# American Institute of Mathematical Sciences

December  2013, 3(4): 447-466. doi: 10.3934/mcrf.2013.3.447

## Sparse stabilization and optimal control of the Cucker-Smale model

 1 Conservatoire National des Arts et Métiers, Département Ingénierie Mathématique (IMATH), Équipe M2N, 292 rue Saint-Martin, 75003, Paris,, France 2 Technische Universität München, Facultät Mathematik, Boltzmannstrasse 3, D-85748, Garching bei München, Germany 3 Rutgers University, Department of Mathematics, Business & Science Building Room 325, Camden, NJ 08102, United States 4 Université Pierre et Marie Curie (Univ. Paris 6) and Institut Universitaire de France and Team GECO Inria Saclay, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

Received  November 2012 Revised  June 2013 Published  September 2013

This article is mainly based on the work [7], and it is dedicated to the 60th anniversary of B. Bonnard, held in Dijon in June 2012.
We focus on a controlled Cucker--Smale model in finite dimension. Such dynamics model self-organization and consensus emergence in a group of agents. We explore how it is possible to control this model in order to enforce or facilitate pattern formation or convergence to consensus. In particular, we are interested in designing control strategies that are componentwise sparse in the sense that they require a small amount of external intervention, and also time sparse in the sense that such strategies are not chattering in time. These sparsity features are desirable in view of practical issues.
We first show how very simple sparse feedback strategies can be designed with the use of a variational principle, in order to steer the system to consensus. These feedbacks are moreover optimal in terms of decay rate of some functional, illustrating the general principle according to which sparse is better''. We then combine these results with local controllability properties to get global controllability results. Finally, we explore the sparsity properties of the optimal control minimizing a combination of the distance from consensus and of a norm of the control.
Citation: Marco Caponigro, Massimo Fornasier, Benedetto Piccoli, Emmanuel Trélat. Sparse stabilization and optimal control of the Cucker-Smale model. Mathematical Control & Related Fields, 2013, 3 (4) : 447-466. doi: 10.3934/mcrf.2013.3.447
##### References:
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Google Scholar [11] Y. Chuang, Y. Huang, M. D'Orsogna and A. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials,, in, (2007), 2292. doi: 10.1109/ROBOT.2007.363661. Google Scholar [12] F. H. Clarke, Y. S. Ledyaev, E. D. Sontag and A. I. Subbotin, Asymptotic controllability implies feedback stabilization,, IEEE Trans. Automat. Control, 42 (1997), 1394. doi: 10.1109/9.633828. Google Scholar [13] C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces,, ESAIM Control Optim. Calc. Var., 17 (2011), 243. doi: 10.1051/cocv/2010003. Google Scholar [14] C. Clason and K. Kunisch, A measure space approach to optimal source placement,, Comput. Optim. Appl., 53 (2012), 155. doi: 10.1007/s10589-011-9444-9. Google Scholar [15] J.-M. Coron and E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equations,, SIAM J. 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Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar [37] A. Lachapelle and M. T. Wolfram, On a mean field game approach modeling congestion and aversion in pedestrian crowds,, Trans. Res.: Part B: Methodological, 45 (2011), 1572. doi: 10.1016/j.trb.2011.07.011. Google Scholar [38] 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 [39] S. Lemercier, A. Jelic, R. Kulpa, J. Hua, J. Fehrenbach, P. Degond, C. Appert-Rolland, S. Donikian and J. Pettré, Realistic following behaviors for crowd simulation,, Computer Graphics Forum, 31 (2012), 489. doi: 10.1111/j.1467-8659.2012.03028.x. Google Scholar [40] N. Leonard and E. Fiorelli, Virtual leaders, artificial potentials and coordinated control of groups,, in, (2001), 2968. Google Scholar [41] S. Mallat, "A Wavelet Tour of Signal Processing. The Sparse Way,", Third edition, (2009). Google Scholar [42] M. Moussaïd, E. G. Guillot, M. Moreau, J. Fehrenbach, O. Chabiron, S. Lemercier, J. Pettré, C. Appert-Rolland, P. Degond