On a mean field optimal control problem

In this paper we consider a mean field optimal control problem with an aggregation-diffusion constraint, where agents interact through a potential, in the presence of a Gaussian noise term. Our analysis focuses on a PDE system coupling a Hamilton-Jacobi and a Fokker-Planck equation, describing the optimal control aspect of the problem and the evolution of the population of agents, respectively. The main contribution of the paper is a result on the existence of solutions for the aforementioned system. We notice this model is in close connection with the theory of mean-field games systems. However, a distinctive feature concerns the nonlocal character of the interaction; it affects the drift term in the Fokker-Planck equation as well as the Hamiltonian of the system, leading to new difficulties to be addressed.

[1]  Yves Achdou,et al.  Mean Field Games: Numerical Methods for the Planning Problem , 2012, SIAM J. Control. Optim..

[2]  M. Chaplain,et al.  Mathematical modelling of cancer cell invasion of tissue: local and non-local models and the effect of adhesion. , 2008 .

[3]  Pierre Cardaliaguet,et al.  Long time behavior of the master equation in mean field game theory , 2017, Analysis & PDE.

[4]  Marco Cirant,et al.  Multi-population Mean Field Games systems with Neumann boundary conditions , 2015 .

[5]  Marco Cirant,et al.  Stationary focusing mean-field games , 2016, 1602.04231.

[6]  Massimo Fornasier,et al.  Mean Field Control Hierarchy , 2016, Applied Mathematics & Optimization.

[7]  T. Kolokolnikov,et al.  PREDICTING PATTERN FORMATION IN PARTICLE INTERACTIONS , 2012 .

[8]  P. C. D. Raynal Strong existence and uniqueness for degenerate SDE with Hölder drift , 2017 .

[9]  Radek Erban,et al.  Mathematical Modelling of Turning Delays in Swarm Robotics , 2014, ArXiv.

[10]  Massimo Fornasier,et al.  Un)conditional consensus emergence under perturbed and decentralized feedback controls , 2015 .

[11]  Pierre Degond,et al.  Kinetic Theory of Particle Interactions Mediated by Dynamical Networks , 2016, Multiscale Model. Simul..

[12]  Massimo Fornasier,et al.  Particle, kinetic, and hydrodynamic models of swarming , 2010 .

[13]  Diogo A. Gomes,et al.  Time dependent mean-field games in the superquadratic case , 2013, 1311.6684.

[14]  Pierre Cardaliaguet,et al.  Mean field games systems of first order , 2014, 1401.1789.

[15]  P. Cardaliaguet,et al.  Mean Field Games , 2020, Lecture Notes in Mathematics.

[16]  Lincoln Chayes,et al.  The McKean–Vlasov Equation in Finite Volume , 2009, 0910.4615.

[17]  Benedetto Piccoli,et al.  Control to Flocking of the Kinetic Cucker-Smale Model , 2014, SIAM J. Math. Anal..

[18]  Pierre Degond,et al.  Particle Interactions Mediated by Dynamical Networks: Assessment of Macroscopic Descriptions , 2017, Journal of Nonlinear Science.

[19]  Wilfrid Gangbo,et al.  Existence of a solution to an equation arising from the theory of Mean Field Games , 2015 .

[20]  B. Perthame Transport Equations in Biology , 2006 .

[21]  A. Bensoussan,et al.  On The Interpretation Of The Master Equation , 2015, 1503.07754.

[22]  Benjamin Moll,et al.  Optimal Development Policies with Financial Frictions , 2014, Econometrica.

[23]  M. Fornasier,et al.  Mean-Field Optimal Control , 2013, 1306.5913.

[24]  Darryl D. Holm,et al.  Aggregation of finite-size particles with variable mobility. , 2005, Physical review letters.

[25]  Marco Scianna,et al.  Adhesion and volume constraints via nonlocal interactions determine cell organisation and migration profiles. , 2018, Journal of theoretical biology.

[26]  Massimo Fornasier,et al.  Mean-field sparse optimal control , 2014, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[27]  S. Martin,et al.  Explicit flock solutions for Quasi-Morse potentials , 2013, European Journal of Applied Mathematics.

[28]  P. Lions Generalized Solutions of Hamilton-Jacobi Equations , 1982 .

[29]  Minyi Huang,et al.  Large-Population Cost-Coupled LQG Problems With Nonuniform Agents: Individual-Mass Behavior and Decentralized $\varepsilon$-Nash Equilibria , 2007, IEEE Transactions on Automatic Control.

[30]  J A Sherratt,et al.  A Nonlocal Model for Contact Attraction and Repulsion in Heterogeneous Cell Populations , 2015, Bulletin of Mathematical Biology.

[31]  조준학,et al.  Growth of human bronchial epithelial cells at an air-liquid interface alters the response to particle exposure , 2013, Particle and Fibre Toxicology.

