Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-23T06:37:03.208Z Has data issue: false hasContentIssue false
  • English
  • Français

Mean-Field Optimal Control

Published online by Cambridge University Press:  08 August 2014

Massimo Fornasier  and
Francesco Solombrino
Massimo Fornasier
Affiliation:
Technische Universität München, Fakultät Mathematik, 3 85748 Boltzmannstrasse, Garching bei München, Germany. massimo.fornasier@ma.tum.de; francesco.solombrino@ma.tum.de
Francesco Solombrino
Affiliation:
Technische Universität München, Fakultät Mathematik, 3 85748 Boltzmannstrasse, Garching bei München, Germany. massimo.fornasier@ma.tum.de; francesco.solombrino@ma.tum.de
Get access

Abstract

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.

Keywords

Sparse optimal controlmean-field limitΓ-limitoptimal control with ODE constraintsoptimal control with PDE constraints
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 4 , October 2014 , pp. 1123 - 1152
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahmed, N. and Ding, X., Controlled McKean-Vlasov equations. Commun. Appl. Anal. 5 (2001) 183–206. Google Scholar
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford, Clarendon Press (2000). Google Scholar
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lect. Math. ETH Zürich 2nd, edition. Birkhäuser Verlag, Basel (2008). Google Scholar
Andersson, D. and Djehiche, B., A maximum principle for SDEs of mean-field type. Appl. Math. Opt. 63 (2011) 341–356. Google Scholar
Ballerini, M., Cabibbo, N., Candelier, R., Cavagna, A., Cisbani, E., Giardina, L., Lecomte, L., Orlandi, A., Parisi, G., Procaccini, A., Viale, M. and Zdravkovic, V., Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study. Proc. National Academy of Sci. 105 (2008) 1232–1237. Google ScholarPubMed
A. Bensoussan, J. Frehse and P. Yam, Mean field games and mean field type control theory. Springer, New York (2013). CrossRefGoogle Scholar
A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, vol. 2 of AIMS Ser. Appl. Math.. American Institute of Mathematical Sciences (AIMS), Springfield, MO (2007). Google Scholar
Buckdahn, R., Djehiche, B. and Li, J., A general stochastic maximum principle for sdes of mean-field type. Appl. Math. Opt. 64 (2011) 197–216. Google Scholar
S. Camazine, J. Deneubourg, N. Franks, J. Sneyd, G. Theraulaz and E. Bonabeau, Self-Organization in Biological Systems. Princeton University Press (2003). Google Scholar
Cañizo, J.A., Carrillo, J.A. and Rosado, J., A well-posedness theory in measures for some kinetic models of collective motion. Math. Model. Meth. Appl. Sci. 21 (2011) 515–539. Google Scholar
M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat, Sparse stabilization and control of the Cucker−Smale model. Preprint: arXiv:1210.5739 (2012). Google Scholar
J.A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: mean-field limit and Wasserstein distances. Preprint: arXiv:1304.5776 (2013). CrossRefGoogle Scholar
Carrillo, J.A., D’Orsogna, M.R. and Panferov, V., Double milling in self-propelled swarms from kinetic theory. Kinet. Relat. Models 2 (2009) 363–378. Google Scholar
J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Math. Modeling of Collective Behavior in Socio-Economic and Life Sci., edited by G. Naldi, L. Pareschi, G. Toscani and N. Bellomo. Model. Simul. Sci. Engrg. Technol. Birkhäuser, Boston (2010) 297–336. Google Scholar
Casas, E., Clason, C. and Kunisch, K., Approximation of elliptic control problems in measure spaces with sparse solutions. SIAM J. Control Optim. 50 (2012) 1735–1752. Google Scholar
Chuang, Y., D’Orsogna, M., Marthaler, D., Bertozzi, A. and Chayes, L., State transition and the continuum limit for the 2D interacting, self-propelled particle system. Physica D 232 (2007) 33–47. Google Scholar
