Sparse stabilization and optimal control of the Cucker-Smale model

Marco Caponigro; Massimo Fornasier; Benedetto Piccoli; Emmanuel Trélat

doi:10.3934/mcrf.2013.3.447

Article Contents

2013, Volume 3, Issue 4: 447-466. Doi: 10.3934/mcrf.2013.3.447

This issue Previous Article Controllability with quadratic drift Next Article Ricci curvatures in Carnot groups

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,
2.
Technische Universität München, Facultät Mathematik, Boltzmannstrasse 3, D-85748, Garching bei München
3.
Rutgers University, Department of Mathematics, Business & Science Building Room 325, Camden, NJ 08102
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

Received: October 31, 2012

Revised: May 31, 2013

Published: November 2013

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 49J15, 93D15; Secondary: 65K10, 93B05.

Citation:

Related Papers

Cited by

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), 1232-1237.doi: 10.1073/pnas.0711437105.
[2]	N. Bellomo, M. A. Herrero and A. Tosin, On the dynamics of social conflict: Looking for the Black Swan, arXiv:1202.4554, 2012.
[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), 2142-2230.doi: 10.1016/j.jfa.2011.12.012.
[4]	A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control," AIMS Series on Applied Mathematics, 2, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.
[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, Princeton Studies in Complexity, Princeton University Press, Princeton, NJ, 2003.
[6]	E. J. Candès, J. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Appl. Math., 59 (2006), 1207-1223.doi: 10.1002/cpa.20124.
[7]	M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat, Sparse stabilization and control of the Cucker-Smale model, arXiv:1210.5739.
[8]	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, Inc., Boston, MA, (2010), 297-336.doi: 10.1007/978-0-8176-4946-3_12.
[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), 1735-1752.doi: 10.1137/110843216.
[10]	L. Cesari, "Optimization-Theory and Applications. Problems with Ordinary Differential Equations," Applications of Mathematics (New York), 17, Springer-Verlag, New York, 1983.
[11]	Y. Chuang, Y. Huang, M. D'Orsogna and A. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials, in "IEEE International Conference on Robotics and Automation," Roma, (2007), 2292-2299.doi: 10.1109/ROBOT.2007.363661.
[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), 1394-1407.doi: 10.1109/9.633828.
[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), 243-266.doi: 10.1051/cocv/2010003.
[14]	C. Clason and K. Kunisch, A measure space approach to optimal source placement, Comput. Optim. Appl., 53 (2012), 155-171.doi: 10.1007/s10589-011-9444-9.
[15]	J.-M. Coron and E. Trélat, Global steady-state controllability of one-dimensional semilinear heat equations, SIAM J. Control Optim., 43 (2004), 549-569 (electronic).doi: 10.1137/S036301290342471X.
[16]	J.-M. Coron and E. Trélat, Global steady-state stabilization and controllability of 1D semilinear wave equations, Commun. Contemp. Math., 8 (2006), 535-567.doi: 10.1142/S0219199706002209.
[17]	I. Couzin and N. Franks, Self-organized lane formation and optimized traffic flow in army ants, Proc. R. Soc. Lond. B, 270 (2002), 139-146.doi: 10.1098/rspb.2002.2210.
[18]	I. Couzin, J. Krause, N. Franks and S. Levin, Effective leadership and decision making in animal groups on the move, Nature, 433 (2005), 513-516.doi: 10.1038/nature03236.
[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), 39-57.doi: 10.1115/1.3650527.
[20]	E. Cristiani, B. Piccoli and A. Tosin, Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints, 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, Inc., Boston, MA, 2010.doi: 10.1007/978-0-8176-4946-3_13.
[21]	E. Cristiani, B. Piccoli and A. Tosin, Multiscale modeling of granular flows with application to crowd dynamics, Multiscale Model. Simul., 9 (2011), 155-182.doi: 10.1137/100797515.
[22]	F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.doi: 10.1109/TAC.2007.895842.
[23]	F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.doi: 10.1007/s11537-007-0647-x.
[24]	F. Cucker, S. Smale and D. Zhou, Modeling language evolution, Found. Comput. Math., 4 (2004), 315-343.doi: 10.1007/s10208-003-0101-2.
[25]	D. L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory, 52 (2006), 1289-1306.doi: 10.1109/TIT.2006.871582.
[26]	B. Düring, D. Matthes and G. Toscani, Kinetic equations modelling wealth redistribution: A comparison of approaches, Phys. Rev. E, 78 (2008), 056103, 12 pp.doi: 10.1103/PhysRevE.78.056103.
[27]	Y. Eldar and H. Rauhut, Average case analysis of multichannel sparse recovery using convex relaxation, IEEE Trans. Inform. Theory, 56 (2010), 505-519.doi: 10.1109/TIT.2009.2034789.
[28]	M. Fornasier and H. Rauhut, Recovery algorithms for vector-valued data with joint sparsity constraints, SIAM J. Numer. Anal., 46 (2008), 577-613.doi: 10.1137/0606668909.
[29]	M. Fornasier and H. Rauhut, "Handbook of Mathematical Methods in Imaging," chapter Compressive Sensing, Springer-Verlag, 2010.
[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), 1679-1683.doi: 10.1109/TAC.2010.2046113.
