An optimal control problem for the continuity equation arising in smart charging

Article Mathematics

Optimal control of the Fokker-Planck equation under state constraints in the Wasserstein space

Samuel Daudin

JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES (2023)

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Article Mathematics, Applied

Mean Field Approximation of an Optimal Control Problem for the Continuity Equation Arising in Smart Charging

Adrien Séguret

APPLIED MATHEMATICS AND OPTIMIZATION (2023)

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Article Mathematics, Applied

Classical and weak solutions to local first-order mean field games through elliptic regularity

Sebastian Munoz

Summary: We study the regularity and well-posedness of the local, first-order forward-backward mean field games system with a polynomially growing cost function and a Hamiltonian of quadratic growth. Depending on the presence of a lower bound for the running cost function, we consider different regimes and study systems and terminal data that are strictly monotone in density. By using a transformation introduced by P.-L. Lions, we obtain an elliptic partial differential equation with oblique boundary conditions, which is strictly elliptic when the coupling is unbounded from below. We prove the smoothness of the solution in this case. When the problem is degenerate elliptic, we establish the existence and uniqueness of weak solutions similar to those obtained by P. Cardaliaguet and P. J. Graber for the case of a terminal condition that is independent of the density. The weak solutions are shown to be the viscous limits of classical solutions to strictly elliptic problems.

ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE (2022)

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Article Automation & Control Systems

A class of hybrid LQG mean field games with state-invariant switching and stopping strategies

Dena Firoozi et al.

Summary: A novel framework that combines Mean Field Game (MFG) theory and Hybrid Optimal Control (HOC) theory is proposed to obtain a unique epsilon-Nash equilibrium for a non-cooperative game with switching and stopping times. The framework considers a major agent and a large number of minor agents, where each agent has stochastic linear dynamics with quadratic costs. The optimal switching and stopping times are shown to be state-invariant and only depend on the dynamical parameters of each agent. Optimal strategies for switchings and stopping times are established using Hybrid LQG MFG theory.

AUTOMATICA (2022)

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Article Mathematics, Applied

Global Well-Posedness of Master Equations for Deterministic Displacement Convex Potential Mean Field Games

Wilfrid Gangbo et al.

Summary: This manuscript constructs global in time solutions to master equations for potential mean field games. The study explores a class of Lagrangians and initial data functions that are displacement convex, which may be in contrast with the widely considered property of Lasry-Lions monotonicity. Solutions to both the scalar and vectorial master equations are constructed in potential mean field games when the underlying space is Rd, which is not compact.

COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS (2022)

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Article Operations Research & Management Science

Optimal Control of Diffusion Processes with Terminal Constraint in Law

Samuel Daudin

Summary: This paper considers stochastic optimal control problems with constraints on the probability distribution of the final output. Necessary conditions for optimality in the form of a coupled system of partial differential equations involving a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation are proved using convex duality techniques.

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS (2022)

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Article Mathematics, Applied

Finite state mean field games with Wright-Fisher common noise

Erhan Bayraktar et al.

Summary: In this study, uniqueness is enforced in finite state mean field games by introducing a Wright-Fisher common noise. The analysis of the master equation reveals a non-linear version of the Kimura type equation, with a unique smooth solution when the drift at the boundary is strong enough. This requires a priori estimates of Holder type for the corresponding Kimura operator with merely continuous drift.

JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES (2021)

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Article Mathematics, Applied

Necessary Optimality Conditions for Optimal Control Problems in Wasserstein Spaces

Benoit Bonnet et al.

Summary: This article derives first-order necessary optimality conditions for a constrained optimal control problem in the Wasserstein space of probability measures, introduces a new notion of localised metric subdifferential, and investigates intrinsic linearised Cauchy problems associated with non-local continuity equations. By using these concepts, the Pontryagin Maximum Principle for optimal control problems with inequality final-point constraints is proven synthetically and geometrically, and sufficient conditions for the normality of the maximum principle are proposed.

APPLIED MATHEMATICS AND OPTIMIZATION (2021)

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Article Automation & Control Systems

INTRINSIC LIPSCHITZ REGULARITY OF MEAN-FIELD OPTIMAL CONTROLS

Benoit Bonnet et al.

Summary: In this article, sufficient conditions for the Lipschitz regularity of the controlled vector fields solution of optimal control problems formulated on continuity equations are provided. The approach involves mean-field approximations and a careful extension of the existence result of locally optimal Lipschitz feedbacks.

SIAM JOURNAL ON CONTROL AND OPTIMIZATION (2021)

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Article Mathematics

Short time solution to the master equation of a first order mean field game

Sergio Mayorga

JOURNAL OF DIFFERENTIAL EQUATIONS (2020)

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Article Mathematics

A variational approach to the mean field planning problem

Carlo Orrieri et al.