Y. Chuang, Y. Huang, M. D’Orsogna and A. Bertozzi, Multi-vehicle flocking: scalability of cooperative control algorithms using pairwise potentials. IEEE Int. Conference on Robotics and Automation (2007) 2292–2299. Google Scholar
Clason, C. and Kunisch, K., A duality-based approach to elliptic control problems in non-reflexive Banach spaces. ESAIM: COCV 17 (2011) 243–266. Google Scholar
Clason, C. and Kunisch, K., A measure space approach to optimal source placement. Comput. Optim. Appl. 53 (2012) 155–171. Google Scholar
Couzin, I. and Franks, N., Self-organized lane formation and optimized traffic flow in army ants. Proc. R. Soc. London B 270 (2002) 139–146. Google Scholar
Couzin, I., Krause, J., Franks, N. and Levin, S., Effective leadership and decision making in animal groups on the move. Nature 433 (2005) 513–516. Google Scholar
Craig, A. J. and Flügge-Lotz, I., Investigation of optimal control with a minimum-fuel consumption criterion for a fourth-order plant with two control inputs; synthesis of an efficient suboptimal control. J. Basic Engrg. 87 (1965) 39–58. Google Scholar
E. Cristiani, B. Piccoli and A. Tosin, Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, edited by G. Naldi, L. Pareschi, G. Toscani and N. Bellomo. Model. Simul. Sci. Engrg. Technol. Birkhäuser, Boston (2010). Google Scholar
Cristiani, E., Piccoli, B. and Tosin, A., Multiscale modeling of granular flows with application to crowd dynamics. Multiscale Model. Simul. 9 (2011) 155–182. Google Scholar
Cucker, F. and Dong, J.-G., A general collision-avoiding flocking framework. IEEE Trans. Automat. Control 56 (2011) 1124–1129. Google Scholar
Cucker, F. and Mordecki, E., Flocking in noisy environments. J. Math. Pures Appl. 89 (2008) 278–296. Google Scholar
Cucker, F. and Smale, S., Emergent behavior in flocks. IEEE Trans. Automat. Control 52 (2007) 852–862,. Google Scholar
Cucker, F. and Smale, S., On the mathematics of emergence. Japan J. Math. 2 (2007) 197–227. Google Scholar
Cucker, F., Smale, S. and Zhou, D., Modeling language evolution. Found. Comput. Math. 4 (2004) 315–343. Google Scholar
G. Dal Maso, An Introduction to Γ-Convergence. Progress in Nonlinear Differ. Eqs. Appl., vol. 8. Birkhäuser Boston Inc., Boston, MA (1993). Google Scholar
H. Federer, Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, vol. 153. Springer-Verlag, Berlin, Heidelberg, New York (1969). Google Scholar
A.F. Filippov, Differential equations with Discontinuous Righthand Sides. Vol. 18 of Math. Appl. (Soviet Series). Translated from the Russian. Kluwer Academic Publishers Group, Dordrecht (1988). Google Scholar
R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Scientific Comput. Springer-Verlag, Berlin (2008). Reprint of the 1984 original. CrossRefGoogle Scholar
G. Grégoire and H. Chaté, Onset of collective and cohesive motion. Phys. Rev. Lett. 92 (2004). Google Scholar
Herzog, R., Stadler, G. and Wachsmuth, G., Directional sparsity in optimal control of partial differential equations. SIAM J. Control Optim. 50 (2012) 943–963. Google Scholar
M. Huang, P. Caines and R. Malhamé, Individual and mass behaviour in large population stochastic wireless power control problems: centralized and Nash equilibrium solutions. Proc. of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA (2003) 98–103. Google Scholar
Jadbabaie, A., Lin, J. and Morse, A.S., Correction to: “Coordination of groups of mobile autonomous agents using nearest neighbor rules” [48 (2003) 988–1001; MR 1986266]. IEEE Trans. Automat. Control 48 (2003) 1675. Google Scholar
Ke, J., Minett, J., Au, C.-P. and Wang, W.-Y., Self-organization and selection in the emergence of vocabulary. Complexity 7 (2002) 41–54. Google Scholar
Keller, E. F. and Segel, L.A., Initiation of slime mold aggregation viewed as an instability. J. Theoret. Biol. 26 (1970) 399–415. Google Scholar
Koch, A. and White, D., The social lifestyle of myxobacteria. Bioessays 20 (1998) 1030–1038. Google Scholar
