Université de Tours
Résumés
Ali ASSI
Homogenization of Non-Convex Hamilton–Jacobi Equations with Infinitely Many Traffic Lights.
In this talk, I present results on the homogenization of first-order, time-dependent Hamilton–Jacobi equations. The problem is posed on a one-dimensional domain with infinitely many traffic lights located at the integer points. The Hamiltonian is non-convex and discontinuous in both space and time. The spatial discontinuities occur at the integer points, corresponding to the locations of the traffic lights. The time discontinuities arise from the switching of the traffic lights and form a dense set in time. The main difficulty lies in the absence of a standard viscosity solution theory adapted to this densely time-discontinuous setting. We overcome this difficulty by developing a framework that requires global Lipschitz regularity in order to obtain a suitable comparison principle in the presence of time discontinuities that are dense in time. A key step is therefore the construction of globally Lipschitz correctors together with uniform control of time–space oscillations. Finally, convergence is proved using the perturbed test function method, adapting classical homogenization techniques to the discontinuous-in-time setting.
Fabio CAMILLI
\(L^p\)-estimates for numerical schemes of Hamilton--Jacobi equations
We establish \(L^p\) error estimates for monotone numerical schemes approximating Hamilton--Jacobi equations on the \(d\)-dimensional torus. Using the adjoint method, we first prove an \(L^1\) error bound of order one for finite-difference and semi-Lagrangian schemes under standard convexity assumptions on the Hamiltonian. By interpolation, we also obtain \(L^p\) estimates for every finite \(p>1\). Our analysis covers a broad class of schemes, improves several existing results, and provides a unified framework for discrete error estimates (joint with Alessio Basti).
Elisabetta CARLINI
An Algorithm Based on Monotone Operators for Deterministic Mean Field Games
We present a semi-Lagrangian numerical scheme for first-order, time-dependent, and non-local Mean Field Game systems, based on a key monotonicity property that ensures stability and convergence. The method leads to a fully discrete MFG formulation in which the coupling is driven by discrete optimal controls.To efficiently solve the resulting nonlinear system, we introduce a Learning Value Algorithm and an accelerated strategy based on policy iteration. We discuss the convergence properties of both approaches and show how the accelerated algorithm significantly improves computational performance. Finally, numerical experiments are presented to illustrate the effectiveness of the proposed approach. Joint work with V. Coscetti
Lucas COEURET
Existence of BV-regular solutions for the 1D Hughes’ model with affine cost function
Hughes' model is a traffic model aiming to describe the behavior of a crowd of rational pedestrians wanting to evacuate an area by exits located at its border. Just like many traffic flow models, Hughes' model enters within the much larger framework of conservation laws with moving interfaces acting as spatial discontinuities for the flux. Those interfaces allow to separate the crowd in groups having different goals, for instance aiming for different exits. One of the main difficulties of Hughes' model stems from the dynamic of the moving interfaces which depends nonlocally on the density of pedestrians. Well-posedness of Hughes’ model has seen a growing interest during the last decade. We present a novel result of existence of BV-regular solutions of the 1D Hughes’ model in the so-called affine cost function case where pedestrians choose their exit based on the distance to the exits and the number of people in the way. This result is achieved by constructing approximate solutions via a wave-front tracking algorithm, proving subtle compactness estimates. Notably, the developed algorithm is implementable into a numerical scheme, potentially benefiting the numerical analysis community interested in reliable approximations of Hughes model solutions.
Paola GOATIN
Dissipation of stop-and-go waves in traffic flows using controlled vehicles: a macroscopic approach
We study the boundary stabilization of Generic Second Order Macroscopic traffic models in Lagrangian coordinates. These consist in 2x2 nonlinear hyperbolic systems of balance equations with a relaxation type source term. We provide the existence of weak solutions of the Initial Boundary Value problem for generic relaxation terms. In particular, we do not require the "sub-characteristic" stability condition to hold, so that equilibria are unstable and perturbations may lead to the formation of large oscillations, modeling the appearance and persistence of stop-and-go waves. Moreover, since the largest eigenvalue of the system is null, the boundaries are characteristic, and the available results on boundary controllability do not apply. Therefore, we perform a detailed analysis of the Wave Front Tracking approximate solutions to show that weak solutions can be steered to the corresponding equilibrium state by prescribing the equilibrium speed at the right boundary. This corresponds to controlling the speed of one vehicle to stabilize the upstream traffic flow. The result is illustrated through numerical examples.
Claude Le Bris
Defects in homogenization theory: from linear diffusion to Hamilton-Jacobi
We review a series of works that address homogenization for partial differential equations with highly oscillatory coefficients. A prototypical setting is that of periodic coefficients that are locally, or more globally perturbed by a « defect ». We investigate the homogenization limits obtained, first for linear elliptic equations, both in conservative and non conservative forms, and next for nonlinear equations such as Hamilton-Jacobi type equations. The talk is based on a series of joint works with various colleagues: Xavier Blanc and Pierre-Louis Lions, Yves Achdou, Andrea Braides and Gianni Dal Maso.
Abraham SYLLA
Differential Games for a Mixed ODE-PDE System
We present an epidemic model with vaccination. The talk is devoted to a zero-sum differential game governed by a mixed system consisting of an ODE and a hyperbolic balance law. Two competing players act through their respective controls to influence the choice of people towards the vaccination. We are interested in both the well posedness of the system, for fixed control functions, and in the resulting differential game, based on a dynamic information pattern. This is a joint work with Mauro Garavello (Milano-Bicocca) and Elena Rossi (Modena and Reggio Emilia).
Erwin TOPP
Degenerate elliptic equations on networks with Kirchhoff vertex conditions.
In this talk, I will report some results about the existence, uniqueness and qualitative properties for solutions of Hamilton-Jacobi equations on networks, in the case the vertex condition is of Kirchhoff type. By means of PDE methods in the context of viscosity solutions, we extend basic results already addressed for (coercive) first-order PDE to more general equations, including degenerate elliptic equations with rather general second-order and integro-differential ingredients. This is a joint work with Olivier Ley (Rennes) and Guy Barles (Tours).
Denis ULLMO
A Mean Field Games description of pedestrian dynamics : anticipation and local interactions.
In most models, the local navigation of pedestrians is assumed to involve no anticipation beyond the most imminent collisions. However, these models typically fail to reproduce some key features experimentally evidenced in dense crowds crossed by an intruder, namely, transverse displacements toward regions of higher density due to the anticipation of the intruder's crossing. I will show how a minimal mean-field game model that emulates agents planning out a global strategy that minimizes their overall discomfort can actually address this experimental scenario. By solving the problem in the permanent regime thanks to an analogy with the nonlinear Schrödinger's equation, I will demonstrate how it is possible to identify the two main variables governing the model's behavior and to exhaustively investigate its phase diagram, showing that the model is remarkably successful in replicating the experimental observations associated with the intruder experiment. However, it is rather clear that other experimental configurations, in which local interactions between pedestrians play an important role, will require an extension to this minimal mean-field game model. In a second part, I will introduce a simple example of a new kind of mean-field game in which we can investigate these issues.
