A Pontryagin Maximum Principle in Wasserstein Spaces for Constrained Optimal Control Problems

In this paper, we prove a Pontryagin Maximum Principle for constrained optimal control problems in the Wasserstein space of probability measures. The dynamics, is described by a transport equation with non-local velocities and is subject to end-point and running state constraints. Building on our previous work, we combine the classical method of needle-variations from geometric control theory and the metric differential structure of the Wasserstein spaces to obtain a maximum principle stated in the so-called Gamkrelidze form.Comment: 35 page

First-Order Pontryagin Maximum Principle for Risk-Averse Stochastic Optimal Control Problems

In this paper, we derive a set of first-order Pontryagin optimality conditions for a risk-averse stochastic optimal control problem subject to final time inequality constraints, and whose cost is a general finite coherent risk measure. Unlike previous contributions in the literature, our analysis holds for classical stochastic differential equations driven by standard Brownian motions. Moreover, it presents the advantages of neither involving second-order adjoint equations, nor leading to the so-called weak version of the PMP, in which the maximization condition with respect to the control variable is replaced by the stationarity of the Hamiltonian

Consensus and Flocking Under Communication Failures for a Class of Cucker-Smale Systems

We study sufficient conditions for the emergence of consensus and flocking in a class of strongly cooperative non-linear multi-agent systems subject to arbitrary communication failures. Our approach is based on a combination of Lyapunov analysis along with the formulation of a novel persistence of excitation condition for cooperative systems. This assumption can be interpreted in terms of average connectedness of the interaction graph of the system, and provides quantitative convergence rates towards consensus and flocking

Carath\'eodory Theory and A Priori Estimates for Continuity Inclusions in the Space of Probability Measures

In this article, we extend the foundations of the theory of differential inclusions in the space of probability measures with compact support, laid down recently in one of our previous work, to the setting of general Wasserstein spaces. Anchoring our analysis on novel estimates for solutions of continuity equations, we propose a new existence result ``\`a la Peano'' for this class of dynamics, under mere Carath\'eodory regularity assumptions. The latter is based on a set-valued generalisation of the semi-discrete Euler scheme proposed by Filippov to study ordinary differential equations with measurable right-hand sides. We also bring substantial improvements to the earlier versions of the Filippov theorem, compactness and relaxation properties of the solution sets of continuity inclusions which are derived in the more restrictive Cauchy-Lipschitz setting

Generic Singularities of the 3D-Contact sub-Riemannian Conjugate Locus

In this paper, we extend and complete the classification of the generic singularities of the 3D-contact sub-Riemmanian conjugate locus in a neighbourhood of the origin.Comment: 7 pages, 2 figure

Consensus Formation in First-Order Graphon Models with Time-Varying Topologies

In this article, we investigate the asymptotic formation of consensus for several classes of time-dependent cooperative graphon dynamics. After motivating the use of this type of macroscopic models to describe multi-agent systems, we adapt the classical notion of scrambling coefficient to this setting, leverage it to establish sufficient conditions ensuring the exponential convergence to consensus with respect to the $L^{\infty}$-norm topology. We then shift our attention to consensus formation expressed in terms of the $L^2$-norm, and prove three different consensus result for symmetric, balanced and strongly connected topologies, which involve a suitable generalisation of the notion of algebraic connectivity to this infinite-dimensional framework. We then show that, just as in the finite-dimensional setting, the notion of algebraic connectivity that we propose encodes information about the connectivity properties of the underlying interaction topology. We finally use the corresponding results to shed some light on the relation between $L^2$- and $L^{\infty}$-consensus formation, and illustrate our contributions by a series of numerical simulations.Comment: 48 pages, 16 figure

6000 ans d’histoire environnementale de la lagune de Villepey (Fréjus, Var) : Premiers résultats de l’étude multi-proxies de la carotte Vil2

Les Étangs de Villepey correspondent à une lagune occupant la partie méridionale de la basse vallée de l’Argens. Afin de préciser la dynamique hydro-sédimentaire et l’évolution de ce milieu sur le temps long, un carottage de 8,75m de profondeur a été réalisé en bordure de la lagune en 2004. Les découvertes anciennes de vestiges d’épave(s) antique(s) dans les étangs posaient la question de l’évolution du littoral dans ce secteur depuis l’Antiquité. L’analyse, stratigraphique appuyée par l’étude des lithofaciès, (faciès sédimentaire, granulométrie des sables) et des biofaciès (ostracofaune et malacofaune benthique), a permis d’identifier précisément les paléo-biotopes et les différentes dynamiques sédimentaires, dont le calage chronologique est assuré par 4 dates radiocarbone.Les paléoenvironnements révélés par la carotte Vil2 mettent en évidence la présence d’un milieu marin vers 3900 av. J.-C. au moins, qui évolue en lagune à partir de l’Âge du Fer. À l’époque romaine, la lagune est protégée et communique avec la mer ; le tirant d’eau qui peut y être restitué n’exclue pas qu’elle ait pu faire l’objet d’un aménagement portuaire. La persistance de milieux lagunaires à Villepey depuis environ 2400 ans met également en évidence une progradation dissymétrique de la ria de l’Argens, plus rapide sur le flanc nord (secteur de Fréjus) que sur le flanc sud de la vallée, dans le secteur de Villepey.The lagoon of Villepey is situated in the southern part of the lower Argens valley. In order to specify the long term hydro-sedimentary dynamics and coastal landscape evolution, a 8.75m depth core on the edge of the lagoon has been studied. The discovery, in the 50’s, of remains of roman wreck(s) in the lagoon, brought the question of the morphology of the shore during the Antiquity. Sedimentary analysis and bio-indicators (ostracoda and macofauna) identification, allowed to precisely identify the palaeo-biotopes and sedimentary dynamics, whose chronology is determined by 4 radiocarbon dates. The Vil2 core shows the presence of the open sea at Villepey around 3900 B. C. at least. From the Iron Age, this environment evolves into a lagoon. During the roman period, the site is protected (pocket beach) and communicate with the sea. It is possible to evaluate the draught at this time, which would have been sufficient to install a harbour. The results also evidence a dissymetrical progradation of the Argens ria, faster on the northern side of the valley (around Fréjus), while the southern side, at Villepey, shows a lagoonal environment since around 2400 years

Consensus and Flocking under Communication Failures for a Class of Cucker-Smale Systems

In this paper, we study sufficient conditions for the emergence of asymptotic consensus and flocking in a certain class of non-linear generalised Cucker-Smale systems subject to multiplicative communication failures. Our approach is based on the combination of strict Lyapunov design together with the formulation of a suitable persistence condition for multi-agent systems. The latter can be interpreted as a lower bound on the algebraic connectivity of the time-average of the interaction graph generated by the communication weights, and provides quantitative decay estimates for the variance functional along the solutions of the system