Lasry, J.-M. and Lions, P.-L., Mean field games. Japan J. Math. 2 (2007) 229–260. Google Scholar
N. Leonard and E. Fiorelli, Virtual leaders, artificial potentials and coordinated control of groups. Proc. of 40th IEEE Conf. Decision Contr. (2001) 2968–2973. Google Scholar
Niwa, H., Self-organizing dynamic model of fish schooling. J. Theoret. Biol. 171 (1994) 123–136. Google Scholar
M. Nuorian, P. Caines and R. Malhamé, Synthesis of Cucker−Smale type flocking via mean field stochastic control theory: Nash equilibria. Proc. of 48th Allerton Conf. Comm., Cont. Comp., Monticello, Illinois (2010) 814–815. Google Scholar
M. Nuorian, P. Caines and R. Malhamé, Mean field analysis of controlled Cucker−Smale type flocking: Linear analysis and perturbation equations. Proc. of 18th IFAC World Congress Milano, Italy (2011) 4471–4476. Google Scholar
Parrish, J. and Edelstein-Keshet, L., Complexity, pattern and evolutionary trade-offs in animal aggregation. Science 294 (1999) 99–101. Google Scholar
Parrish, J., Viscido, S. and Gruenbaum, D., Self-organized fish schools: An examination of emergent properties. Biol. Bull. 202 (2002) 296–305. Google ScholarPubMed
Perea, L., Gómez, G. and Elosegui, P., Extension of the Cucker–Smale control law to space flight formations. AIAA J. Guidance, Control, and Dynamics 32 2009 527–537. Google Scholar
Perthame, B., Mathematical tools for kinetic equations. Bull. Am. Math. Soc., New Ser. 41 (2004) 205–244. Google Scholar
B. Perthame, Transport Equations in Biology. Basel, Birkhäuser (2007). Google Scholar
Pieper, K. and Vexler, B., A priori error analysis for discretization of sparse elliptic optimal control problems in measure space. SIAM J. Control Optim. 51 (2013) 2788–2808. Google Scholar
Y. Privat, E. Trélat and E. Zuazua, Complexity and regularity of maximal energy domains for the wave equation with fixed initial data. Discrete Contin. Dyn. Syst. Ser. A. Google Scholar
Rannacher, R. and Vexler, B., Adaptive finite element discretization in PDE-based optimization. GAMM-Mitt. 33 (2010) 177–193. Google Scholar
Romey, W., Individual differences make a difference in the trajectories of simulated schools of fish. Ecol. Model. 92 (1996) 65–77. Google Scholar
Short, M.B., D’Orsogna, M. R., Pasour, V.B., Tita, G.E., Brantingham, P.J., Bertozzi, A.L. and Chayes, L.B., A statistical model of criminal behavior. Math. Models Methods Appl. Sci. 18 (2008) 1249–1267. Google Scholar
Stadler, G., Elliptic optimal control problems with L 1-control cost and applications for the placement of control devices. Comput. Optim. Appl. 44 (2009) 159–181. Google Scholar
Sugawara, K. and Sano, M., Cooperative acceleration of task performance: Foraging behavior of interacting multi-robots system. Phys. D 100 (1997) 343–354. Google Scholar
Toner, J. and Tu, Y., Long-range order in a two-dimensional dynamical xy model: How birds fly together. Phys. Rev. Lett. 75 (1995) 4326–4329. Google Scholar
Vicsek, T., Czirok, A., Ben-Jacob, E., Cohen, I. and Shochet, O., Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75 (1995) 1226–1229. Google Scholar
Vicsek, T. and Zafeiris, A., Collective motion. Phys. Rep. 517 (2012) 71–140. Google Scholar
C. Villani, Optimal Transport, vol. 338. Grundlehren der Math. Wissenschaften, [Fundamental Principles of Mathematical Science]. Springer-Verlag, Berlin (2009). Old and new. Google Scholar
Vossen, G. and Maurer, H., L 1 minimization in optimal control and applications to robotics. Optim. Control Appl. Methods 27 (2006) 301–321. Google Scholar
Wachsmuth, G. and Wachsmuth, D., Convergence and regularization results for optimal control problems with sparsity functional. ESAIM: COCV 17 (2011) 858–886. Google Scholar
Yates, C., Erban, R., Escudero, C., Couzin, L., Buhl, J., Kevrekidis, L., Maini, P. and Sumpter, D., Inherent noise can facilitate coherence in collective swarm motion. Proc. Natl. Acad. Sci. 106 (2009) 5464–5469. Google ScholarPubMed