Sparse stabilization of dynamical systems driven by attraction and avoidance forces

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March 2014, 9(1): 1-31. doi: 10.3934/nhm.2014.9.1

Sparse stabilization of dynamical systems driven by attraction and avoidance forces

Mattia Bongini ^1, and Massimo Fornasier ^2,

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

Received September 2013 Revised November 2013 Published April 2014

Abstract
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Conditional self-organization and pattern-formation are relevant phenomena arising in biological, social, and economical contexts, and received a growing attention in recent years in mathematical modeling. An important issue related to optimal government strategies is how to design external parsimonious interventions, aiming at enforcing systems to converge to specific patterns. This is in contrast to other models where the players of the systems are allowed to interact freely and are supposed autonomously, either by game rules or by embedded decentralized feedback control rules, to converge to patterns. In this paper we tackle the problem of designing optimal centralized feedback controls for systems of moving particles, subject to mutual attraction and repulsion forces, and friction. Under certain conditions on the attraction and repulsion forces, if the total energy of the system, composed of the sum of its kinetic and potential parts, is below a certain critical threshold, then such systems are known to converge autonomously to the stable configuration of keeping confined and collision avoiding in space, uniformly in time. If the energy is above such a critical level, then the space coherence can be lost. We show that in the latter situation of lost self-organization, one can nevertheless steer the system to return to stable energy levels by feedback controls defined as the minimizers of a certain functional with $l_1$-norm penalty and constraints. Additionally we show that the optimal strategy in this class of controls is necessarily sparse, i.e., the control acts on at most one agent at each time. This is another remarkable example of how homophilious systems, i.e., systems where agents tend to be strongly more influenced by near agents than far ones, are naturally prone to sparse stabilization, explaining the effectiveness of parsimonious interventions of governments in societies.

Keywords: optimal control with ODE constraints, Sparse stabilization, $l_1$-norm minimization., consensus emergence, cohesion and avoidance forces.

Mathematics Subject Classification: Primary: 37B25, 49J15; Secondary: 68P3.

Citation: Mattia Bongini, Massimo Fornasier. Sparse stabilization of dynamical systems driven by attraction and avoidance forces. Networks & Heterogeneous Media, 2014, 9 (1) : 1-31. doi: 10.3934/nhm.2014.9.1

References:

[1]	L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems,, Clarendon Press, (2000). Google Scholar
[2]	J.-P. Aubin and A. Cellina, Differential Inclusions,, Set-valued maps and viability theory, (1984). doi: 10.1007/978-3-642-69512-4. Google Scholar
[3]	M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat, Sparse stabilization and control of alignment models,, Math. Control Relat. Fields, 3 (2013), 447. doi: 10.3934/mcrf.2013.3.447. Google Scholar
[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, 553 (2014), 1. doi: 10.1007/978-3-7091-1785-9_1. Google Scholar
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[8]	F. Cucker and J.-G. Dong, A general collision-avoiding flocking framework,, IEEE Trans. Automat. Control, 56 (2011), 1124. doi: 10.1109/TAC.2011.2107113. Google Scholar
[9]	F. Cucker and J.-G. Dong, A conditional, collision-avoiding, model for swarming,, Discrete and Continuous Dynamical Systems, 34 (2014), 1009. doi: 10.3934/dcds.2014.34.1009. Google Scholar
[10]	F. Cucker and S. Smale, Emergent behavior in flocks,, IEEE Trans. Automat. Control, 52 (2007), 852. doi: 10.1109/TAC.2007.895842. Google Scholar
[11]	F. Cucker and S. Smale, On the mathematics of emergence,, Jpn. J. Math., 2 (2007), 197. doi: 10.1007/s11537-007-0647-x. Google Scholar
[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. Google Scholar
[13]	A. F. Filippov, Differential Equations with Discontinuous Righthand Sides,, Translated from the Russian, (1988). Google Scholar
[14]	M. Fornasier and F. Solombrino, Mean-field optimal control,, preprint, (2013). Google Scholar
[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. doi: 10.1109/TAC.2010.2046113. Google Scholar
[16]	J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics,, Cambridge University Press, (1998). Google Scholar
[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, (2003), 98. Google Scholar
[18]	J.-M. Lasry and P.-L. Lions, Mean field games,, Jpn. J. Math. (3), 2 (2007), 229. doi: 10.1007/s11537-007-0657-8. Google Scholar
[19]	S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus,, SIAM Rev., (). Google Scholar
[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., (2010), 814. doi: 10.1109/ALLERTON.2010.5706992. Google Scholar
[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. Google Scholar
[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. doi: 10.1137/060674909. Google Scholar
[23]	H. G. Tanner, On the controllability of nearest neighbor interconnections,, in Proceedings of the 43rd IEEE Conference on Decision and Control, (2004), 2467. doi: 10.1109/CDC.2004.1428782. Google Scholar
[24]	T. Vicsek and A. Zafeiris, Collective motion,, Physics Reports, 517 (2012), 71. doi: 10.1016/j.physrep.2012.03.004. Google Scholar

