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Sparse stabilization of dynamical systems driven by attraction and avoidance forces
1. | Technische Universität München, Fakultät Mathematik, Boltzmannstraße 3, D-85748 Garching, Germany |
2. | Technische Universität München, Facultät Mathematik, Boltzmannstrasse 3, D-85748, Garching bei München |
References:
[1] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford University Press, New York, 2000. |
[2] |
J.-P. Aubin and A. Cellina, Differential Inclusions, Set-valued maps and viability theory, Grundlehren der Mathematischen Wissenschaften, 264, Springer-Verlag, Berlin, 1984.
doi: 10.1007/978-3-642-69512-4. |
[3] |
M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat, Sparse stabilization and control of alignment models, Math. Control Relat. Fields, 3 (2013), 447-466. Available from: http://www-m15.ma.tum.de/foswiki/pub/M15/Allgemeines/PublicationsEN/flocking_V9.pdf.
doi: 10.3934/mcrf.2013.3.447. |
[4] |
J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, in Collective Dynamics from Bacteria to Crowds, CISM International Centre for Mechanical Sciences, 553, Springer, 2014, 1-46.
doi: 10.1007/978-3-7091-1785-9_1. |
[5] |
J. A. Carrillo, M. R. D'Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinet. Relat. Models, 2 (2009), 363-378.
doi: 10.3934/krm.2009.2.363. |
[6] |
J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences (eds. G. Naldi, L. Pareschi, G. Toscani and N. Bellomo), Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, 2010, 297-336.
doi: 10.1007/978-0-8176-4946-3_12. |
[7] |
Y. Chuang, M. D'Orsogna, D. Marthaler, A. Bertozzi and L. Chayes, State transition and the continuum limit for the 2D interacting, self-propelled particle system, Physica D, 232 (2007), 33-47.
doi: 10.1016/j.physd.2007.05.007. |
[8] |
F. Cucker and J.-G. Dong, A general collision-avoiding flocking framework, IEEE Trans. Automat. Control, 56 (2011), 1124-1129.
doi: 10.1109/TAC.2011.2107113. |
[9] |
F. Cucker and J.-G. Dong, A conditional, collision-avoiding, model for swarming, Discrete and Continuous Dynamical Systems, 34 (2014), 1009-1020.
doi: 10.3934/dcds.2014.34.1009. |
[10] |
F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[11] |
F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.
doi: 10.1007/s11537-007-0647-x. |
[12] |
M. D'Orsogna, Y. Chuang, A. Bertozzi and L. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006).
doi: 10.1103/PhysRevLett.96.104302. |
[13] |
A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Translated from the Russian, Mathematics and its Applications (Soviet Series), 18, Kluwer Academic Publishers Group, Dordrecht, 1988. |
[14] |
M. Fornasier and F. Solombrino, Mean-field optimal control, preprint, arXiv:1306.5913, (2013). |
[15] |
S.-Y. Ha, T. Ha and J.-H. Kim, Emergent behavior of a Cucker-Smale type particle model with nonlinear velocity couplings, IEEE Trans. Automat. Control, 55 (2010), 1679-1683.
doi: 10.1109/TAC.2010.2046113. |
[16] |
J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. |
[17] |
M. Huang, P. Caines and R. Malhamé, Individual and mass behaviour in large population stochastic wireless power control problems: Centralized and nash equilibrium solutions, in Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii, USA, December, 2003, 98-103. |
[18] |
J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math. (3), 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
[19] |
S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Rev., to appear. |
[20] |
M. Nuorian, P. Caines and R. Malhamé, Synthesis of Cucker-Smale type flocking via mean field stochastic control theory: Nash equilibria, in Proceedings of the 48th Allerton Conf. on Comm., Cont. and Comp., Monticello, Illinois, 2010, 814-819.
doi: 10.1109/ALLERTON.2010.5706992. |
[21] |
M. Nuorian, P. Caines and R. Malhamé, Mean field analysis of controlled Cucker-Smale type flocking: Linear analysis and perturbation equations, in Proceedings of 18th IFAC World Congress Milano (Italy) August 28-September 2, 2011, 4471-4476. |
[22] |
A. Rahmani, M. Ji, M. Mesbahi and M. Egerstedt, Controllability of multi-agent systems from a graph-theoretic perspective, SIAM J. Control and Optimization, 48 (2009), 162-186.
doi: 10.1137/060674909. |
[23] |
H. G. Tanner, On the controllability of nearest neighbor interconnections, in Proceedings of the 43rd IEEE Conference on Decision and Control, IEEE Press, Piscataway, NJ, 2004, 2467-2472.
