Mean-Field Optimal Control
ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 4, pp. 1123-1152.

We introduce the concept of mean-field optimal control which is the rigorous limit process connecting finite dimensional optimal control problems with ODE constraints modeling multi-agent interactions to an infinite dimensional optimal control problem with a constraint given by a PDE of Vlasov-type, governing the dynamics of the probability distribution of interacting agents. While in the classical mean-field theory one studies the behavior of a large number of small individuals freely interacting with each other, by simplifying the effect of all the other individuals on any given individual by a single averaged effect, we address the situation where the individuals are actually influenced also by an external policy maker, and we propagate its effect for the number N of individuals going to infinity. On the one hand, from a modeling point of view, we take into account also that the policy maker is constrained to act according to optimal strategies promoting its most parsimonious interaction with the group of individuals. This will be realized by considering cost functionals including L1-norm terms penalizing a broadly distributed control of the group, while promoting its sparsity. On the other hand, from the analysis point of view, and for the sake of generality, we consider broader classes of convex control penalizations. In order to develop this new concept of limit rigorously, we need to carefully combine the classical concept of mean-field limit, connecting the finite dimensional system of ODE describing the dynamics of each individual of the group to the PDE describing the dynamics of the respective probability distribution, with the well-known concept of Γ-convergence to show that optimal strategies for the finite dimensional problems converge to optimal strategies of the infinite dimensional problem.

DOI : 10.1051/cocv/2014009
Classification : 49J15, 49J20, 35Q83, 35Q91, 37N25
Mots-clés : Sparse optimal control, mean-field limit, Γ-limit, optimal control with ODE constraints, optimal control with PDE constraints 
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Fornasier, Massimo; Solombrino, Francesco. Mean-Field Optimal Control. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 4, pp. 1123-1152. doi : 10.1051/cocv/2014009. https://www.numdam.org/articles/10.1051/cocv/2014009/

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  • Huang, Minyi; Yang, Xuwei Linear quadratic mean field social optimization: Asymptotic solvability and decentralized control, Applied Mathematics and Optimization, Volume 84 (2021), pp. 1969-2010 | DOI:10.1007/s00245-021-09817-0 | Zbl:1486.49045
  • Albi, Giacomo; Ferrarese, Federica; Segala, Chiara Optimized leaders strategies for crowd evacuation in unknown environments with multiple exits, Crowd dynamics. Volume 3. Modeling and social applications in the time of COVID-19, Cham: Birkhäuser, 2021, pp. 97-131 | DOI:10.1007/978-3-030-91646-6_5 | Zbl:1513.76032
  • Burger, Martin; Kreusser, Lisa Maria; Totzeck, Claudia Mean-field optimal control for biological pattern formation, European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations, Volume 27 (2021), p. 24 (Id/No 40) | DOI:10.1051/cocv/2021034 | Zbl:1467.49014
  • Gong, Xiaoqian; Piccoli, Benedetto; Visconti, Giuseppe Mean-Field of Optimal Control Problems for Hybrid Model of Multilane Traffic, IEEE Control Systems Letters, Volume 5 (2021) no. 6, p. 1964 | DOI:10.1109/lcsys.2020.3046540
