arXiv subjects

Nicoletta Tchou

Publications and source records attributed to Nicoletta Tchou.

At least 19 recordsLinked to original sources

arXiv2026

Constrained Mean Field Games with Grushin type dynamics

This paper is devoted to a class of finite horizon deterministic mean field games with Grushin type dynamics, state constraints and nonlocal coupling. First, we consider the optimal control problem that each agent aims to solve when the evolution of the population is given and we establish some properties as: the existence of an optimal trajectory for any starting point $(x,t)$, the closed graph property for the multivalued map which associates to each point $(x,t)$ the set of optimal trajectories starting from that point, endowed with a suitable notion of convergence, the continuity of the value function. The main issue to overcome is due to the local interplay at boundary points between the set of state constraints and the degenerate dynamics. To this end, we shall point out two different sets of assumptions which are both sufficient for these properties. Afterwards, we tackle the mean field games; taking advantage of the aforementioned properties, we prove the existence of a relaxed equilibrium (which describes the evolution of the game in terms of a probability on the set of admissible trajectories) and derive the existence of a mild solution (which is a couple formed by the value function for the generic player and a family of time dependent measures on the state).

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arXiv2025

Deterministic Mean Field Games on Networks and Related Optimal Control Problems

We study a class of deterministic mean field games and related optimal control problems, with a finite time horizon and in which the state space is a network. An agent controls her velocity, and, when she occupies a vertex, she can either remain still or enter any adjacent edge. The running and terminal costs are assumed to be continuous in each edge, but may jump at the vertices. Compared to the companion paper [4], we make more general assumptions about the costs and consider networks with an arbitrary number of vertices; this higher degree of generality brings new difficulties. For the optimal control problems mentioned above, we obtain in particular the existence of optimal trajectories and regularity results concerning the optimal trajectories and the value function. These control theoretic results make it possible to address a class of mean field games on networks, with costs that do not depend separately on the control and on the distribution of states, and that are non-local with respect to the latter. Focusing on a Lagrangian formulation, we obtain the existence of relaxed equilibria consisting of probability measures on admissible trajectories. To any relaxed equilibrium corresponds a mild solution, i.e. a pair $(u, m)$ made of the value function $u$ of a related optimal control problem and a family $m = (m(t))_t$ of probability measures on the network. Given $m$, the value function $u$ is a viscosity solution of a Hamilton-Jacobi problem on the network. We then investigate the regularity properties of $u$ and a weak form of a Fokker-Planck equation satisfied by $m$.

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arXiv2024

The continuity equation in the Heisenberg-periodic case: a representation formula and an application to Mean Field Games

We provide a representation of the weak solution of the continuity equation on the Heisenberg group $\mathbb H^1$ with periodic data (the periodicity is suitably adapted to the group law). This solution is the push forward of a measure concentrated on the flux associated with the drift of the continuity equation. Furthermore, we shall use this interpretation for proving that weak solutions to first order Mean Field Games on $\mathbb H^1$ are also mild solutions.

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arXiv2023

First order Mean Field Games on networks

This paper is devoted to finite horizon deterministic mean field games in which the state space is a network. The agents control their velocity, and when they occupy a vertex, they can enter into any incident edge. The running and terminal costs are assumed to be continuous in each edge but not necessarily globally continuous on the network. A Lagrangian formulation is proposed and studied. It leads to relaxed equilibria consisting of probability measures on admissible trajectories. The existence of such relaxed equilibria is obtained. The proof requires the existence of optimal trajectories and a closed graph property for the map which associates to each point the set of optimal trajectories starting from that point. To any relaxed equilibrium corresponds a mild solution of the mean field game, i.e. a pair $(u,m)$ made of the value function $u$ of a related optimal control problem, and a family $m= (m(t))_t$ of probability measures on the network. Given $m$, the value function $u$ is characterized by a Hamilton-Jacobi problem on the network. Regularity properties of $u$ and a weak form of a Fokker-Planck equation satisfied by $m$ are investigated.

