Mean-field optimal control as Gamma-limit of finite agent controls

We summarize the relations of optimality systems for an interacting particle dynamic in the microscopic and in the kinetic description. In particular, we answer the question if the passing to the mean-field limit and deriving the first order optimality system can be interchanged without affecting the results. The answer is affirmative, if one derives the optimality system on the kinetic level in the metric space (P2, W2). Moreover, we discuss the relation of to the adjoint PDE derived in the L 2 -sense. Here, the gradient can be derived as expected from the calculus in Wasserstein space.

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“…Moreover, a convergence rate for the convergence of the controls in the limit N → ∞ can be shown [13]. Recent results can be also found in [26].…”

Section: Optimal Control Problem and Relations Of The Optimality Systemsmentioning

confidence: 78%

Optimization Problems for Interacting Particle Systems and Corresponding Mean‐field Limits

Pinnau

Totzeck

2019

Proc Appl Math and Mech

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“…From the point of view of the necessary conditions, in [6] the authors provide a Pontryagin Maximum principle for a problem with interactions encoded in the dynamics by mean of a nonlocal vector field, under strong regularity on the control function. Finally, in [23] the authors investigate Γ-convergence results for a controlled nonlocal dynamics, proving thus the consistency of the mean field model in a measure-theoretic setting with the corresponding one in finite dimension.…”

Section: Introductionmentioning

confidence: 96%

Generalized dynamic programming principle and sparse mean-field control problems

Cavagnari

Marigonda

Piccoli

2020

Journal of Mathematical Analysis and Applications

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In this paper we study optimal control problems in Wasserstein spaces, which are suitable to describe macroscopic dynamics of multi-particle systems. The dynamics is described by a parametrized continuity equation, in which the Eulerian velocity field is affine w.r.t. some variables. Our aim is to minimize a cost functional which includes a control norm, thus enforcing a control sparsity constraint. More precisely, we consider a nonlocal restriction on the total amount of control that can be used depending on the overall state of the evolving mass. We treat in details two main cases: an instantaneous constraint on the control applied to the evolving mass and a cumulative constraint, which depends also on the amount of control used in previous times. For both constraints, we prove the existence of optimal trajectories for general cost functions and that the value function is viscosity solution of a suitable Hamilton-Jacobi-Bellmann equation. Finally, we discuss an abstract Dynamic Programming Principle, providing further applications in the Appendix.

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“…The investigation of (deterministic) optimal control problems in the space of measures is attracting an increasing interest by the mathematical community in the last years, due to the potential applications in the study of complex systems, or multi-agent systems (see, e.g., [16,18,19]). Indeed, in the framework of mean field approximation of multi agent system, starting from a control problem for a large number of the (discrete) agents, the problem is recasted in the framework of probability measures (see the recent [15] or the preprint [12] for -convergence results for optimal control problems with nonlocal dynamics). This procedure reduces the dimensionality and the number of equations, possibly leading to a simpler and treatable problem from the point of view of numerics.…”

Section: Introductionmentioning

confidence: 99%

Compatibility of state constraints and dynamics for multiagent control systems

2021

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This study concerns the problem of compatibility of state constraints with a multiagent control system. Such a system deals with a number of agents so large that only a statistical description is available. For this reason, the state variable is described by a probability measure on $${\mathbb {R}}^d$$ R d representing the density of the agents and evolving according to the so-called continuity equation which is an equation stated in the Wasserstein space of probability measures. The aim of the paper is to provide a necessary and sufficient condition for a given constraint (a closed subset of the Wasserstein space) to be compatible with the controlled continuity equation. This new condition is characterized in a viscosity sense as follows: the distance function to the constraint set is a viscosity supersolution of a suitable Hamilton–Jacobi–Bellman equation stated on the Wasserstein space. As a byproduct and key ingredient of our approach, we obtain a new comparison theorem for evolutionary Hamilton–Jacobi equations in the Wasserstein space.

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Mean-field optimal control as Gamma-limit of finite agent controls

Cited by 66 publications

References 61 publications

Optimization Problems for Interacting Particle Systems and Corresponding Mean‐field Limits

Optimization Problems for Interacting Particle Systems and Corresponding Mean‐field Limits

Generalized dynamic programming principle and sparse mean-field control problems

Compatibility of state constraints and dynamics for multiagent control systems

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