  • Bonnet, Benoît; Rossi, Francesco Variance Optimization and Control Regularity for Mean-Field Dynamics, IFAC-PapersOnLine, Volume 54 (2021) no. 19, p. 13 | DOI:10.1016/j.ifacol.2021.11.048
  • Caflisch, Russel; Silantyev, Denis; Yang, Yunan Adjoint DSMC for nonlinear Boltzmann equation constrained optimization, Journal of Computational Physics, Volume 439 (2021), p. 29 (Id/No 110404) | DOI:10.1016/j.jcp.2021.110404 | Zbl:1537.76146
  • Bonnet, Benoît; Frankowska, Hélène Differential inclusions in Wasserstein spaces: the Cauchy-Lipschitz framework, Journal of Differential Equations, Volume 271 (2021), pp. 594-637 | DOI:10.1016/j.jde.2020.08.031 | Zbl:1454.49018
  • Azmi, Behzad; Kalise, Dante; Kunisch, Karl Optimal feedback law recovery by gradient-augmented sparse polynomial regression, Journal of Machine Learning Research (JMLR), Volume 22 (2021), p. 32 (Id/No 48) | Zbl:1539.65072
  • Cesaroni, Annalisa; Cirant, Marco One-dimensional multi-agent optimal control with aggregation and distance constraints: qualitative properties and mean-field limit, Nonlinearity, Volume 34 (2021) no. 3, pp. 1408-1447 | DOI:10.1088/1361-6544/abc795 | Zbl:1467.82048
  • Burger, Martin; Pinnau, René; Totzeck, Claudia; Tse, Oliver Mean-field optimal control and optimality conditions in the space of probability measures, SIAM Journal on Control and Optimization, Volume 59 (2021) no. 2, pp. 977-1006 | DOI:10.1137/19m1249461 | Zbl:1460.49019
  • Bonnet, Benoît; Rossi, Francesco Intrinsic Lipschitz regularity of mean-field optimal controls, SIAM Journal on Control and Optimization, Volume 59 (2021) no. 3, pp. 2011-2046 | DOI:10.1137/20m1321474 | Zbl:1466.35068
  • Bonnet, Benoît; Frankowska, Hélène, 2020 59th IEEE Conference on Decision and Control (CDC) (2020), p. 470 | DOI:10.1109/cdc42340.2020.9303835
  • Zhou, Yuyang; Herzallah, Randa Probabilistic message passing control and FPD based decentralised control for stochastic complex systems, AIMS Electronics and Electrical Engineering, Volume 4 (2020) no. 2, p. 216 | DOI:10.3934/electreng.2020.2.216
  • Jimenez, Chloé; Marigonda, Antonio; Quincampoix, Marc Optimal control of multiagent systems in the Wasserstein space, Calculus of Variations and Partial Differential Equations, Volume 59 (2020) no. 2, p. 45 (Id/No 58) | DOI:10.1007/s00526-020-1718-6 | Zbl:1436.49003
  • Keerthana, Ganeshan; Anandan, Panneerselvam; Nandhagopal, Nachimuthu Robust Hybrid Artificial Fish Swarm Simulated Annealing Optimization Algorithm for Secured Free Scale Networks against Malicious Attacks, Computers, Materials Continua, Volume 66 (2020) no. 1, p. 903 | DOI:10.32604/cmc.2020.012255
  • Banda, M. K.; Herty, M.; Trimborn, T. Recent developments in controlled crowd dynamics, Crowd Dynamics, Volume 2. Theory, models, and applications, Cham: Birkhäuser, 2020, pp. 133-157 | DOI:10.1007/978-3-030-50450-2_7 | Zbl:1490.93056
  • Albi, Giacomo; Cristiani, Emiliano; Pareschi, Lorenzo; Peri, Daniele Mathematical Models and Methods for Crowd Dynamics Control, Crowd Dynamics, Volume 2 (2020), p. 159 | DOI:10.1007/978-3-030-50450-2_8
  • Burger, Martin; Pinnau, René; Totzeck, Claudia; Tse, Oliver; Roth, Andreas Instantaneous control of interacting particle systems in the mean-field limit, Journal of Computational Physics, Volume 405 (2020), p. 20 (Id/No 109181) | DOI:10.1016/j.jcp.2019.109181 | Zbl:1454.82025
