Optimal transport with convex obstacle

We consider the Monge transportation problem when the cost is the squared geodesic distance around a convex obstacle. We show that there exists at least one—and in general infinitely many—optimal transport maps.     

Optimal times for constrained nonlinear control problems without local controllability

We study optimal times to reach a given closed target for controlled systems with a state constraint. Our goal is to characterize these optimal time functions in such a way that it is possible to compute them numerically and we do not need to compute trajectories of the controlled system. In this paper we provide new results using viability theory. This allows us to study optimal time functions free from the controllability assumptions classically made in the partial differential equations approach.     

Front Propagation Problems with Nonlocal Terms, II

We study the propagation of hyper surfaces Σ_t of ^N satisfying the equation V = h(x, Ω_t), where V is the normal velocity of Σ_t at x, Ω_t is the interior of Σ_t, and h is a given evolution law. We prove the existence of generalized solutions and an inclusion principle for these generalized solutions. Copyright 2001 Academic Press.Dans cet article nous étudions certains problèmes de propagation d'hypersurfaces Σ_t de ^N suivant des lois de la forme: V = h(x, Ω_t), où V est la vitesse normale à Σ_t en x, Ω_t est l'intérieur de Σ_t et h est une loi d'évolution. On définit une notion de solutions généralisées, et on montre l'existence de telles solutions. De plus on démontre un principe d'inclusion pour ces solutions.     

Nonsmooth semipermeable barriers, Isaacs’ equation, and application to a differential game with one target and two players

We study the target problem which is a differential game where one of the players aims at reaching a target while the other player aims at avoiding this target forever. We characterize the victory domains of the players by means of geometric conditions and prove that the boundary of the victory domains is a nonsmooth semipermeable surface, i.e., is a solution (in a weak sense) of the Isaacs equation: sup _u inf _v 〈f (x, u, v),p〉 = 0, wheref is the dynamic of the system,u andv are the respective controls of the players, andp is a normal to the boundary of the victory domains at the pointx.     

Geodesics for a class of distances in the space of probability measures

In this paper, we study the characterization of geodesics for a class of distances between probability measures introduced by Dolbeault, Nazaret and Savaré. We first prove the existence of a potential function and then give necessary and sufficient optimality conditions that take the form of a coupled system of PDEs somehow similar to the Mean-Field-Games system of Lasry and Lions. We also consider an equivalent formulation posed in a set of probability measures over curves.     

A discussion about the homogenization of moving interfaces

There has been considerable interest lately in the homogenization theory for first- and second-order partial differential equations in periodic/almost periodic and random, stationary, ergodic environments. Of special interest is the study of the averaged behavior of moving interfaces. In this note we revisit the last issue. We present several new results concerning interfaces moving by either oscillatory first-order or curvature dependent coupled with oscillatory forcing normal velocity in periodic environments and analyze in detail their behavior. Under sharp assumptions we show that such fronts may homogenize, get trapped or oscillate.     

Mean field game of controls and an application to trade crowding

In this paper we formulate the now classical problem of optimal liquidation (or optimal trading) inside a mean field game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader facing a “background noise” (or “mean field”). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price. In this paper the trader faces the uncertainty of fair price changes too but not only. He also has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically. Our MFG formulation of this problem belongs to the class of “extended MFG”, we hence provide generic results to address these “MFG of controls”, before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of “heterogenous preferences” (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can “learn” it day after day, observing others’ behaviors.     

Existence and uniqueness of a Nash equilibrium feedback for a simple nonzero-sum differential game

Existence and uniqueness of a Nash equilibrium feedback is established for a simple class nonzero-sum differential games on the line.