arXiv subjects

Benoît Bonnet-Weill

Publications and source records attributed to Benoît Bonnet-Weill.

8 recordsLinked to original sources

arXiv2026

Exponential Turnpike Theorems for Nonlinear Deterministic Meanfield Optimal Control Problems

In this article, we establish exponential turnpike theorems for a class of nonlinear deterministic meanfield optimal control problems. We carry out our analysis simultaneously in the so-called Lagrangian and Eulerian frameworks. In the Lagrangian setting, the problem is lifted to a Hilbert space of random variables, and we prove an exponential turnpike theorem by combining first-order optimality conditions, a second-order expansion of the lifted Hamiltonian, and an operator Riccati diagonalization argument. In the Eulerian setting, we derive intrinsic KKT conditions for the static constrained problem, and show how the Eulerian second-order hypotheses split into a horizontal part, transferred by unitary conjugation to the lifted space, and a vertical part which reduces to uniform pointwise stabilizability and detectability conditions on multiplication operators. This yields an exponential turnpike theorem in the Wasserstein space for optimal Pontryagin triples. Along the way, we %provide explicit the link between Wasserstein Hessians and their Lagrangian lifts, and provide several remarks clarifying the role of occupation measures, local Eulerian minimizers, and control constraints in our results.

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arXiv2025

A Characterization of Law-Invariant and Coherent Risk Measures through Optimal Transport

In this article, we propose a novel characterization of law-invariant and coherent risk measures, based on a generalized optimal transport problem in which the second marginal of the admissible plans is not fixed, but required to lie within a target set of probability measures. One of the main contributions of this work is a general representation formula for such risk measures, which is closely related to Kusuoka's theorem. When the aforementioned target set is convex, our representation result allows for the systematic derivation of general duality formulas. To illustrate our findings, we explicitly compute the target sets associated with several classical law-invariant coherent risk measures, including the prototypical conditional value at risk and higher moment measures.

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arXiv2025

Viability Theory in the $1$-Wasserstein Space

In this article, we establish necessary and sufficient viability conditions for continuity inclusions over the 1-Wasserstein space. Depending on the regularity properties of the dynamics, we derive two results which are based on fairly different proof strategies. When the admissible velocities are Lipschitz in the measure variable, we show that it is necessary and sufficient for viable solutions to exist that the latter intersect the graphical derivative of the constraints. On the other hand, when the admissible velocities are merely upper semicontinuous in the measure variable, we provide a sufficient condition for viability involving the infinitesimal behaviour of their Aumann integral over a neighbouring set of measures.

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arXiv2025

Structured Continuity Equations in Fibred Wasserstein Spaces

In this article, we develop a comprehensive ODE-theory for structured continuity equations in fibred probability spaces, which represent a class of heterogeneous PDEs arising as the meanfield limit nonexchangeable particle systems. After investigating in depth the topologies induced by the so-called fibred and classical Wasserstein metrics on such probability spaces, we establish quantitative Cauchy-Lipschitz and qualitative Carathéodory-Peano well-posedness results for structured continuity equations, along with precise correspondences between this class of evolutions, classical Lagrangian dynamics, and continuity equations. In keeping with what has long been known for exchangeable dynamics, we derive a general meanfield approximation result by solutions of nonexchangeable particle systems, along with a quantitative variant thereof under practically reasonable regularity assumptions on the driving field and initial data.

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arXiv2025

Set-Valued Koopman Theory for Control Systems

In this paper, we introduce a new notion of Koopman operator which faithfully encodes the dynamics of controlled systems by leveraging the tools of set-valued analysis. In this context, we propose generalisations of the Liouville and Perron-Frobenius operators, and show that they respectively coincide with proper set-valued analogues of the infinitesimal generator and dual operator of the Koopman semigroup. We also give meaning to the spectra of these set-valued maps and prove an adapted version of the classical spectral mapping theorem relating the eigenvalues of a semigroup with those of its generator. Our approach provides theoretical justifications for existing practical methods in the Koopman community that study control systems by bundling together the Koopman and Liouville operators associated with different control inputs.

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arXiv2024

Carathéodory 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 compactly supported probability measures, introduced recently by the authors, to the setting of general Wasserstein spaces. In this context, we prove a novel existence result à la Peano for this class of dynamics under mere Carathéodory regularity assumptions. The latter is based on a natural, yet previously unexplored set-valued adaptation of the semi-discrete Euler scheme proposed by Filippov to study ordinary differential equations whose right-hand sides are measurable in the time variable. By leveraging some of the underlying methods along with new estimates for solutions of continuity equations, we also bring substantial improvements to the earlier versions of the Filippov estimates, compactness and relaxation properties of the solution sets of continuity inclusions, which are derived in the Cauchy-Lipschitz framework.

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arXiv2024

On the Viability and Invariance of Proper Sets under Continuity Inclusions in Wasserstein Spaces

In this article, we derive conditions for the existence of solutions to state-constrained continuity inclusions in Wasserstein spaces whose right-hand sides may be discontinuous in time. These latter are based on a fine investigation of the infinitesimal behaviour of the underlying reachable sets, through which we show that up to a negligible set of times, every admissible velocity of a continuity inclusion can be approximately realised as the metric derivative of a solution of the dynamics, and vice versa. Building on these results, we are able to establish necessary and sufficient geometric conditions for the viability and invariance of stationary and time-dependent constraint sets which involve a suitable notion of contingent cones in Wasserstein spaces, and presented in ascending order of generality. We then close the article by exhibiting two prototypical examples of constraints sets appearing in applications for which one can compute relevant subfamilies of contingent directions.

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arXiv2024

Exponential Consensus Formation in Time-Varying Multiagent Systems via Compactification Methods

In this article, we establish exponential contraction results for the diameter and variance of general first-order multiagent systems. Our approach is based on compactification techniques, and works under rather mild assumptions. Namely, we posit that either the scrambling coefficient, or the algebraic connectivity of the averaged interaction graphs of the system over all time windows of a given length are uniformly positive.

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