September  2015, 10(3): 699-715. doi: 10.3934/nhm.2015.10.699

Mean--field control and Riccati equations

1. 

RWTH Aachen University, IGPM, Templergraben 55, 52062 Aachen, Germany, Germany

2. 

University of Ferrara, Department of Mathematics and Computer Science, Via Machiavelli 35, 44121 Ferrara, Italy

Received  October 2014 Revised  January 2015 Published  July 2015

We present a control approach for large systems of interacting agents based on the Riccati equation. If the agent dynamics enjoys a strong symmetry the arising high dimensional Riccati equation is simplified and the resulting coupled system allows for a formal mean--field limit. The steady--states of the kinetic equation of Boltzmann and Fokker Planck type can be studied analytically. In case of linear dynamics and quadratic objective function the presented approach is optimal and is compared to the model predictive control approach introduced in [2].
Citation: Michael Herty, Lorenzo Pareschi, Sonja Steffensen. Mean--field control and Riccati equations. Networks and Heterogeneous Media, 2015, 10 (3) : 699-715. doi: 10.3934/nhm.2015.10.699
References:
[1]

G. Albi, Kinetic Approximation, Stability and Control of Collective Behavior in Self-Organized Systems, Ph.D. thesis, Università degli Studi di Ferrara, 2013.

[2]

G. Albi, M. Herty and L. Pareschi, Kinetic description of optimal control problems and applications to opinion consensus, Comm. Math. Sciences, to appear, (2015).

[3]

G. Albi and L. Pareschi, Binary interaction algorithms for the simulation of flocking and swarming dynamics, Multiscale Model. Simul., 11 (2013), 1-29. doi: 10.1137/120868748.

[4]

________, Modeling of self-organized systems interacting with a few individuals: From microscopic to macroscopic dynamics, Appl. Math. Lett., 26 (2013), 397-401.

[5]

G. Albi, L. Pareschi and M. Zanella, Boltzmann-type control of opinion consensus through leaders, Phil. Trans. R. Soc. A, 372 (2014), 20140138, 18pp. doi: 10.1098/rsta.2014.0138.

[6]

D. Armbruster and C. Ringhofer, Thermalized kinetic and fluid models for re-entrant supply chains, SIAM J. Multiscale Modeling and Simulation, 3 (2005), 782-800. doi: 10.1137/030601636.

[7]

N. Bellomo, G. Ajmone Marsan and A. Tosin, Complex Systems and Society. Modeling and Simulation, Springer Briefs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-7242-1.

[8]

N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci., 22 (2012), 1140006, 29pp. doi: 10.1142/S0218202511400069.

[9]

A. Bensoussan, J. Frehse and P. Yam, Mean Field Games and Mean Field Type Control Theory, Springer Briefs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-8508-7.

[10]

A. Borzí and S. Wongkaew, Modeling and control through leadership of a refined flocking system, Math. Models Methods Appl. Sci., 25 (2015), 255-282. doi: 10.1142/S0218202515500098.

[11]

L. Boudin and F. Salvarani, A kinetic approach to the study of opinion formation, ESAIM: Math. Mod. Num. Anal., 43 (2009), 507-522. doi: 10.1051/m2an/2009004.

[12]

R. M. Colombo and N. Pogodaev, Confinement strategies in a model for the interaction between individuals and a continuum, SIAM J. Appl. Dyn. Syst., 11 (2012), 741-770. doi: 10.1137/110854321.

[13]

_______, On the control of moving sets: Positive and negative confinement results, SIAM J. Control Optim., 51 (2013), 380-401.

[14]

S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Stat. Phys., 120 (2005), 253-277. doi: 10.1007/s10955-005-5456-0.

[15]

I. Couzin, J. Krause, N. Franks and S. Levin, Effective leadership and decision-making in animal groups on the move, Nature, 433 (2005), 513-516. doi: 10.1038/nature03236.

[16]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.

[17]

P. Degond, M. Herty and J.-G. Liu, Flow on sweeping networks, Multiscale Modeling & Simulation, 12 (2014), 538-565. doi: 10.1137/130927061.

[18]

P. Degond, J. Liu and C. Ringhofer, Evolution of the distribution of wealth in economic neighborhood by local nash equilibrium closure, preprint, 2013.

[19]

P. Degond, J.-G. Liu, S. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods and Applications of Analysis, 20 (2013), 89-114. doi: 10.4310/MAA.2013.v20.n2.a1.

[20]

P. Degond, J.-G. Liu and C. Ringhofer, Large-scale dynamics of mean-field games driven by local nash equilibria, Journal of Nonlinear Science, 24 (2014), 93-115. doi: 10.1007/s00332-013-9185-2.

[21]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215. doi: 10.1142/S0218202508003005.

[22]

B. Düring, P. Markowich, J.-F. Pietschmann and M.-T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proc. R. Soc. A, 465 (2009), 3687-3708. doi: 10.1098/rspa.2009.0239.

[23]

A. Fleig, L. Grüne and R. Guglielmi, Some results on model predictive control for the Fokker-Planck equation, in Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014), 2014, 1203-1206.

