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 L^{1}-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 control, mean-field limit, Γ-limit, optimal control with ODE constraints, optimal control with PDE constraints

@article{COCV_2014__20_4_1123_0, author = {Fornasier, Massimo and Solombrino, Francesco}, title = {Mean-Field {Optimal} {Control}}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1123--1152}, publisher = {EDP-Sciences}, volume = {20}, number = {4}, year = {2014}, doi = {10.1051/cocv/2014009}, mrnumber = {3264236}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2014009/} }

TY - JOUR AU - Fornasier, Massimo AU - Solombrino, Francesco TI - Mean-Field Optimal Control JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 1123 EP - 1152 VL - 20 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2014009/ DO - 10.1051/cocv/2014009 LA - en ID - COCV_2014__20_4_1123_0 ER -

%0 Journal Article %A Fornasier, Massimo %A Solombrino, Francesco %T Mean-Field Optimal Control %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 1123-1152 %V 20 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2014009/ %R 10.1051/cocv/2014009 %G en %F COCV_2014__20_4_1123_0

Fornasier, Massimo; Solombrino, Francesco. Mean-Field Optimal Control. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 4, pp. 1123-1152. doi : 10.1051/cocv/2014009. http://archive.numdam.org/articles/10.1051/cocv/2014009/

[1] Controlled McKean-Vlasov equations. Commun. Appl. Anal. 5 (2001) 183-206. | MR | Zbl

and ,[2] Functions of Bounded Variation and Free Discontinuity Problems. Oxford, Clarendon Press (2000). | MR | Zbl

, and ,[3] Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lect. Math. ETH Zürich 2nd, edition. Birkhäuser Verlag, Basel (2008). | MR | Zbl

, and ,[4] A maximum principle for SDEs of mean-field type. Appl. Math. Opt. 63 (2011) 341-356. | MR | Zbl

and ,[5] 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.

, , , , , , , , , , and ,[6] Mean field games and mean field type control theory. Springer, New York (2013). | MR | Zbl

, and ,[7] Introduction to the Mathematical Theory of Control, vol. 2 of AIMS Ser. Appl. Math.. American Institute of Mathematical Sciences (AIMS), Springfield, MO (2007). | MR | Zbl

and ,[8] A general stochastic maximum principle for sdes of mean-field type. Appl. Math. Opt. 64 (2011) 197-216. | MR | Zbl

, and ,[9] Self-Organization in Biological Systems. Princeton University Press (2003). | MR | Zbl

, , , , and ,[10] A well-posedness theory in measures for some kinetic models of collective motion. Math. Model. Meth. Appl. Sci. 21 (2011) 515-539. | Zbl

, and ,[11] Sparse stabilization and control of the Cucker−Smale model. Preprint: arXiv:1210.5739 (2012).

, , and ,[12] The derivation of swarming models: mean-field limit and Wasserstein distances. Preprint: arXiv:1304.5776 (2013).

, and ,[13] Double milling in self-propelled swarms from kinetic theory. Kinet. Relat. Models 2 (2009) 363-378. | MR | Zbl

, and ,[14] 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. | MR | Zbl

, , and ,[15] Approximation of elliptic control problems in measure spaces with sparse solutions. SIAM J. Control Optim. 50 (2012) 1735-1752. | MR

, and ,[16] State transition and the continuum limit for the 2D interacting, self-propelled particle system. Physica D 232 (2007) 33-47. | MR

, , , and ,[17] Multi-vehicle flocking: scalability of cooperative control algorithms using pairwise potentials. IEEE Int. Conference on Robotics and Automation (2007) 2292-2299.

, , and ,[18] A duality-based approach to elliptic control problems in non-reflexive Banach spaces. ESAIM: COCV 17 (2011) 243-266. | Numdam | MR | Zbl

and ,[19] A measure space approach to optimal source placement. Comput. Optim. Appl. 53 (2012) 155-171. | MR | Zbl

and ,[20] Self-organized lane formation and optimized traffic flow in army ants. Proc. R. Soc. London B 270 (2002) 139-146.

and ,[21] Effective leadership and decision making in animal groups on the move. Nature 433 (2005) 513-516.