[31]	R. Herzog, G. Stadler and G. Wachsmuth, Directional sparsity in optimal control of partial differential equations, SIAM J. Control and Optimization, 50 (2012), 943-963.doi: 10.1137/100815037.
[32]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Dtsch. Math.-Ver., 105 (2003), 103-165.
[33]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II, Jahresber. Dtsch. Math.-Ver., 106 (2004), 51-69.
[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), 988-1001; MR 1986266] IEEE Trans. Automat. Control, 48 (2003), 1675.
[35]	J. Ke, J. Minett, C.-P. Au and W.-Y. Wang, Self-organization and selection in the emergence of vocabulary, Complexity, 7 (2002), 41-54.doi: 10.1002/cplx.10030.
[36]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.doi: 10.1016/0022-5193(70)90092-5.
[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), 1572-1589.doi: 10.1016/j.trb.2011.07.011.
[38]	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.
[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), 489-498.doi: 10.1111/j.1467-8659.2012.03028.x.
[40]	N. Leonard and E. Fiorelli, Virtual leaders, artificial potentials and coordinated control of groups, in "Proc. 40th IEEE Conf. Decision Contr.," (2001), 2968-2973.
[41]	S. Mallat, "A Wavelet Tour of Signal Processing. The Sparse Way," Third edition, Elsevier/Academic Press, Amsterdam, 2009.
[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), e1002442.
[43]	H. Niwa, Self-organizing dynamic model of fish schooling, J. Theor. Biol., 171 (1994), 123-136.doi: 10.1006/jtbi.1994.1218.
[44]	J. Parrish and L. Edelstein-Keshet, Complexity, pattern, and evolutionary trade-offs in animal aggregation, Science, 294 (1999), 99-101.doi: 10.1126/science.284.5411.99.
[45]	J. Parrish, S. Viscido and D. Gruenbaum, Self-organized fish schools: An examination of emergent properties, Biol. Bull., 202 (2002), 296-305.doi: 10.2307/1543482.
[46]	C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.doi: 10.1007/BF02476407.
[47]	L. Perea, G. Gómez and P. Elosegui, Extension of the Cucker-Smale control law to space flight formations, AIAA Journal of Guidance, Control, and Dynamics, 32 (2009), 527-537.doi: 10.2514/1.36269.
[48]	B. Perthame, "Transport Equations in Biology," Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.
[49]	K. Pieper and B. Vexler, A priori error analysis for discretization of sparse elliptic optimal control problems in measure space, SIAM J. Control Optim., 51 (2013), 2788-2808.doi: 10.1137/120889137.
[50]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962.
[51]	Y. Privat, E. Trélat and E. Zuazua, Complexity and regularity of maximal energy domains for the wave equation with fixed initial data, hal-00813647, version 1, 2013.
[52]	Y. Privat, E. Trélat and E. Zuazua, Optimal location of controllers for the one-dimensional wave equation, to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire, (2013).
[53]	Y. Privat, E. Trélat and E. Zuazua, Optimal observation of the one-dimensional wave equation, J. Fourier Anal. Appl., 19 (2013), 514-544.doi: 10.1007/s00041-013-9267-4.
[54]	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.
[55]	W. Romey, Individual differences make a difference in the trajectories of simulated schools of fish, Ecol. Model., 92 (1996), 65-77.doi: 10.1016/0304-3800(95)00202-2.
[56]	R. Sepulchre, D. Paley and N. E. Leonard, Stabilization of planar collective motion: All-to-all communication, IEEE Transactions on Automatic Control, 52 (2007), 811-824.doi: 10.1109/TAC.2007.898077.
[57]	M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci., 18 (2008), 1249-1267.doi: 10.1142/S0218202508003029.
[58]	G. Stadler, Elliptic optimal control problems with $L^1$-control cost and applications for the placement of control devices, Comput. Optim. Appl., 44 (2009), 159-181.doi: 10.1007/s10589-007-9150-9.
[59]	K. Sugawara and M. Sano, Cooperative acceleration of task performance: Foraging behavior of interacting multi-robots system, Physica D, 100 (1997), 343-354.
[60]	N. Taleb, "The Black Swan," Penguin, 2010.
[61]	J. Toner and Y. Tu, Long-range order in a two-dimensional dynamical XY model: How birds fly together, Phys. Rev. Lett., 75 (1995), 4326-4329.doi: 10.1103/PhysRevLett.75.4326.
[62]	E. Trélat, "Contrôle Optimal. Théorie & Applications," Vuibert, Paris, 2005.
[63]	T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.doi: 10.1103/PhysRevLett.75.1226.
[64]	G. Vossen and H. Maurer, $L^1$ minimization in optimal control and applications to robotics, Optimal Control Applications and Methods, 27 (2006), 301-321.doi: 10.1002/oca.781.
[65]	G. Wachsmuth and D. Wachsmuth, Convergence and regularization results for optimal control problems with sparsity functional, ESAIM, Control Optim. Calc. Var., 17 (2011), 858-886.doi: 10.1051/cocv/2010027.
[66]	C. Yates, R. Erban, C. Escudero, L. Couzin, J. Buhl, L. Kevrekidis, P. Maini and D. Sumpter, Inherent noise can facilitate coherence in collective swarm motion, Proceedings of the National Academy of Sciences, 106 (2009), 5464-5469.doi: 10.1073/pnas.0811195106.
[67]	M. I. Zelikin and V. F. Borisov, "Theory of Chattering Control. With Applications to Astronautics, Robotics, Economics, and Engineering," Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994.doi: 10.1007/978-1-4612-2702-1.