Solutions to Hamilton–Jacobi equation on a Wasserstein space

Abstract

We consider a Hamilton-Jacobi equation associated to the Mayer optimal control problem in the Wasserstein space P 2 (R d ) and define its solutions in terms of the Hadamard generalized differentials. Continuous solutions are unique whenever we focus our attention on solutions defined on explicitly described time dependent compact valued tubes of probability measures. We also prove some viability and invariance theorems in the Wasserstein space and discuss a new notion of proximal normal. Mathematics Subject ClassificationRecently, there is an increasing interest in optimal control problems stated in the Wasserstein spaces P p (R d ) of probability measures, see for instance [6,10,17,19,27,29,30] and the bibliographies contained therein. Wasserstein spaces being merely metric spaces, results of the classical control theory, developed in vector spaces, are not valid anymore in this new framework. Quite a number of counterparts of these classical results, like existence of optimal controls, first order necessary optimality conditions, regularity of the value function, sensitivity relations, links with differential inclusions, etc., were recently obtained in [7][8][9] for measures in P c (R d ), that is probability measures on R d having a compact support. In particular, it was discussed in [8] how the Bolza optimal control problems on P c (R d ) can be reduced to those of the Mayer type, which allows us, in the present work, to restrict the attention to these simpler (in writing) control problems.Communicated by A. Mondino.

Viability and Exponentially Stable Trajectories for Differential Inclusions in Wasserstein Spaces

“…stands for the seminormed space of Borel measurable and νintegrable maps, and whose expression is akin to that recently introduced in [12]. Building on this notion, we show in Theorem 4.4 that if the following geometric compatibility condition…”

“…From a technical standpoint, these inquiries often boil down to studying variational problems in the space of probability measures, and are commonly approached using optimal transport techniques and Wasserstein geometry, in the spirit of the reference treatises [5,78,79]. Without aiming at full exhaustivity, we point the reader to the manuscripts [26,40,52,54] for various existence and qualitative regularity results on deterministic mean-field optimal control problems, as well as to the following broad series of works dealing with optimality conditions, either in the form of Pontryagin's maximum principle [17,19,20,22,25,75,76] or of Hamilton-Jacobi-Bellman equations [12,41,66]. We also mention the references [2,34,39,74] which propose astute control strategies to stir collective systems towards specific asymptotic patterns, and finally [1,20,29,30] for general numerical methods in the context of mean-field optimal control.…”

“…Motivated by this blooming interest for variational problems in measure spaces, several research groups have been investigating relevant generalisations of the core concepts of set-valued analysis to the setting of mean-field control [12,21,22,27,23,42,43,66], a lively trend that reached more recently other closely related topics such as mean-field games [6,31] and the study of sufficient conditions for the well-posedness of measure dynamics [44,67]. It is de facto widely accepted that the language of correspondences, differential inclusions and generalised gradients provides in many cases a synthetic and powerful framework in which most problems stemming from the calculus of variations, games and control theory can be encompassed, as supported e.g.…”

“…[28,60], as well as to investigate sufficient stability conditions for differential inclusions [7]. These viewpoints -like many others stemming from set-valued analysis -present the advantage of being readily transposable beyond the setting of finite-dimensional vector spaces, as illustrated by their recent applications to problems in metric spaces [10,12,13,43].…”

“…In terms of bibliographical positioning, this work can be seen as a far-reaching extension of [12] by the second author, in which viability and invariance properties are established as a means to prove the well-posedness of general Hamilton-Jacobi-Bellman equations for optimal control problems in Wasserstein spaces, under more restrictive regularity assumptions on the data. We also point to the independent works [10,11,43] which focus on the study of viability properties for another class of set-valued dynamics in Wasserstein spaces to which we already alluded earlier.…”