, , and ,[22] 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.

and ,[23] 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). | MR | Zbl

, and ,[24] Multiscale modeling of granular flows with application to crowd dynamics. Multiscale Model. Simul. 9 (2011) 155-182. | MR | Zbl

, and ,[25] A general collision-avoiding flocking framework. IEEE Trans. Automat. Control 56 (2011) 1124-1129. | MR

and ,[26] Flocking in noisy environments. J. Math. Pures Appl. 89 (2008) 278-296. | MR | Zbl

and ,[27] Emergent behavior in flocks. IEEE Trans. Automat. Control 52 (2007) 852-862,. | MR

and ,[28] On the mathematics of emergence. Japan J. Math. 2 (2007) 197-227. | MR | Zbl

and ,[29] Modeling language evolution. Found. Comput. Math. 4 (2004) 315-343. | MR | Zbl

, and ,[30] An Introduction to Γ-Convergence. Progress in Nonlinear Differ. Eqs. Appl., vol. 8. Birkhäuser Boston Inc., Boston, MA (1993). | MR | Zbl

,[31] Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, vol. 153. Springer-Verlag, Berlin, Heidelberg, New York (1969). | MR | Zbl

,[32] Differential equations with Discontinuous Righthand Sides. Vol. 18 of Math. Appl. (Soviet Series). Translated from the Russian. Kluwer Academic Publishers Group, Dordrecht (1988). | MR | Zbl

,[33] Numerical Methods for Nonlinear Variational Problems. Scientific Comput. Springer-Verlag, Berlin (2008). Reprint of the 1984 original. | MR | Zbl

,[34] Onset of collective and cohesive motion. Phys. Rev. Lett. 92 (2004).

and ,[35] Directional sparsity in optimal control of partial differential equations. SIAM J. Control Optim. 50 (2012) 943-963. | MR | Zbl

, and ,[36] 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.

, and ,[37] 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. | MR

, and ,[38] Self-organization and selection in the emergence of vocabulary. Complexity 7 (2002) 41-54. | MR

, , and ,[39] Initiation of slime mold aggregation viewed as an instability. J. Theoret. Biol. 26 (1970) 399-415. | Zbl

and ,[40] The social lifestyle of myxobacteria. Bioessays 20 (1998) 1030-1038.

and ,[41] Mean field games. Japan J. Math. 2 (2007) 229-260. | MR | Zbl

and ,[42] Virtual leaders, artificial potentials and coordinated control of groups. Proc. of 40th IEEE Conf. Decision Contr. (2001) 2968-2973.

and ,[43] Self-organizing dynamic model of fish schooling. J. Theoret. Biol. 171 (1994) 123-136.

,[44] 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.

, and ,[45] 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.

, and ,[46] Complexity, pattern and evolutionary trade-offs in animal aggregation. Science 294 (1999) 99-101.

and ,[47] Self-organized fish schools: An examination of emergent properties. Biol. Bull. 202 (2002) 296-305.

, and ,[48] Extension of the Cucker-Smale control law to space flight formations. AIAA J. Guidance, Control, and Dynamics 32 2009 527-537.

, and ,[49] Mathematical tools for kinetic equations. Bull. Am. Math. Soc., New Ser. 41 (2004) 205-244. | MR | Zbl

,[50] Transport Equations in Biology. Basel, Birkhäuser (2007). | MR | Zbl

,[51] A priori error analysis for discretization of sparse elliptic optimal control problems in measure space. SIAM J. Control Optim. 51 (2013) 2788-2808. | MR

and ,[52] Complexity and regularity of maximal energy domains for the wave equation with fixed initial data. Discrete Contin. Dyn. Syst. Ser. A.

, and ,[53] Adaptive finite element discretization in PDE-based optimization. GAMM-Mitt. 33 (2010) 177-193. | MR | Zbl

and ,[54] Individual differences make a difference in the trajectories of simulated schools of fish. Ecol. Model. 92 (1996) 65-77.

,[55] A statistical model of criminal behavior. Math. Models Methods Appl. Sci. 18 (2008) 1249-1267. | MR | Zbl

, , , , , and ,[56] Elliptic optimal control problems with L1-control cost and applications for the placement of control devices. Comput. Optim. Appl. 44 (2009) 159-181. | MR | Zbl

,[57] Cooperative acceleration of task performance: Foraging behavior of interacting multi-robots system. Phys. D 100 (1997) 343-354. | Zbl

and ,[58] Long-range order in a two-dimensional dynamical xy model: How birds fly together. Phys. Rev. Lett. 75 (1995) 4326-4329.

and ,[59] Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75 (1995) 1226-1229.

, , , and ,[60] Collective motion. Phys. Rep. 517 (2012) 71-140.

and ,[61] Optimal Transport, vol. 338. Grundlehren der Math. Wissenschaften, [Fundamental Principles of Mathematical Science]. Springer-Verlag, Berlin (2009). Old and new. | MR | Zbl

,[62] L1 minimization in optimal control and applications to robotics. Optim. Control Appl. Methods 27 (2006) 301-321. | MR

and ,[63] Convergence and regularization results for optimal control problems with sparsity functional. ESAIM: COCV 17 (2011) 858-886. | Numdam | MR | Zbl

and ,[64] Inherent noise can facilitate coherence in collective swarm motion. Proc. Natl. Acad. Sci. 106 (2009) 5464-5469.

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