and G. Theraulaz, Traffic instabilities in self-organized pedestrian crowds,, PLoS Computational Biology, 8 (2012). Google Scholar [43] H. Niwa, Self-organizing dynamic model of fish schooling,, J. Theor. Biol., 171 (1994), 123. doi: 10.1006/jtbi.1994.1218. Google Scholar [44] J. Parrish and L. Edelstein-Keshet, Complexity, pattern, and evolutionary trade-offs in animal aggregation,, Science, 294 (1999), 99. doi: 10.1126/science.284.5411.99. Google Scholar [45] J. Parrish, S. Viscido and D. Gruenbaum, Self-organized fish schools: An examination of emergent properties,, Biol. Bull., 202 (2002), 296. doi: 10.2307/1543482. Google Scholar [46] C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311. doi: 10.1007/BF02476407. Google Scholar [47] L. Perea, G. 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##### References:
 [1] M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, L. Giardina, L. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study,, PNAS, 105 (2008), 1232. doi: 10.1073/pnas.0711437105. Google Scholar [2] N. Bellomo, M. A. Herrero and A. Tosin, On the dynamics of social conflict: Looking for the Black Swan,, , (2012). Google Scholar [3] A. Blanchet, E. A. Carlen and J. A. Carrillo, Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model,, J. Funct. Anal., 262 (2012), 2142. doi: 10.1016/j.jfa.2011.12.012. Google Scholar [4] A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control,", AIMS Series on Applied Mathematics, 2 (2007). Google Scholar [5] S. Camazine, J.-L. Deneubourg, N. Franks, J. Sneyd, G. Theraulaz and E. Bonabeau, "Self-Organization in Biological Systems,", Reprint of the 2001 original, (2001). Google Scholar [6] E. J. Candès, J. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements,, Comm. Pure Appl. Math., 59 (2006), 1207. doi: 10.1002/cpa.20124. Google Scholar [7] M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat, Sparse stabilization and control of the Cucker-Smale model,, , (). Google Scholar [8] J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming,, in, (2010), 297. doi: 10.1007/978-0-8176-4946-3_12. Google Scholar [9] E. Casas, C. Clason and K. Kunisch, Approximation of elliptic control problems in measure spaces with sparse solutions,, SIAM J. Control Optim., 50 (2012), 1735. doi: 10.1137/110843216. Google Scholar [10] L. Cesari, "Optimization-Theory and Applications. Problems with Ordinary Differential Equations,", Applications of Mathematics (New York), 17 (1983). Google Scholar [11] Y. Chuang, Y. Huang, M. D'Orsogna and A. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials,, in, (2007), 2292. doi: 10.1109/ROBOT.2007.363661. Google Scholar [12] F. H. Clarke, Y. S. Ledyaev, E. D. Sontag and A. I. Subbotin, Asymptotic controllability implies feedback stabilization,, IEEE Trans. Automat. Control, 42 (1997), 1394. doi: 10.1109/9.633828. Google Scholar [13] C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces,, ESAIM Control Optim. Calc. Var., 17 (2011), 243. doi: 10.1051/cocv/2010003. Google Scholar [14] C. Clason and K. Kunisch, A measure space approach to optimal source placement,, Comput. Optim. Appl., 53 (2012), 155. doi: 10.1007/s10589-011-9444-9. Google Scholar [15] J.-M. Coron and E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equations,, SIAM J. Control Optim., 43 (2004), 549. doi: 10.1137/S036301290342471X. Google Scholar [16] J.-M. Coron and E. Trélat, Global steady-state stabilization and controllability of 1D semilinear wave equations,, Commun. Contemp. Math., 8 (2006), 535. doi: 10.1142/S0219199706002209. Google Scholar [17] I. Couzin and N. Franks, Self-organized lane formation and optimized traffic flow in army ants,, Proc. R. Soc. Lond. B, 270 (2002), 139. doi: 10.1098/rspb.2002.2210. Google Scholar [18] I. Couzin, J. Krause, N. Franks and S. Levin, Effective leadership and decision making in animal groups on the move,, Nature, 433 (2005), 513. doi: 10.1038/nature03236. Google Scholar [19] A. J. Craig and I. Flügge-Lotz, 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 Engineering, 87 (1965), 39. doi: 10.1115/1.3650527. Google Scholar [20] E. Cristiani, B. Piccoli and