doi: 10.1109/CDC.2004.1428782. |
[24] |
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] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford University Press, New York, 2000. |
[2] |
J.-P. Aubin and A. Cellina, Differential Inclusions, Set-valued maps and viability theory, Grundlehren der Mathematischen Wissenschaften, 264, Springer-Verlag, Berlin, 1984.
doi: 10.1007/978-3-642-69512-4. |
[3] |
M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat, Sparse stabilization and control of alignment models, Math. Control Relat. Fields, 3 (2013), 447-466. Available from: http://www-m15.ma.tum.de/foswiki/pub/M15/Allgemeines/PublicationsEN/flocking_V9.pdf.
doi: 10.3934/mcrf.2013.3.447. |
[4] |
J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, in Collective Dynamics from Bacteria to Crowds, CISM International Centre for Mechanical Sciences, 553, Springer, 2014, 1-46.
doi: 10.1007/978-3-7091-1785-9_1. |
[5] |
J. A. Carrillo, M. R. D'Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinet. Relat. Models, 2 (2009), 363-378.
doi: 10.3934/krm.2009.2.363. |
[6] |
J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences (eds. G. Naldi, L. Pareschi, G. Toscani and N. Bellomo), Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, 2010, 297-336.
doi: 10.1007/978-0-8176-4946-3_12. |
[7] |
Y. Chuang, M. D'Orsogna, D. Marthaler, A. Bertozzi and L. Chayes, State transition and the continuum limit for the 2D interacting, self-propelled particle system, Physica D, 232 (2007), 33-47.
doi: 10.1016/j.physd.2007.05.007. |
[8] |
F. Cucker and J.-G. Dong, A general collision-avoiding flocking framework, IEEE Trans. Automat. Control, 56 (2011), 1124-1129.
doi: 10.1109/TAC.2011.2107113. |
[9] |
F. Cucker and J.-G. Dong, A conditional, collision-avoiding, model for swarming, Discrete and Continuous Dynamical Systems, 34 (2014), 1009-1020.
doi: 10.3934/dcds.2014.34.1009. |
[10] |
F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[11] |
F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.
doi: 10.1007/s11537-007-0647-x. |
[12] |
M. D'Orsogna, Y. Chuang, A. Bertozzi and L. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006).
doi: 10.1103/PhysRevLett.96.104302. |
[13] |
A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Translated from the Russian, Mathematics and its Applications (Soviet Series), 18, Kluwer Academic Publishers Group, Dordrecht, 1988. |
[14] |
M. Fornasier and F. Solombrino, Mean-field optimal control, preprint, arXiv:1306.5913, (2013). |
[15] |
S.-Y. Ha, T. Ha and J.-H. Kim, Emergent behavior of a Cucker-Smale type particle model with nonlinear velocity couplings, IEEE Trans. Automat. Control, 55 (2010), 1679-1683.
doi: 10.1109/TAC.2010.2046113. |
[16] |
J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. |
[17] |
M. Huang, P. Caines and R. Malhamé, Individual and mass behaviour in large population stochastic wireless power control problems: Centralized and nash equilibrium solutions, in Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii, USA, December, 2003, 98-103. |
[18] |
J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math. (3), 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
[19] |
S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Rev., to appear. |
[20] |
M. Nuorian, P. Caines and R. Malhamé, Synthesis of Cucker-Smale type flocking via mean field stochastic control theory: Nash equilibria, in Proceedings of the 48th Allerton Conf. on Comm., Cont. and Comp., Monticello, Illinois, 2010, 814-819.
doi: 10.1109/ALLERTON.2010.5706992. |
[21] |
M. Nuorian, P. Caines and R. Malhamé, Mean field analysis of controlled Cucker-Smale type flocking: Linear analysis and perturbation equations, in Proceedings of 18th IFAC World Congress Milano (Italy) August 28-September 2, 2011, 4471-4476. |
[22] |
A. Rahmani, M. Ji, M. Mesbahi and M. Egerstedt, Controllability of multi-agent systems from a graph-theoretic perspective, SIAM J. Control and Optimization, 48 (2009), 162-186.
doi: 10.1137/060674909. |
[23] |
H. G. Tanner, On the controllability of nearest neighbor interconnections, in Proceedings of the 43rd IEEE Conference on Decision and Control, IEEE Press, Piscataway, NJ, 2004, 2467-2472.
doi: 10.1109/CDC.2004.1428782. |
[24] |
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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