  • Duprez, Michel; Morancey, Morgan; Rossi, Francesco Minimal time for the continuity equation controlled by a localized perturbation of the velocity vector field, Journal of Differential Equations, Volume 269 (2020) no. 1, pp. 82-124 | DOI:10.1016/j.jde.2019.11.098 | Zbl:1436.93021
  • Carrillo, José A.; Pimentel, Edgard A.; Voskanyan, Vardan K. On a mean field optimal control problem, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 199 (2020), p. 13 (Id/No 112039) | DOI:10.1016/j.na.2020.112039 | Zbl:1448.35526
  • Herty, Michael; Pareschi, Lorenzo; Steffensen, Sonja Control strategies for the dynamics of large particle systems, Active particles, Volume 2. Advances in theory, models, and applications, Cham: Birkhäuser, 2019, pp. 149-171 | DOI:10.1007/978-3-030-20297-2_5 | Zbl:1453.93007
  • Bonnet, Benoît; Rossi, Francesco The Pontryagin Maximum Principle in the Wasserstein space, Calculus of Variations and Partial Differential Equations, Volume 58 (2019) no. 1, p. 36 (Id/No 11) | DOI:10.1007/s00526-018-1447-2 | Zbl:1404.49016
  • Bonnet, Benoît A Pontryagin maximum principle in Wasserstein spaces for constrained optimal control problems, European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations, Volume 25 (2019), p. 38 (Id/No 52) | DOI:10.1051/cocv/2019044 | Zbl:1442.49025
  • Fornasier, M.; Lisini, S.; Orrieri, C.; Savaré, G. Mean-field optimal control as gamma-limit of finite agent controls, European Journal of Applied Mathematics, Volume 30 (2019) no. 6, pp. 1153-1186 | DOI:10.1017/s0956792519000044 | Zbl:1427.35302
  • Albi, G.; Bellomo, N.; Fermo, L.; Ha, S.-Y.; Kim, J.; Pareschi, L.; Poyato, D.; Soler, J. Vehicular traffic, crowds, and swarms: from kinetic theory and multiscale methods to applications and research perspectives, M3AS. Mathematical Models Methods in Applied Sciences, Volume 29 (2019) no. 10, pp. 1901-2005 | DOI:10.1142/s0218202519500374 | Zbl:1431.35211
  • Albi, Giacomo; Bongini, Mattia; Rossi, Francesco; Solombrino, Francesco Leader formation with mean-field birth and death models, M3AS. Mathematical Models Methods in Applied Sciences, Volume 29 (2019) no. 4, pp. 633-679 | DOI:10.1142/s0218202519400025 | Zbl:1430.35238
  • Staritsyn, Maxim; Pogodaev, Nikolay Impulsive Relaxation of Continuity Equations and Modeling of Colliding Ensembles, Optimization and Applications, Volume 974 (2019), p. 367 | DOI:10.1007/978-3-030-10934-9_26
  • Pinnau, René; Totzeck, Claudia Optimization Problems for Interacting Particle Systems and Corresponding Mean‐field Limits, PAMM, Volume 19 (2019) no. 1 | DOI:10.1002/pamm.201900148
  • E, Weinan; Han, Jiequn; Li, Qianxiao A mean-field optimal control formulation of deep learning, Research in the Mathematical Sciences, Volume 6 (2019) no. 1, p. 41 (Id/No 10) | DOI:10.1007/s40687-018-0172-y | Zbl:1421.49021
  • Duprez, Michel; Morancey, Morgan; Rossi, Francesco Approximate and exact controllability of the continuity equation with a localized vector field, SIAM Journal on Control and Optimization, Volume 57 (2019) no. 2, pp. 1284-1311 | DOI:10.1137/17m1152917 | Zbl:1411.93028
  • Piccoli, Benedetto; Duteil, Nastassia Pouradier; Trélat, Emmanuel Sparse Control of Hegselmann–Krause Models: Black Hole and Declustering, SIAM Journal on Control and Optimization, Volume 57 (2019) no. 4, p. 2628 | DOI:10.1137/18m1168911
  • Piccoli, Benedetto; Duteil, Nastassia Pouradier; Trélat, Emmanuel Sparse control of Hegselmann-Krause models: black hole and declustering, SIAM Journal on Control and Optimization, Volume 57 (2019) no. 4, pp. 2628-2659 | DOI:10.1137/18m1168911; | Zbl:1422.91620