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arXiv2021

Non coercive unbounded first order Mean Field Games: the Heisenberg example

In this paper we study evolutive first order Mean Field Games in the Heisenberg group; each agent can move in the whole space but it has to follow "horizontal" trajectories which are given in terms of the vector fields generating the group and the kinetic part of the cost depends only on the horizontal velocity. The Hamiltonian is not coercive in the gradient term and the coefficients of the first order term in the continuity equation may have a quadratic growth at infinity. The main results of this paper are two: the former is to establish the existence of a weak solution to the Mean Field Game systems while the latter is to represent this solution following the Lagrangian formulation of the Mean Field Games. We also provide some generalizations to Heisenberg-type structures.

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arXiv2021

Deterministic mean field games with control on the acceleration and state constraints

We consider deterministic mean field games in which the agents control their acceleration and are constrained to remain in a domain of R n. We study relaxed equilibria in the Lagrangian setting; they are described by a probability measure on trajectories. The main results of the paper concern the existence of relaxed equilibria under suitable assumptions. The fact that the optimal trajectories of the related optimal control problem solved by the agents do not form a compact set brings a difficulty in the proof of existence. The proof also requires closed graph properties of the map which associates to initial conditions the set of optimal trajectories.

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arXiv2021

First order Mean Field Games in the Heisenberg group: periodic and non periodic case

In this paper we study evolutive first order Mean Field Games in the Heisenberg group~$\He^1$; each agent can move only along "horizontal" trajectories which are given in terms of the vector fields generating~$\He^1$ and the kinetic part of the cost depends only on the horizontal velocity. The Hamiltonian is not coercive in the gradient term and the coefficients of the first order term in the continuity equation may have a quadratic growth at infinity.The main results of this paper are two: the former is to establish the existence of a weak solution to the Mean Field Game system while the latter is to represent this solution following the Lagrangian formulation of the Mean Field Games.We shall tackle both the Heisenberg-periodic and the non periodic case following two different approaches. To get these results, we prove some properties which have their own interest: uniqueness results for a second order Fokker-Planck equation and a probabilistic representation of the solution to the continuity equation.\end{abstract}

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arXiv2020

Deterministic mean field games with control on the acceleration

In the present work, we study deterministic mean field games (MFGs) with finite time horizon in which the dynamics of a generic agent is controlled by the acceleration. They are described by a system of PDEs coupling a continuity equation for the density of the distribution of states (forward in time) and a Hamilton-Jacobi (HJ) equation for the optimal value of a representative agent (backward in time). The state variable is the pair $(x, v)\in R^N\times R^N$ where x stands for the position and v stands for the velocity. The dynamics is often referred to as the double integrator. In this case, the Hamiltonian of the system is neither strictly convex nor coercive, hence the available results on MFGs cannot be applied. Moreover, we will assume that the Hamiltonian is unbounded w.r.t. the velocity variable v. We prove the existence of a weak solution of the MFG system via a vanishing viscosity method and we characterize the distribution of states as the image of the initial distribution by the flow associated with the optimal control.

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arXiv2019

A note on non-coercive first order Mean Field Games with analytic data

We study first order evolutive Mean Field Games whose operators are non-coercive. This situation occurs, for instance, when some directions are `forbidden' to the generic player at some points. Under some regularity assumptions, we establish existence of a weak solution of the system. Mainly, we shall describe the evolution of the population's distribution as the push-forward of the initial distribution through a flow, suitably defined in terms of the underlying optimal control problem.

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arXiv2019

Finite Horizon Mean Field Games on Networks

We consider finite horizon stochastic mean field games in which the state space is a network. They are described by a system coupling a backward in time Hamilton-Jacobi-Bellman equation and a forward in time Fokker-Planck equation. The value function u is continuous and satisfies general Kirchhoff conditions at the vertices. The density m of the distribution of states satisfies dual transmission conditions: in particular, m is generally discontinuous across the vertices, and the values of m on each side of the vertices satisfy special compatibility conditions. The stress is put on the case when the Hamiltonian is Lipschitz continuous. Existence and uniqueness are proven.

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arXiv2018

Non-coercive first order Mean Field Games

We study first order evolutive Mean Field Games where the Hamiltonian is non-coercive. This situation occurs, for instance, when some directions are "forbidden" to the generic player at some points. We establish the existence of a weak solution of the system via a vanishing viscosity method and, mainly, we prove that the evolution of the population's density is the push-forward of the initial density through the flow characterized almost everywhere by the optimal trajectories of the control problem underlying the Hamilton-Jacobi equation. As preliminary steps, we need that the optimal trajectories for the control problem are unique (at least for a.e. starting points) and that the optimal controls can be expressed in terms of the horizontal gradient of the value function.