[32]  Guy Theraulaz,et al.  Self-Organization in Biological Systems , 2001, Princeton studies in complexity.

[33]  Yann Brenier,et al.  Extended Monge-Kantorovich Theory , 2003 .

[34]  Pierre-Louis Lions,et al.  Partial differential equation models in macroeconomics , 2014, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[35]  Pierre-Louis Lions,et al.  The Dynamics of Inequality , 2015 .

[36]  Yves Achdou,et al.  Iterative strategies for solving linearized discrete mean field games systems , 2012, Networks Heterog. Media.

[37]  Diogo A. Gomes,et al.  Regularity Theory for Mean-Field Game Systems , 2016 .

[38]  C. Hemelrijk,et al.  Self-organised complex aerial displays of thousands of starlings: a model , 2009, 0908.2677.

[39]  Diogo A. Gomes,et al.  Time-Dependent Mean-Field Games with Logarithmic Nonlinearities , 2014, SIAM J. Math. Anal..

[40]  Yves Achdou,et al.  Mean Field Games: Numerical Methods , 2010, SIAM J. Numer. Anal..

[41]  P. Lions,et al.  Jeux à champ moyen. I – Le cas stationnaire , 2006 .

[42]  Diogo A. Gomes,et al.  Time dependent mean-field games in the superquadratic case , 2013 .

[43]  Pierre Cardaliaguet,et al.  Weak Solutions for First Order Mean Field Games with Local Coupling , 2013, 1305.7015.

[44]  A. Bensoussan,et al.  Mean Field Games and Mean Field Type Control Theory , 2013 .

[45]  A. Bertozzi,et al.  Self-propelled particles with soft-core interactions: patterns, stability, and collapse. , 2006, Physical review letters.

[46]  P. Lions,et al.  Jeux à champ moyen. II – Horizon fini et contrôle optimal , 2006 .

[47]  Diogo A. Gomes,et al.  A stochastic Evans-Aronsson problem , 2013 .

[48]  Edgard A. Pimentel,et al.  Regularity for second order stationary mean-field games , 2015, 1503.06445.

[49]  Diogo A. Gomes,et al.  On the existence of classical solutions for stationary extended mean field games , 2013, 1305.2696.

[50]  Pierre-Louis Lions,et al.  Long time average of mean field games , 2012, Networks Heterog. Media.

[51]  Martino Bardi,et al.  Uniqueness of solutions in Mean Field Games with several populations and Neumann conditions , 2018 .

[52]  Massimo Fornasier,et al.  Sparse Stabilization and Control of Alignment Models , 2012, 1210.5739.

[53]  Pierre-Louis Lions,et al.  Long Time Average of Mean Field Games with a Nonlocal Coupling , 2013, SIAM J. Control. Optim..

[54]  Alain Bensoussan,et al.  The Master equation in mean field theory , 2014, 1404.4150.

[55]  K. Painter,et al.  A User's Guide to Pde Models for Chemotaxis , 2022 .

[56]  P. Markowich,et al.  Boltzmann and Fokker–Planck equations modelling opinion formation in the presence of strong leaders , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[57]  Lawrence C. Evans,et al.  Adjoint and Compensated Compactness Methods for Hamilton–Jacobi PDE , 2010 .

[58]  Dan Crisan,et al.  CLASSICAL SOLUTIONS TO THE MASTER EQUATION FOR LARGE POPULATION EQUILIBRIA , 2014 .

[59]  C. Villani Topics in Optimal Transportation , 2003 .

[60]  Peter E. Caines,et al.  Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle , 2006, Commun. Inf. Syst..

[61]  M. Fornasier,et al.  Sparse stabilization and optimal control of the Cucker-Smale model , 2013 .

[62]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[63]  Leah Edelstein-Keshet,et al.  Inferring individual rules from collective behavior , 2010, Proceedings of the National Academy of Sciences.

[64]  P. Lions,et al.  Mean field games , 2007 .

[65]  P. Cardaliaguet,et al.  Second order mean field games with degenerate diffusion and local coupling , 2014, 1407.7024.

[66]  P. C. D. Raynal Strong existence and uniqueness for stochastic differential equation with Hölder drift and degenerate noise , 2012, 1205.6688.

[67]  Francois Delarue,et al.  The Master Equation for Large Population Equilibriums , 2014, 1404.4694.

[68]  Diogo A. Gomes,et al.  A-priori estimates for stationary mean-field games , 2012, Networks Heterog. Media.

[69]  Alessio Porretta,et al.  Weak Solutions to Fokker–Planck Equations and Mean Field Games , 2015 .

[70]  M. Novaga,et al.  On stationary fractional mean field games , 2017, Journal de Mathématiques Pures et Appliquées.

[71]  Yves Achdou,et al.  Finite Difference Methods for Mean Field Games , 2013 .