Various supports

A population can be represented as a sum of individuals or as a continuum. Both approaches are unified if one uses probability measures, which are a very convenient tool when endowed with the Wasserstein distance. In this setting, one can study control problems over the dynamic of the population by using roughly the same tools as in classical Euclidian spaces. We present one of such extensions, namely the characterization of the value function of a control problem as the minimal viscosity supersolution of a Hamilton-Jacobi equation.

The space of measure can be topologized in several ways. A very physical choice is the topology induced by the Wasserstein distance, also called the Earth mover's distance. The resulting metric space is very useful in statistics and optimization but also possesses some nice geometric features. This talk will be a simple and graphical introduction to some of these aspects.

We consider a Mayer problem posed in the space $𝒫$₂(ℝ^d) of probability measures with finite second moment, endowed with the Wasserstein distance. Our aim is to write a Hamilton-Jacobi-Bellman equation whose unique solution is the value function of the control problem. This type of problem usually admits nonsmooth solutions, and is treated with the theory of viscosity solutions [Crandall-Lions 1992].
To extend the theory of viscosity solutions to $𝒫$₂(ℝ^d), several lines of investigation are open. One can use semidifferentials [Marigonda-Quincampoix 2018, Gangbo-Tudorascu 2019] or implement metric viscosity techniques in the case of Eikonal-type equations [Ambrosio-Feng 2014, Giga-Hakamuri-Nakasayu 2015]. In this talk, we follow an idea raised in [Jerhaoui 2022] and consider instead the map of directional derivatives as our elementary local approximation of a smooth function, instead of a gradient. This framework allow to define viscosity solutions using quite natural test functions, and supports a strong comparison principle while retaining the link between Hamilton-Jacobi-Bellman equations and control problems. We are interested in understanding in which case can we extend the natural equivalence between semidifferentials and test function definition, that stands in the finite-dimensional setting.

We present two notions of viscosity solutions for first-order Hamilton-Jacobi equations, one using test functions that are directionally differentiable, and another one using generalized sub/superdifferentials. In the classical setting of viscosity over $ℝd$, it is simple to link semidifferentials to the gradients of test functions. We show that a similar equivalence holds in our nonsmooth setting, under appropriate conditions over the Hamiltonian.

The (max,+) semialgebra relies on a particular choice of operations that belongs to the realm of idempotent analysis. In this short talk, we introduce these operations and the dialog between them and classical algebra. As an application, we give the (max,+) parallel to the heat equation.

We present a notion of viscosity solutions adapted to control problems in the space of probability measures over $ℝd$, endowed with the Wasserstein distance. The key idea is to avoid the definition of a gradient of applications going from $𝒫$₂(ℝ^d) to $ℝd$, by working directly with directional derivatives. Modulo adjustments to cope with the absence of local compactness, our definition is then very similar to the finite-dimensional case. We show that such a viscosity notion supports a strong comparison principle, and that the value function of the control problem is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation associated to the control problem. This work follows and generalizes [JeanJerhaouiZidani2022], in which the underlying space is a compact manifold and the dynamic is independant of the measure. Additionally, we are able to consider weaker regularity on the terminal cost and the dynamic, and we take care to only use arguments relying on the geometry of the general tangent cone.

This talk presents a short introduction to Maslov measures and their principal application, the understanding of Hamilton-Jacobi equations. We introduce the (max,+) semialgebra in an elementary way, then define Maslov measures, random variables and stochastic processes, and the class of Maslov chains. Following an article by Del Moral and Doisy, we show that the backward differential equation satisfied by the expectation of some terminal cost with respect to a Maslov chain, conditionned to the initial point, is exactly the Dynamic Programming Principle of Bellman equations. With this link, it is no surprise that HJB equations are (max,+) linear (or (min,+) linear in the classical case of a minimization).

El espacio de Wasserstein es un espacio metrico cuyos elementos se pueden representar como funciones de conjuntos, como leyes de variables aleatorias, o como puntos sobre geodesicas. Cada interpretación tiene sus propias herramientas para definir una teoría de derivación. En estas charlas, vamos a revisar la literatura desde el caso más suave hacia el menos suave.

The space of measures with finite second moment, when endowed with the Wasserstein distance, is a geodesic space but not a vector space. Therefore, building an adequate differential calculus is not straightforward, and currently subject of debate in the literature. We review the main definitions in use in the Hamilton-Jacobi and Mean Fields communities, with examples illustrating the nature of the objects and comments on the range and limitations of each point of view.

We consider optimal control problems over the Wasserstein space. On the one hand, there is now a stable theory of absolutely continuous curves of measures, characterized as the solutions of the continuity equation. In particular, for sufficiently regular dynamics, the trajectories enjoy a nice representation as the pushforward through the semigroup of the underlying ODE. On the other hand, control problems are naturally linked to the viscosity solutions of Hamilton-Jacobi (HJ) equations. Our aim is to contribute to the extension of the viscosity theory over the Wasserstein space. We explore the choice of a particular set of test functions, including the squared distance, that allows us to obtain a weak comparison principle and verify that the value function of the control problem is the unique Lipschitz and bounded solution of the corresponding HJ equation. In addition, this choice bears strong links with subdifferential notions of the literature.

We consider a deterministic control problem with finite horizon. On the one hand, the discretization of the Dynamical Programming Principle (DPP) by mesh-based methods suffers from the curse of dimensionality. On the other hand, the discretized DPP is naturally formulated as a chain of optimization problems. Following the ideas of [Huré, Pham, Bachouch and Langrené 2022] in the stochastic setting, we consider a semi-Lagrangian scheme where the value function at each time step is represented by a neural network. This scheme is applied to an obstacle problem arising from a state-constrained control problem, and some elements of analysis are given in this context. Numerical illustrations are given for dimensions between 2 and 8.

The generalization of viscosity theory to control problems over Wasserstein space is an active topic. This talk will focus on some advances in the direction of test functions. We introduce the Hamilton-Jacobi approach to control problems in a general context in order to highlight the intuition behind it, then decline it in the Wasserstein context. We discuss how to overcome two of the arising difficulties by adapting the notion of viscosity. This allows us to obtain some results of comparison and uniqueness of the solution of a suitable HJ equation.

The first question of optimization is whether some function admits a minimum over some space. The family of Ekeland principles adresses the question "what can we say when there is no minimum?" This talk will present the original principle and some variants in an elementary fashion, with hope to highlight the power of this elegant theory.

In its simplest form, the eikonal equation reads |u’| = 1. The right notion of solution seems to be viscosity solutions, that are, in a sense, maximal. This remark opens on an analytical method of resolution very close to the spectral decomposition of linear operators, and highlights the deep connection between viscosity and control problems.