Optimal Control in Wasserstein Spaces

Abstract : A wealth of mathematical tools allowing to model and analyse multi-agent systems has been brought forth as a consequence of recent developments in optimal transport theory. In this thesis, we extend for the first time several of these concepts to the framework of control theory. The first result presented in this manuscript is the generalization of the Pontryagin Maximum Principle, both in the absence and presence of constraints, to optimal control problems of multi-agent systems studied in the so-called mean-field approximation framework. The proof of this result relies on the generalization of techniques from geometric control theory to the setting of the Riemannian structure of the Wasserstein spaces. Subsequently, we investigate sufficient conditions for the Lipschitz-in-space regularity of mean-field optimal control. These results are generally crucial for ensuring the correspondence between the microscopic multi-agent systems and their macroscopic approximations. We obtain them by combining a mean-field approximation argument with an existence result for Lipschitz optimal feedbacks formulated for microscopic multi-agent models. Later on, we focus our attention on alignment models. We propose a convergence analysis based on Lyapunov-type arguments for cooperative systems subject to random communication failures. We further propose a sparse control strategy which allows to stir weakly-cooperative systems towards a state of almost-alignment. We finally present a result of sub-Riemannian geometry, in which we complete the classification of the generic singularities of the conjugate locus for contact distributions in dimension 3. This result is based on transversality arguments applied to the jets of the metric in a suitable neighbourhood of the origin.
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Benoît Bonnet. Optimal Control in Wasserstein Spaces. Automatic Control Engineering. Aix-Marseille Universite; Laboratoire d'Informatique et Systèmes - LIS; Università degli studi di Padova, 2019. English. ⟨tel-02361353⟩

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