A. Tosin, Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints,, in, (2010). doi: 10.1007/978-0-8176-4946-3_13. Google Scholar [21] E. Cristiani, B. Piccoli and A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics,, Multiscale Model. Simul., 9 (2011), 155. doi: 10.1137/100797515. Google Scholar [22] F. Cucker and S. Smale, Emergent behavior in flocks,, IEEE Trans. Automat. Control, 52 (2007), 852. doi: 10.1109/TAC.2007.895842. Google Scholar [23] 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 [24] F. Cucker, S. Smale and D. Zhou, Modeling language evolution,, Found. Comput. Math., 4 (2004), 315. doi: 10.1007/s10208-003-0101-2. Google Scholar [25] D. L. Donoho, Compressed sensing,, IEEE Trans. Inform. Theory, 52 (2006), 1289. doi: 10.1109/TIT.2006.871582. Google Scholar [26] B. Düring, D. Matthes and G. Toscani, Kinetic equations modelling wealth redistribution: A comparison of approaches,, Phys. Rev. E, 78 (2008). doi: 10.1103/PhysRevE.78.056103. Google Scholar [27] Y. Eldar and H. Rauhut, Average case analysis of multichannel sparse recovery using convex relaxation,, IEEE Trans. Inform. Theory, 56 (2010), 505. doi: 10.1109/TIT.2009.2034789. Google Scholar [28] M. Fornasier and H. Rauhut, Recovery algorithms for vector-valued data with joint sparsity constraints,, SIAM J. Numer. Anal., 46 (2008), 577. doi: 10.1137/0606668909. Google Scholar [29] M. Fornasier and H. Rauhut, "Handbook of Mathematical Methods in Imaging,", chapter Compressive Sensing, (2010). Google Scholar [30] 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 [31] R. Herzog, G. Stadler and G. Wachsmuth, Directional sparsity in optimal control of partial differential equations,, SIAM J. Control and Optimization, 50 (2012), 943. doi: 10.1137/100815037. Google Scholar [32] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I,, Jahresber. Dtsch. Math.-Ver., 105 (2003), 103. Google Scholar [33] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II,, Jahresber. Dtsch. Math.-Ver., 106 (2004), 51. Google Scholar [34] A. Jadbabaie, J. Lin and A. S. Morse, Correction to: "Coordination of groups of mobile autonomous agents using nearest neighbor rules,'', [IEEE Trans. Automat. Control 48 (2003), 48 (2003), 988. Google Scholar [35] J. Ke, J. Minett, C.-P. Au and W.-Y. Wang, Self-organization and selection in the emergence of vocabulary,, Complexity, 7 (2002), 41. doi: 10.1002/cplx.10030. Google Scholar [36] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar [37] A. Lachapelle and M. T. Wolfram, On a mean field game approach modeling congestion and aversion in pedestrian crowds,, Trans. Res.: Part B: Methodological, 45 (2011), 1572. doi: 10.1016/j.trb.2011.07.011. Google Scholar [38] 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 [39] S. Lemercier, A. Jelic, R. Kulpa, J. Hua, J. Fehrenbach, P. Degond, C. Appert-Rolland, S. Donikian and J. Pettré, Realistic following behaviors for crowd simulation,, Computer Graphics Forum, 31 (2012), 489. doi: 10.1111/j.1467-8659.2012.03028.x. Google Scholar [40] N. Leonard and E. Fiorelli, Virtual leaders, artificial potentials and coordinated control of groups,, in, (2001), 2968. Google Scholar [41] S. Mallat, "A Wavelet Tour of Signal Processing. The Sparse Way,", Third edition, (2009). Google Scholar [42] M. Moussaïd, E. G. Guillot, M. Moreau, J. Fehrenbach, O. Chabiron, S. Lemercier, J. Pettré, C. Appert-Rolland, P. Degond and G. Theraulaz, Traffic instabilities in self-organized pedestrian crowds,, PLoS Computational Biology, 8 (2012). Google Scholar [43] H. Niwa, Self-organizing dynamic model of fish schooling,, J. Theor. Biol., 171 (1994), 123. doi: 10.1006/jtbi.1994.1218. Google Scholar [44] J. Parrish and L. Edelstein-Keshet, Complexity, pattern, and evolutionary trade-offs in animal aggregation,, Science, 294 (1999), 99. doi: 10.1126/science.284.5411.99. Google Scholar [45] J. Parrish, S. Viscido and D. Gruenbaum, Self-organized fish schools: An examination of emergent properties,, Biol. Bull., 202 (2002), 296. doi: 10.2307/1543482. Google Scholar [46] C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311. doi: 10.1007/BF02476407. Google Scholar [47] L. Perea, G. 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