The Pontryagin Maximum Principle in the Wasserstein Space

By Benoît Bonnet, Francesco Rossi. Calculus of Variations and Partial Differential Equations, 2017

64 citations recorded by the source corpus.

We prove a Pontryagin Maximum Principle for optimal control problems in the space of probability measures, where the dynamics is given by a transport equation with non-local velocity. We formulate this first-order optimality condition using the formalism of subdifferential calculus in Wasserstein spaces. We show that the geometric approach based on needle variations and on the evolution of the covector (here replaced by the evolution of a mesure on the dual space) can be translated into this formalism.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/d85e404af7cff5eff9bae69ea543b68eae59e2a1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Hélène Frankowska
2. Giuseppe Savar'e
3. S. Lisini
4. Benoît Bonnet
5. Giulia Cavagnari
6. Carlo Orrieri
7. Nicola Gigli
8. Luigi Ambrosio

In Computational Theory and Mathematics

1. Hélène Frankowska
2. Giuseppe Savar'e
3. S. Lisini
4. Benoît Bonnet

In Finance

1. S. Lisini
2. Hélène Frankowska
3. Giuseppe Savar'e
4. Carlo Orrieri
5. Benoît Bonnet

Citations in the corpus

• Lagrangian, Eulerian and Kantorovich formulations of multi-agent optimal control problems: Equivalence and Gamma-convergence (2020, background): “d results of classical optimal control theory to the mean field setting, with a particular attention to the measure-formulation of the problems in Wasserstein spaces. In this direction, let us mention [11,48] for necessary conditions for optimality in the form of a Pontryagin maximum principle, [21] ”
• A Pontryagin Maximum Principle in Wasserstein spaces for constrained optimal control problems (2018, unknown): “t. In this setting, the Pontryagin Maximum Principle ("PMP" for short) is derived by combining Γ-convergence and mean-field limit arguments. Another approach, introduced in our previous work [14] studies the infinite-dimensional problem by means of the classical technique of needle-variation”
• Differential inclusions in Wasserstein spaces: The Cauchy-Lipschitz framework (2020, background): “While a few results have been dealing with controllability properties of continuity equations [36, 37], the major part of the literature has been devoted to the study of optimal control problems, with contributions ranging from existence results [23, 38, 39, 40] and necessary optimality conditions [”