[24]

M. Fornasier, J. Haskovec and G. Toscani, Fluid dynamic description of flocking via Povzner-Boltzmann equation, Physica D, 240 (2011), 21-31. doi: 10.1016/j.physd.2010.08.003.

[25]

M. Fornasier, B. Piccoli and F. Rossi, Mean-field sparse optimal control, Phil. Trans. R. Soc. A, 372 (2014), 20130400, 21pp. doi: 10.1098/rsta.2013.0400.

[26]

M. Fornasier and F. Solombrino, Mean-field optimal control, ESAIM Control Optim. Calc. Var., 20 (2014), 1123-1152. doi: 10.1051/cocv/2014009.

[27]

S. Galam, Y. Gefen and Y. Shapir, Sociophysics: A new approach of sociological collective behavior, J. Math. Sociology, 9 (1982), 1-13.

[28]

J. Gómez-Serrano, C. Graham and J.-Y. Le Boudec, The bounded confidence model of opinion dynamics, Math. Models Methods Appl. Sci., 22 (2012), 1150007, 46pp. doi: 10.1142/S0218202511500072.

[29]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415.

[30]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence, models, analysis and simulation, Journal of Artificial Societies and Social Simulation, 5 (2002), p2.

[31]

M. Herty and L. Pareschi, Fokker-planck asymptotics for traffic flow models, Kinet. Relat. Models, 3 (2010), 165-179. doi: 10.3934/krm.2010.3.165.

[32]

M. Herty and C. Ringhofer, Averaged kinetic models for flows on unstructured networks, Kinet. Relat. Models, 4 (2011), 1081-1096. doi: 10.3934/krm.2011.4.1081.

[33]

_______, Feedback controls for continuous priority models in supply chain management, Comput. Methods Appl. Math., 11 (2011), 206-213.

[34]

J.-M. Lasry and P.-L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.

[35]

D. Maldarella and L. Pareschi, Kinetic models for socio-economic dynamics of speculative markets, Physica A, 391 (2012), 715-730. doi: 10.1016/j.physa.2011.08.013.

[36]

P. Markowich, M. Burger and L. Caffarelli, eds., Partial differential equation models in the socio-economic sciences, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130406, 8pp. doi: 10.1098/rsta.2013.0406.

[37]

D. Q. Mayne and H. Michalska, Receding horizon control of nonlinear systems, IEEE Trans. Automat. Control, 35 (1990), 814-824. doi: 10.1109/9.57020.

[38]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621. doi: 10.1137/120901866.

[39]

G. Naldi, L. Pareschi and G. Toscani, eds., Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, 2010. doi: 10.1007/978-0-8176-4946-3.

[40]

L. Pareschi and G. Toscani, Interacting Multi-Agent Systems. Kinetic Equations & Monte Carlo Methods, Oxford University Press, 2013.

[41]

E. D. Sontag, Mathematical Control Theory, Texts in Applied Mathematics, 6, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7.

[42]

G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496. doi: 10.4310/CMS.2006.v4.n3.a1.

[43]

T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140. doi: 10.1016/j.physrep.2012.03.004.

show all references

References:
[1]

G. Albi, Kinetic Approximation, Stability and Control of Collective Behavior in Self-Organized Systems, Ph.D. thesis, Università degli Studi di Ferrara, 2013.

[2]

G. Albi, M. Herty and L. Pareschi, Kinetic description of optimal control problems and applications to opinion consensus, Comm. Math. Sciences, to appear, (2015).

[3]

G. Albi and L. Pareschi, Binary interaction algorithms for the simulation of flocking and swarming dynamics, Multiscale Model. Simul., 11 (2013), 1-29. doi: 10.1137/120868748.

[4]

________, Modeling of self-organized systems interacting with a few individuals: From microscopic to macroscopic dynamics, Appl. Math. Lett., 26 (2013), 397-401.

[5]

G. Albi, L. Pareschi and M. Zanella, Boltzmann-type control of opinion consensus through leaders, Phil. Trans. R. Soc. A, 372 (2014), 20140138, 18pp. doi: 10.1098/rsta.2014.0138.

[6]

D. Armbruster and C. Ringhofer, Thermalized kinetic and fluid models for re-entrant supply chains, SIAM J. Multiscale Modeling and Simulation, 3 (2005), 782-800. doi: 10.1137/030601636.

[7]

N. Bellomo, G. Ajmone Marsan and A. Tosin, Complex Systems and Society. Modeling and Simulation, Springer Briefs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-7242-1.

[8]

N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci., 22 (2012), 1140006, 29pp. doi: 10.1142/S0218202511400069.

[9]

A. Bensoussan, J. Frehse and P. Yam, Mean Field Games and Mean Field Type Control Theory, Springer Briefs in Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-8508-7.

[10]

A. Borzí and S. Wongkaew, Modeling and control through leadership of a refined flocking system, Math. Models Methods Appl. Sci., 25 (2015), 255-282. doi: 10.1142/S0218202515500098.

[11]

L. Boudin and F. Salvarani, A kinetic approach to the study of opinion formation, ESAIM: Math. Mod. Num. Anal., 43 (2009), 507-522. doi: 10.1051/m2an/2009004.

