# American Institute of Mathematical Sciences

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

## 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

Received  September 2013 Revised  November 2013 Published  April 2014

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.
Citation: Mattia Bongini, Massimo Fornasier. Sparse stabilization of dynamical systems driven by attraction and avoidance forces. Networks and 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, 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., (). [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., (). [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.
 [1] Lei Wu, Zhe Sun. A new spectral method for $l_1$-regularized minimization. Inverse Problems and Imaging, 2015, 9 (1) : 257-272. doi: 10.3934/ipi.2015.9.257 [2] Yingying Li, Stanley Osher, Richard Tsai. Heat source identification based on $l_1$ constrained minimization. Inverse Problems and Imaging, 2014, 8 (1) : 199-221. doi: 10.3934/ipi.2014.8.199 [3] Pia Heins, Michael Moeller, Martin Burger. Locally sparse reconstruction using the $l^{1,\infty}$-norm. Inverse Problems and Imaging, 2015, 9 (4) : 1093-1137. doi: 10.3934/ipi.2015.9.1093 [4] Lidan Li, Hongwei Zhang, Liwei Zhang. Inverse quadratic programming problem with $l_1$ norm measure. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2425-2437. doi: 10.3934/jimo.2019061 [5] Zhaohui Guo, Stanley Osher. Template matching via $l_1$ minimization and its application to hyperspectral data. Inverse Problems and Imaging, 2011, 5 (1) : 19-35. doi: 10.3934/ipi.2011.5.19 [6] Jiying Liu, Jubo Zhu, Fengxia Yan, Zenghui Zhang. Compressive sampling and $l_1$ minimization for SAR imaging with low sampling rate. Inverse Problems and Imaging, 2013, 7 (4) : 1295-1305. doi: 10.3934/ipi.2013.7.1295 [7] Vladimir Gaitsgory, Tanya Tarnopolskaya. Threshold value of the penalty parameter in the minimization of $L_1$-penalized conditional value-at-risk. Journal of Industrial and Management Optimization, 2013, 9 (1) : 191-204. doi: 10.3934/jimo.2013.9.191 [8] Duo Wang, Zheng-Fen Jin, Youlin Shang. A penalty decomposition method for nuclear norm minimization with l1 norm fidelity term. Evolution Equations and Control Theory, 2019, 8 (4) : 695-708. doi: 10.3934/eect.2019034 [9] Eduardo Casas, Fredi Tröltzsch. Sparse optimal control for the heat equation with mixed control-state constraints. Mathematical Control and Related Fields, 2020, 10 (3) : 471-491. doi: 10.3934/mcrf.2020007 [10] Marco Caponigro, Massimo Fornasier, Benedetto Piccoli, Emmanuel Trélat. Sparse stabilization and optimal control of the Cucker-Smale model. Mathematical Control and Related Fields, 2013, 3 (4) : 447-466. doi: 10.3934/mcrf.2013.3.447 [11] Zhen-Zhen Tao, Bing Sun. Galerkin spectral method for elliptic optimal control problem with $L^2$-norm control constraint. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021220 [12] Gautier Picot. Energy-minimal transfers in the vicinity of the lagrangian point $L_1$. Conference Publications, 2011, 2011 (Special) : 1196-1205. doi: 10.3934/proc.2011.2011.1196 [13] Braxton Osting, Jérôme Darbon, Stanley Osher. Statistical ranking using the $l^{1}$-norm on graphs. Inverse Problems and Imaging, 2013, 7 (3) : 907-926. doi: 10.3934/ipi.2013.7.907 [14] Heinz Schättler, Urszula Ledzewicz, Helmut Maurer. Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2657-2679. doi: 10.3934/dcdsb.2014.19.2657 [15] Zhen-Zhen Tao, Bing Sun. Error estimates for spectral approximation of flow optimal control problem with $L^2$-norm control constraint. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022030 [16] P. R. Zingano. Asymptotic behavior of the $L^1$ norm of solutions to nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (1) : 151-159. doi: 10.3934/cpaa.2004.3.151 [17] Ahmad Mousavi, Zheming Gao, Lanshan Han, Alvin Lim. Quadratic surface support vector machine with L1 norm regularization. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1835-1861. doi: 10.3934/jimo.2021046 [18] Gero Friesecke, Felix Henneke, Karl Kunisch. Frequency-sparse optimal quantum control. Mathematical Control and Related Fields, 2018, 8 (1) : 155-176. doi: 10.3934/mcrf.2018007 [19] Satoshi Ito, Soon-Yi Wu, Ting-Jang Shiu, Kok Lay Teo. A numerical approach to infinite-dimensional linear programming in $L_1$ spaces. Journal of Industrial and Management Optimization, 2010, 6 (1) : 15-28. doi: 10.3934/jimo.2010.6.15 [20] Yupeng Li, Wuchen Li, Guo Cao. Image segmentation via $L_1$ Monge-Kantorovich problem. Inverse Problems and Imaging, 2019, 13 (4) : 805-826. doi: 10.3934/ipi.2019037

2020 Impact Factor: 1.213