  • de Badyn, Mathias Hudoba; Eren, Utku; Acikmese, Behcet; Mesbahi, Mehran, 2018 IEEE Conference on Decision and Control (CDC) (2018), p. 1225 | DOI:10.1109/cdc.2018.8619808
  • Albi, Giacomo; Pareschi, Lorenzo Selective model-predictive control for flocking systems, Communications in Applied and Industrial Mathematics, Volume 9 (2018) no. 2, pp. 4-21 | DOI:10.2478/caim-2018-0009 | Zbl:1423.93113
  • Fornasier, Massimo Learning and sparse control of multiagent systems, European congress of mathematics. Proceedings of the 7th ECM (7ECM) congress, Berlin, Germany, July 18–22, 2016, Zürich: European Mathematical Society (EMS), 2018, pp. 551-581 | DOI:10.4171/176-1/26 | Zbl:1401.93014
  • Zhang, Fangbo; Bertozzi, Andrea L.; Elamvazhuthi, Karthik; Berman, Spring Performance Bounds on Spatial Coverage Tasks by Stochastic Robotic Swarms, IEEE Transactions on Automatic Control, Volume 63 (2018) no. 6, p. 1563 | DOI:10.1109/tac.2017.2747769
  • Bailo, Rafael; Bongini, Mattia; Carrillo, José A.; Kalise, Dante Optimal consensus control of the Cucker-Smale model, IFAC-PapersOnLine, Volume 51 (2018) no. 13, p. 1 | DOI:10.1016/j.ifacol.2018.07.245
  • Staritsyn, Maksim Vladimirovich; Maltugueva, Nadezhda Stanislavovna; Pogodaev, Nikolaĭ Il'ich; Sorokin, Stepan Pavlovich Impulsive control of systems with network structure describing spread of political influence, Izvestiya Irkutskogo Gosudarstvennogo Universiteta. Seriya Matematika, Volume 25 (2018), pp. 126-143 | DOI:10.26516/1997-7670.2018.25.126 | Zbl:1409.49035
  • Albi, Giacomo; Choi, Young-Pil; Häck, Axel-Stefan Pressureless Euler alignment system with control, M3AS. Mathematical Models Methods in Applied Sciences, Volume 28 (2018) no. 9, pp. 1635-1664 | DOI:10.1142/s0218202518400018 | Zbl:1411.82025
  • Carmona, René; Delarue, François MFGs with a Common Noise: Strong and Weak Solutions, Probabilistic Theory of Mean Field Games with Applications II, Volume 84 (2018), p. 107 | DOI:10.1007/978-3-319-56436-4_2
  • Carmona, René; Delarue, François Solving MFGs with a Common Noise, Probabilistic Theory of Mean Field Games with Applications II, Volume 84 (2018), p. 155 | DOI:10.1007/978-3-319-56436-4_3
  • Carmona, René; Delarue, François The Master Field and the Master Equation, Probabilistic Theory of Mean Field Games with Applications II, Volume 84 (2018), p. 239 | DOI:10.1007/978-3-319-56436-4_4
  • Carmona, René; Delarue, François Optimization in a Random Environment, Probabilistic Theory of Mean Field Games with Applications II, Volume 84 (2018), p. 3 | DOI:10.1007/978-3-319-56436-4_1
  • Carmona, René; Delarue, François Classical Solutions to the Master Equation, Probabilistic Theory of Mean Field Games with Applications II, Volume 84 (2018), p. 323 | DOI:10.1007/978-3-319-56436-4_5
  • Carmona, René; Delarue, François Convergence and Approximations, Probabilistic Theory of Mean Field Games with Applications II, Volume 84 (2018), p. 447 | DOI:10.1007/978-3-319-56436-4_6
  • Carmona, René; Delarue, François Extensions for Volume II, Probabilistic Theory of Mean Field Games with Applications II, Volume 84 (2018), p. 541 | DOI:10.1007/978-3-319-56436-4_7
  • Staritsyn, Maxim On “discontinuous” continuity equation and impulsive ensemble control, Systems Control Letters, Volume 118 (2018), pp. 77-83 | DOI:10.1016/j.sysconle.2018.06.001 | Zbl:1402.93138
  • Djehiche, Boualem; Tcheukam, Alain; Tembine, Hamidou Mean-Field-Type Games in Engineering, AIMS Electronics and Electrical Engineering, Volume 1 (2017) no. 1, p. 18 | DOI:10.3934/electreng.2017.1.18