[12]

R. M. Colombo and N. Pogodaev, Confinement strategies in a model for the interaction between individuals and a continuum, SIAM J. Appl. Dyn. Syst., 11 (2012), 741-770. doi: 10.1137/110854321.

[13]

_______, On the control of moving sets: Positive and negative confinement results, SIAM J. Control Optim., 51 (2013), 380-401.

[14]

S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Stat. Phys., 120 (2005), 253-277. doi: 10.1007/s10955-005-5456-0.

[15]

I. Couzin, J. Krause, N. Franks and S. Levin, Effective leadership and decision-making in animal groups on the move, Nature, 433 (2005), 513-516. doi: 10.1038/nature03236.

[16]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.

[17]

P. Degond, M. Herty and J.-G. Liu, Flow on sweeping networks, Multiscale Modeling & Simulation, 12 (2014), 538-565. doi: 10.1137/130927061.

[18]

P. Degond, J. Liu and C. Ringhofer, Evolution of the distribution of wealth in economic neighborhood by local nash equilibrium closure, preprint, 2013.

[19]

P. Degond, J.-G. Liu, S. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods and Applications of Analysis, 20 (2013), 89-114. doi: 10.4310/MAA.2013.v20.n2.a1.

[20]

P. Degond, J.-G. Liu and C. Ringhofer, Large-scale dynamics of mean-field games driven by local nash equilibria, Journal of Nonlinear Science, 24 (2014), 93-115. doi: 10.1007/s00332-013-9185-2.

[21]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215. doi: 10.1142/S0218202508003005.

[22]

B. Düring, P. Markowich, J.-F. Pietschmann and M.-T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proc. R. Soc. A, 465 (2009), 3687-3708. doi: 10.1098/rspa.2009.0239.

[23]

A. Fleig, L. Grüne and R. Guglielmi, Some results on model predictive control for the Fokker-Planck equation, in Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014), 2014, 1203-1206.

[24]

M. Fornasier, J. Haskovec and G. Toscani, Fluid dynamic description of flocking via Povzner-Boltzmann equation, Physica D, 240 (2011), 21-31. doi: 10.1016/j.physd.2010.08.003.

[25]

M. Fornasier, B. Piccoli and F. Rossi, Mean-field sparse optimal control, Phil. Trans. R. Soc. A, 372 (2014), 20130400, 21pp. doi: 10.1098/rsta.2013.0400.

[26]

M. Fornasier and F. Solombrino, Mean-field optimal control, ESAIM Control Optim. Calc. Var., 20 (2014), 1123-1152. doi: 10.1051/cocv/2014009.

[27]

S. Galam, Y. Gefen and Y. Shapir, Sociophysics: A new approach of sociological collective behavior, J. Math. Sociology, 9 (1982), 1-13.

[28]

J. Gómez-Serrano, C. Graham and J.-Y. Le Boudec, The bounded confidence model of opinion dynamics, Math. Models Methods Appl. Sci., 22 (2012), 1150007, 46pp. doi: 10.1142/S0218202511500072.

[29]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415.

[30]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence, models, analysis and simulation, Journal of Artificial Societies and Social Simulation, 5 (2002), p2.

[31]

M. Herty and L. Pareschi, Fokker-planck asymptotics for traffic flow models, Kinet. Relat. Models, 3 (2010), 165-179. doi: 10.3934/krm.2010.3.165.

[32]

M. Herty and C. Ringhofer, Averaged kinetic models for flows on unstructured networks, Kinet. Relat. Models, 4 (2011), 1081-1096. doi: 10.3934/krm.2011.4.1081.

[33]

_______, Feedback controls for continuous priority models in supply chain management, Comput. Methods Appl. Math., 11 (2011), 206-213.

[34]

J.-M. Lasry and P.-L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.

[35]

D. Maldarella and L. Pareschi, Kinetic models for socio-economic dynamics of speculative markets, Physica A, 391 (2012), 715-730. doi: 10.1016/j.physa.2011.08.013.

[36]

P. Markowich, M. Burger and L. Caffarelli, eds., Partial differential equation models in the socio-economic sciences, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130406, 8pp. doi: 10.1098/rsta.2013.0406.

[37]

D. Q. Mayne and H. Michalska, Receding horizon control of nonlinear systems, IEEE Trans. Automat. Control, 35 (1990), 814-824. doi: 10.1109/9.57020.

[38]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621. doi: 10.1137/120901866.

[39]

G. Naldi, L. Pareschi and G. Toscani, eds., Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, 2010. doi: 10.1007/978-0-8176-4946-3.

[40]

L. Pareschi and G. Toscani, Interacting Multi-Agent Systems. Kinetic Equations & Monte Carlo Methods, Oxford University Press, 2013.

[41]

E. D. Sontag, Mathematical Control Theory, Texts in Applied Mathematics, 6, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7.

[42]

G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496. doi: 10.4310/CMS.2006.v4.n3.a1.

[43]

T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140. doi: 10.1016/j.physrep.2012.03.004.

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