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arXiv2018

A Class of Infinite Horizon Mean Field Games on Networks

We consider stochastic mean field games for which the state space is a network. In the ergodic case, they are described by a system coupling a Hamilton-Jacobi-Bellman equation and a Fokker-Planck equation, whose unknowns are the invariant measure m, a value function u, and the ergodic constant $ρ$. The function u is continuous and satisfies general Kirchhoff conditions at the vertices. The invariant measure m satisfies dual transmission conditions: in particular, m is discontinuous across the vertices in general, and the values of m on each side of the vertices satisfy special compatibility conditions. Existence and uniqueness are proven, under suitable assumptions.

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arXiv2017

Homogenization of a transmission problem with Hamilton-Jacobi equations and a two-scale interface. Effective transmission conditions

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across a two-scale oscillatory interface. Typically, the amplitude of the oscillations is of the order of $ε$ while the period is of the order of $ε$ 2. As $ε$ $\rightarrow$ 0, the interfaces tend to a straight line $Γ$. We study the asymptotic behavior of the value function as $ε$ $\rightarrow$ 0. We prove that the value function tends to the solution of Hamilton-Jacobi equations in the two half-planes limited by $Γ$, with an effective transmission condition on $Γ$ keeping track of the oscillations.

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arXiv2017

Singular perturbations for a subelliptic operator

We study some classes of singular perturbation problems where the dynamics of the fast variables evolve in the whole space obeying to an infinitesimal operator which is subelliptic and ergodic. We prove that the corresponding ergodic problem admits a solution which is globally Lipschitz continuous and it has at most a logarithmic growth at infinity. The main result of this paper establishes that as $ε\to 0$, the value functions of the singular perturbation problems converge locally uniformly to the solution of an effective problem whose operator and data are explicitly given in terms of the invariant measure for the ergodic operator.

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arXiv2017

Asymptotic behaviour for operators of Grushin type: invariant measure and singular perturbations

This paper concerns singular perturbation problems where the dynamics of the fast variable evolve in the whole space according to an operator whose infinitesimal generator is formed by a Grushin type second order part and a Ornstein-Uhlenbeck first order part. We prove that the dynamics of the fast variables admits an invariant measure and that the associated ergodic problem has a viscosity solution which is also regular and with logarithmic growth at infinity. These properties play a crucial role in the main theorem which establishes that the value functions of the starting perturbation problems converge to the solution of an effective problem whose operator and initial datum are given in terms of the associated invariant measure.

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arXiv2016

The ergodic problem for some subelliptic operators with unbounded coefficients

We study existence and uniqueness of the invariant measure for a stochastic process with degenerate diffusion, whose infinitesimal generator is a linear subelliptic operator in the whole space R N with coefficients that may be unbounded. Such a measure together with a Liouville-type theorem will play a crucial role in two applications: the ergodic problem studied through stationary problems with vanishing discount and the long time behavior of the solution to a parabolic Cauchy problem. In both cases, the constants will be characterized in terms of the invariant measure.

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arXiv2015

Effective transmission conditions for Hamilton-Jacobi equations defined on two domains separated by an oscillatory interface

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across an oscillatory interface $Γ_ε$. The oscillations of the interface have small period and amplitude, both of the order of $ε$, and the interfaces $Γ_ε$ tend to a straight line $Γ$. We study the asymptotic behavior as $ε\to 0$. We prove that the value function tends to the solution of Hamilton-Jacobi equations in the two half-planes limited by $Γ$, with an effective transmission condition on $Γ$ keeping track of the oscillations of $Γ_ε$.

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arXiv2014

Homogenization Results for a Deterministic Multi-domains Periodic Control Problem

We consider homogenization problems in the framework of deterministic optimal control when the dynamics and running costs are completely different in two (or more) complementary domains of the space $\R^N$. For such optimal control problems, the three first authors have shown that several value functions can be defined, depending, in particular, of the choice is to use only "regular strategies" or to use also "singular strategies". We study the homogenization problem in these two different cases. It is worth pointing out that, if the second one can be handled by usual partial differential equations method " á la Lions-Papanicolaou-Varadhan" with suitable adaptations, the first case has to be treated by control methods (dynamic programming).

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