  • Albi, Giacomo; Pareschi, Lorenzo; Toscani, Giuseppe; Zanella, Mattia Recent Advances in Opinion Modeling: Control and Social Influence, Active Particles, Volume 1 (2017), p. 49 | DOI:10.1007/978-3-319-49996-3_2
  • Aydoğdu, Aylin; Caponigro, Marco; McQuade, Sean; Piccoli, Benedetto; Pouradier Duteil, Nastassia; Rossi, Francesco; Trélat, Emmanuel Interaction Network, State Space, and Control in Social Dynamics, Active Particles, Volume 1 (2017), p. 99 | DOI:10.1007/978-3-319-49996-3_3
  • Albi, Giacomo; Choi, Young-Pil; Fornasier, Massimo; Kalise, Dante Mean field control hierarchy, Applied Mathematics and Optimization, Volume 76 (2017) no. 1, pp. 93-135 | DOI:10.1007/s00245-017-9429-x | Zbl:1378.49024
  • Herty, Michael; Zanella, Mattia Performance bounds for the mean-field limit of constrained dynamics, Discrete and Continuous Dynamical Systems, Volume 37 (2017) no. 4, pp. 2023-2043 | DOI:10.3934/dcds.2017086 | Zbl:1366.35203
  • Albi, Giacomo; Fornasier, Massimo; Kalise, Dante A Boltzmann approach to mean-field sparse feedback control, IFAC-PapersOnLine, Volume 50 (2017) no. 1, p. 2898 | DOI:10.1016/j.ifacol.2017.08.646
  • Bongini, Mattia; Fornasier, Massimo; Rossi, Francesco; Solombrino, Francesco Mean-field Pontryagin maximum principle, Journal of Optimization Theory and Applications, Volume 175 (2017) no. 1, pp. 1-38 | DOI:10.1007/s10957-017-1149-5 | Zbl:1386.49003
  • Bongini, Mattia; Buttazzo, Giuseppe Optimal control problems in transport dynamics, M3AS. Mathematical Models Methods in Applied Sciences, Volume 27 (2017) no. 3, pp. 427-451 | DOI:10.1142/s0218202517500063 | Zbl:1365.49004
  • Bongini, Mattia; Fornasier, Massimo; Hansen, Markus; Maggioni, Mauro Inferring interaction rules from observations of evolutive systems. I: The variational approach., M3AS. Mathematical Models Methods in Applied Sciences, Volume 27 (2017) no. 5, pp. 909-951 | DOI:10.1142/s0218202517500208 | Zbl:1368.37017
  • Lacker, Daniel Limit theory for controlled McKean-Vlasov dynamics, SIAM Journal on Control and Optimization, Volume 55 (2017) no. 3, pp. 1641-1672 | DOI:10.1137/16m1095895 | Zbl:1362.93167
  • Albi, Giacomo; Bongini, Mattia; Cristiani, Emiliano; Kalise, Dante Invisible control of self-organizing agents leaving unknown environments, SIAM Journal on Applied Mathematics, Volume 76 (2016) no. 4, pp. 1683-1710 | DOI:10.1137/15m1017016 | Zbl:1415.91230
  • Bongini, Mattia; Fornasier, Massimo; Kalise, Dante (Un)conditional consensus emergence under perturbed and decentralized feedback controls, Discrete and Continuous Dynamical Systems, Volume 35 (2015) no. 9, pp. 4071-4094 | DOI:10.3934/dcds.2015.35.4071 | Zbl:1335.93006
  • Bongini, Mattia; Fornasier, Massimo; Junge, Oliver; Scharf, Benjamin Sparse control of alignment models in high dimension, Networks and Heterogeneous Media, Volume 10 (2015) no. 3, pp. 647-697 | DOI:10.3934/nhm.2015.10.647 | Zbl:1336.93013
  • Herty, Michael; Pareschi, Lorenzo; Steffensen, Sonja Mean-field control and Riccati equations, Networks and Heterogeneous Media, Volume 10 (2015) no. 3, pp. 699-715 | DOI:10.3934/nhm.2015.10.699 | Zbl:1332.35372
  • Piccoli, Benedetto; Rossi, Francesco; Trélat, Emmanuel Control to flocking of the kinetic Cucker-Smale model, SIAM Journal on Mathematical Analysis, Volume 47 (2015) no. 6, pp. 4685-4719 | DOI:10.1137/140996501 | Zbl:1327.93230
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