depends on space and time. The dynamically evolving stochastic discretization uses a tree-structure representation that allows for the efficient implementation of the various operators needed to perform anisotropic multiresolution analysis. Efficiency of the overall adaptive scheme is assessed on the stochastic traffic equation with uncertain initial conditions and velocity leading to expansion waves and shocks that propagate with random velocities. Numerical tests highlight the computational savings achieved as well as the benefit of using anisotropic discretizations in view of dealing with problems involving a larger number of stochastic parameters.

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