JOURNAL OF FUNCTIONAL ANALYSIS (2019)

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Article Mathematics

New estimates on the regularity of the pressure in density-constrained mean field games

Hugo Lavenant et al.

JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES (2019)

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Article Mathematics, Applied

The planning problem in mean field games as regularized mass transport

P. Jameson Graber et al.

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS (2019)

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Article Mathematics, Applied

The Pontryagin Maximum Principle in the Wasserstein Space

Benoit Bonnet et al.

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS (2019)

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Article Mathematics, Applied

Regularity via duality in calculus of variations and degenerate elliptic PDEs

Filippo Santambrogio

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS (2018)

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Article Mathematics, Applied

First-order, stationary mean-field games with congestion

David Evangelista et al.

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS (2018)

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Article Automation & Control Systems

PROXIMAL METHODS FOR STATIONARY MEAN FIELD GAMES WITH LOCAL COUPLINGS

L. M. Briceno-Arias et al.

SIAM JOURNAL ON CONTROL AND OPTIMIZATION (2018)

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Article Mathematics, Applied

Mean Field Type Control with Congestion

Yves Achdou et al.

APPLIED MATHEMATICS AND OPTIMIZATION (2016)

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Article Automation & Control Systems

FIRST ORDER MEAN FIELD GAMES WITH DENSITY CONSTRAINTS: PRESSURE EQUALS PRICE

Pierre Cardaliaguet et al.

SIAM JOURNAL ON CONTROL AND OPTIMIZATION (2016)

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Article Automation & Control Systems

MEAN FIELD GAMES SYSTEMS OF FIRST ORDER

Pierre Cardaliaguet et al.

ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS (2015)

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Article Mathematics, Applied

A variational approach to second order mean field games with density constraints: The stationary case

Alpar Richard Meszaros et al.

JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES (2015)

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Article Mathematics

Existence of a solution to an equation arising from the theory of Mean Field Games

Wilfrid Gangbo et al.

JOURNAL OF DIFFERENTIAL EQUATIONS (2015)

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Article Automation & Control Systems

Modeling, Control, and Stability Analysis of Heterogeneous Thermostatically Controlled Load Populations Using Partial Differential Equations

Azad Ghaffari et al.

JOURNAL OF DYNAMIC SYSTEMS MEASUREMENT AND CONTROL-TRANSACTIONS OF THE ASME (2015)

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Article Operations Research & Management Science

Augmented Lagrangian Methods for Transport Optimization, Mean Field Games and Degenerate Elliptic Equations

Jean-David Benamou et al.

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS (2015)

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Article Mathematics, Applied

Second order mean field games with degenerate diffusion and local coupling

Pierre Cardaliaguet et al.

NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS (2015)

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Article Automation & Control Systems

MEAN FIELD GAMES SYSTEMS OF FIRST ORDER

Pierre Cardaliaguet et al.

ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS (2015)

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Article Mathematics, Applied

Continuous Time Finite State Mean Field Games

Diogo A. Gomes et al.

APPLIED MATHEMATICS AND OPTIMIZATION (2013)

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Article Mathematics, Applied

Geodesics for a class of distances in the space of probability measures

P. Cardaliaguet et al.

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS (2013)

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Article Mathematics, Interdisciplinary Applications

A MODEST PROPOSAL FOR MFG WITH DENSITY CONSTRAINTS

Filippo Santambrogio

NETWORKS AND HETEROGENEOUS MEDIA (2012)

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Article Mathematics, Applied

Discrete time, finite state space mean field games

Diogo A. Gomes et al.

JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES (2010)

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Article Automation & Control Systems

AN OPTIMIZATION PROBLEM FOR MASS TRANSPORTATION WITH CONGESTED DYNAMICS

G. Buttazzo et al.

SIAM JOURNAL ON CONTROL AND OPTIMIZATION (2009)

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Article Automation & Control Systems

Large-population cost-coupled LQG problems with nonuniform agents:: Individual-mass behavior and decentralized ε-Nash equilibria

Minyi Huang et al.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL (2007)

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Article Mathematics

Mean field games. II - Finite horizon and optimal control.

Jean-Michel Lasry et al.

COMPTES RENDUS MATHEMATIQUE (2006)

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Article Mathematics

Mean field games. I - The stationary case.

Jean-Michel Lasry et al.

COMPTES RENDUS MATHEMATIQUE (2006)

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Article Mathematics, Applied

Approximation of the marginal distributions of a semi-Markov process using a finite volume scheme

C Cocozza-Thivent et al.

ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE (2004)

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Article Economics

Envelope theorems for arbitrary choice sets

P Milgrom et al.

ECONOMETRICA (2002)

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Article Mathematics, Applied

A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem

JD Benamou et al.

NUMERISCHE MATHEMATIK (2000)

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An optimal control problem for the continuity equation arising in smart charging

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