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References:

[1]	L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems,, Clarendon Press, (2000). Google Scholar
[2]	J.-P. Aubin and A. Cellina, Differential Inclusions,, Set-valued maps and viability theory, (1984). doi: 10.1007/978-3-642-69512-4. Google Scholar
[3]	M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat, Sparse stabilization and control of alignment models,, Math. Control Relat. Fields, 3 (2013), 447. doi: 10.3934/mcrf.2013.3.447. Google Scholar
[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, 553 (2014), 1. doi: 10.1007/978-3-7091-1785-9_1. Google Scholar
[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. doi: 10.3934/krm.2009.2.363. Google Scholar
[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*, (2010), 297. doi: 10.1007/978-0-8176-4946-3_12. Google Scholar
[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. doi: 10.1016/j.physd.2007.05.007. Google Scholar
[8]	F. Cucker and J.-G. Dong, A general collision-avoiding flocking framework,, IEEE Trans. Automat. Control, 56 (2011), 1124. doi: 10.1109/TAC.2011.2107113. Google Scholar
[9]	F. Cucker and J.-G. Dong, A conditional, collision-avoiding, model for swarming,, Discrete and Continuous Dynamical Systems, 34 (2014), 1009. doi: 10.3934/dcds.2014.34.1009. Google Scholar
[10]	F. Cucker and S. Smale, Emergent behavior in flocks,, IEEE Trans. Automat. Control, 52 (2007), 852. doi: 10.1109/TAC.2007.895842. Google Scholar
[11]	F. Cucker and S. Smale, On the mathematics of emergence,, Jpn. J. Math., 2 (2007), 197. doi: 10.1007/s11537-007-0647-x. Google Scholar
[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. Google Scholar
[13]	A. F. Filippov, Differential Equations with Discontinuous Righthand Sides,, Translated from the Russian, (1988). Google Scholar
[14]	M. Fornasier and F. Solombrino, Mean-field optimal control,, preprint, (2013). Google Scholar
[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. doi: 10.1109/TAC.2010.2046113. Google Scholar
[16]	J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics,, Cambridge University Press, (1998). Google Scholar
[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, (2003), 98. Google Scholar
[18]	J.-M. Lasry and P.-L. Lions, Mean field games,, Jpn. J. Math. (3), 2 (2007), 229. doi: 10.1007/s11537-007-0657-8. Google Scholar
[19]	S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus,, SIAM Rev., (). Google Scholar
[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., (2010), 814. doi: 10.1109/ALLERTON.2010.5706992. Google Scholar
[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. Google Scholar
[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. doi: 10.1137/060674909. Google Scholar
[23]	H. G. Tanner, On the controllability of nearest neighbor interconnections,, in Proceedings of the 43rd IEEE Conference on Decision and Control, (2004), 2467. doi: 10.1109/CDC.2004.1428782. Google Scholar
[24]	T. Vicsek and A. Zafeiris, Collective motion,, Physics Reports, 517 (2012), 71. doi: 10.1016/j.physrep.2012.03.004. Google Scholar

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