Abstract

Some optimization or equilibrium problems involving somehow the concept of optimal transport are presented in these notes, mainly devoted to applications to economic and game theory settings. A variant model of transport, taking into account traffic congestion effects is the first topic, and it shows various links with Monge-Kantorovich theory and PDEs. Then, two models for urban planning are introduced. The last section is devoted to two problems from economics and their translation in the language of optimal transport.

Abstract

International audience; Numerous slurry transportation pipeline systems have been built in the past 10 years. At the same time, T. Chakkour & F. Benkhaldoun study in [1] the hydraulic transport of particles in tubes. We investigated in [2] the hydraulic transport of slurry system in horizontal tubes (The Khouribga mines). This mineral pipeline has often been referred to as one of the most challenging projects in terms of operating complexity and system configuration in Morocco. This physical model features a mass and momentum balance for three-fluid model in 1D. It allows to predict the pressure drop and flow patterns. The originality of this work is to present in simplified form a homogeneous single-phase model. The most important advantage of this model is the considerably smaller number of variables to be solved compared to the multiphase model. In this presentation, we give the asymptotic behavior of friction-disharge term f Q 2 that is involved in the last term of motions equation, taking into account the Reynolds number. This allows to understand how the elevation varies, when the flow is very laminar.

Abstract

International audience; In this paper we develop a boundary state feedback control law for a cascaded traffic flow network system: one incoming and one outgoing road connected by a junction. The macroscopic traffic dynamics on each road segment are governed by Aw-Rascle-Zhang (ARZ) model, consisting of second-order nonlinear partial differential equations (PDEs) for traffic density and velocity. Different equilibrium road conditions are considered for the two segments. For stabilization of stop-and-go traffic congestion on the two roads, we consider a ramp metering located at the connecting junction. The traffic flow rate entering from the on-ramp to the mainline junction is actuated. The objective is to simultaneously stabilize the upstream and downstream traffic to given spatially-uniform constant steady states. We design a full state feedback control law for this under-actuated network of two systems of two hetero-directional linear first-order hyperbolic PDEs interconnected through the junction boundary. Exponential Convergence to steady states in L 2 sense is validated by a numerical simulation.

Abstract

Congested traffic problems on very dense networks lead, at the limit, to minimization problems posed on measures on curves as shown in Baillon and Carlier (Netw. Heterog. Media 7:219–241, 2012 ). Here, we go one step further by showing that these problems can be reformulated in terms of the minimization of an integral functional over a set of vector fields with prescribed divergence as in Beckmann (Econometrica 20:643–660, 1952 ). We prove a Sobolev regularity result for their minimizers despite the fact that the Euler–Lagrange equation of the dual is highly degenerate and anisotropic. This somehow extends the analysis of Brasco et al. (J. Math. Pures Appl. 93:652–671, 2010 ) to the anisotropic case. Copyright Springer Science+Business Media New York 2013

Abstract

Motivated by applications to congested optimal transport problems, we prove higher integrability results for the gradient of solutions to some anisotropic elliptic equations, exhibiting a wide range of degeneracy. The model case we have in mind is the following: \\[ \\partial_x \\left[(|u_{x}|-\\delta_1)_+^{q-1}\\, \\frac{u_{x}}{|u_{x}|}\\right]+\\partial_y \\left[(|u_{y}|-\\delta_2)_+^{q-1}\\, \\frac{u_{y}}{|u_{y}|}\\right]=f, \\] for $2\\le q"\\infty$ and some non negative parameters $\\delta_1,\\delta_2$. Here $(\\,\\cdot\\,)_+$ stands for the positive part. We prove that if $f\\in L^\\infty_{loc}$, then $\ abla u\\in L^r_{loc}$ for every $r\\ge 1$.