Optimal transportation for a quadratic cost with convex constraints and applications

We prove existence of an optimal transport map in the Monge-Kantorovich problem associated to a cost $c(x,y)$ which is not finite everywhere, but coincides with $|x-y|^2$ if the displacement $y-x$ belongs to a given convex set $C$ and it is $+\infty$ otherwise. The result is proven for $C$ satisfying some technical assumptions allowing any convex body in $\R^2$ and any convex polyhedron in $\R^d$, $d>2$. The tools are inspired by the recent Champion-DePascale-Juutinen technique. Their idea, based on density points and avoiding disintegrations and dual formulations, allowed to deal with $L^\infty$ problems and, later on, with the Monge problem for arbitrary norms

Duality theory and optimal transport for sand piles growing in a silos

We prove existence and uniqueness of solutions for a system of PDEs which describes the growth of a sandpile in a silos with flat bottom under the action of a vertical, measure source. The tools we use are a discrete approximation of the source and the duality theory for optimal transport (or Monge-Kantorovich) problems

The \infty eigenvalue problem and a problem of optimal transportation

The so-called eigenvalues and eigenfunctions of the infinite Laplacian $\Delta_\infty$ are defined through an asymptotic study of that of the usual $p$-Laplacian $\Delta_p$, this brings to a characterization via a non-linear eigenvalue problem for a PDE satisfied in the viscosity sense. In this paper, we obtain an other characterization of the first eigenvalue via a problem of optimal transportation, and recover properties of the first eigenvalue and corresponding positive eigenfunctions

Optimum and equilibrium in a transport problem with queue effects

Consider a distribution of citizens in an urban area in which some services (supermarkets, post offices...) are present. Each citizen, in order to use a service, spends an amount of time which is due both to the travel time to the service and to the queue time waiting in the service. The choice of the service to be used is made by every citizen in order to be served more quickly. Two types of problems can be considered: a global optimization of the total time spent by the citizens of the whole city (we define a global optimum and we study it with techniques from optimal mass transportation) and an individual optimization, in which each citizen chooses the service trying to minimize just his own time expense (we define the concept of equilibrium and we study it with techniques from game theory). In this framework we are also able to exhibit two time-dependent strategies (based on the notions of prudence and memory respectively) which converge to the equilibrium

Dynamical Systems and Hamilton–Jacobi–Bellman Equations on the Wasserstein Space and their L2 Representations

Several optimal control problems in \BbbR d, like systems with uncertainty, control of flock dynamics, or control of multiagent systems, can be naturally formulated in the space of probability measures in \BbbR d . This leads to the study of dynamics and viscosity solutions to the Hamilton-- Jacobi--Bellman equation satisfied by the value functions of those control problems, both stated in the Wasserstein space of probability measures. Since this space can be also viewed as the set of the laws of random variables in a suitable L2 space, the main aim of the paper is to study such control systems in the Wasserstein space and to investigate the relations between dynamical systems in Wasserstein space and their representations by dynamical systems in L2, both from the points of view of trajectories and of (first-order) Hamilton--Jacobi--Bellman equations

EQUIVALENCE BETWEEN STRICT VISCOSITY SOLUTION AND VISCOSITY SOLUTION IN THE SPACE OF WASSERSTEIN AND REGULAR EXTENSION OF THE HAMILTONIAN IN L^2_IP

This article aims to build bridges between several notions of viscosity solution of first order dynamic Hamilton-Jacobi equations. The first main result states that, under assumptions, the definitions of Gangbo-Nguyen-Tudorascu and Marigonda-Quincampoix are equivalent. Secondly, to make the link with Lions' definition of solution, we build a regular extension of the Hamiltonian in L^2_IP ×L^2_IP. This extension allows to give an existence result of viscosity solution in the sense of Gangbo-Nguyen-Tudorascu, as a corollary of the existence result in L^2_ IP × L^2_IP. We also give a comparison principle for rearrangement invariant solutions of the extended equation. Finally we illustrate the interest of the extended equation by an example in Multi-Agent Control

A zero sum differential game with correlated informations on the initial position. A case with a continuum of initial positions

International audienceWe study a two player zero sum game where the initial position z 0 is not communicated to any player. The initial position is a function of a couple (x 0 , y 0) where x 0 is communicated to player I while y 0 is communicated to player II. The couple (x 0 , y 0) is chosen according a probability measure dm(x, y) = h(x, y)dµ(x)dν(y). We show that the game has a value and, under additional regularity assumptions, that the value is a solution of Hamilton Jacobi Isaacs equation in a dual sense

Optimisation de problèmes de transport

L'essentiel de la thèse est consacré à l'asymptotique d'une suite de problèmes de transport. Dans chacun de ces problèmes dont diverses variantes apparaissent en économie mathématique et en théorie du signal, il s'agit de trouver une mesure discrète minimisant le transport vers une mesure absolument continue donn\'ee f et vérifiant une contrainte du type H(u)£m où H est une entropie donnée et m est un paramètre que nous faisons tendre vers + . Dans un premier temps, nous étudions le cas où f est une densité uniforme sur un cube et H(u)=S (u(x))a avec aÎ[0,1[. Dans le cas général, nous ramenons l'asymptotique de tels problèmes à la détermination de la G-limite de fonctionnelles naturellement associées à la distance de Wasserstein. Dans une deuxième partie de la thèse, nous présentons des perspectives nouvelles de problèmes de transport avec coût dépendant du temps. Le cas particulier de coûts "homogènes" (ne dépendant que de la vitesse moyenne) permet d'écrire des conditions d'optimalité pour le transport de Wasserstein Wp (p>1) sous forme d'un système d'équations eikonale-diffusion (écrit au sens des mesures). Ceci généralise les résultats du cas p=1 ( L. Evans et W. Gangbo, G. Bouchitté et G. Buttazzo) et ceux de Brenier (p>1) au cas où les mesures transportées sont singulières.The main issue of the thesis is the study of the asymptotic behaviour of optimal transportation problems. Such problems occurs in economy and signal theory. Each of them consists in finding the best discrete measure u wich minimizes the transport to an absolute continuous measure f, subject to a constraint of the kind H(u)£m where H is a given entropy functional. In a first step, we study the case where f is a uniform density on a cube and H(u)=S (u(x))a with aÎ[0,1[. In the general case, we reduce the question of the asymptotic behaviour to the description of the G-limit of a suitable functionnal naturally associated to the Wasserstein distance. In the second part of the thesis, we present new applications of transport problems with time depending cost. The particular case of homogeneous cost (depending only on the average speed) allows us to write optimality conditions for the Wasserstein transport Wp (p>1) as a system of equations (eikonal-diffusion) written in the sense of measures. This generalizes the results obtained in the case p=1 ( L. Evans and W. Gangbo, G. Bouchitté and G. Buttazzo) and those of Brenier (p>1) to the case where the transported measures are singular.TOULON-BU Centrale (830622101) / SudocSudocFranceF