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:
[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. Gómez and P. Elosegui, Extension of the Cucker-Smale control law to space flight formations,, AIAA Journal of Guidance, 32 (2009), 527. doi: 10.2514/1.36269. Google Scholar

[48]

B. Perthame, "Transport Equations in Biology,", Frontiers in Mathematics, (2007). Google Scholar

[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. doi: 10.1137/120889137. Google Scholar

[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, (1962). Google Scholar

[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, (2013). Google Scholar

[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). Google Scholar

[53]

Y. Privat, E. Trélat and E. Zuazua, Optimal observation of the one-dimensional wave equation,, J. Fourier Anal. Appl., 19 (2013), 514. doi: 10.1007/s00041-013-9267-4. Google Scholar

[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. doi: 10.1137/060674909. Google Scholar

[55]

W. Romey, Individual differences make a difference in the trajectories of simulated schools of fish,, Ecol. Model., 92 (1996), 65. doi: 10.1016/0304-3800(95)00202-2. Google Scholar

[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. doi: 10.1109/TAC.2007.898077. Google Scholar

[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. doi: 10.1142/S0218202508003029. Google Scholar

[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. doi: 10.1007/s10589-007-9150-9. Google Scholar

[59]

K. Sugawara and M. Sano, Cooperative acceleration of task performance: Foraging behavior of interacting multi-robots system,, Physica D, 100 (1997), 343. Google Scholar

[60]

N. Taleb, "The Black Swan,", Penguin, (2010). Google Scholar

[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. doi: 10.1103/PhysRevLett.75.4326. Google Scholar

[62]

E. Trélat, "Contrôle Optimal. Théorie & Applications,", Vuibert, (2005). Google Scholar

[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. doi: 10.1103/PhysRevLett.75.1226. Google Scholar

[64]

G. Vossen and H. Maurer, $L^1$ minimization in optimal control and applications to robotics,, Optimal Control Applications and Methods, 27 (2006), 301. doi: 10.1002/oca.781. Google Scholar

[65]

G. Wachsmuth and D. Wachsmuth, Convergence and regularization results for optimal control problems with sparsity functional,, ESAIM, 17 (2011), 858. doi: 10.1051/cocv/2010027. Google Scholar

[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. doi: 10.1073/pnas.0811195106. Google Scholar

[67]

M. I. Zelikin and V. F. Borisov, "Theory of Chattering Control. With Applications to Astronautics, Robotics, Economics, and Engineering,", Systems & Control: Foundations & Applications, (1994). doi: 10.1007/978-1-4612-2702-1. Google Scholar

show all references

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. Gómez and P. Elosegui, Extension of the Cucker-Smale control law to space flight formations,, AIAA Journal of Guidance, 32 (2009), 527. doi: 10.2514/1.36269. Google Scholar

[48]

B. Perthame, "Transport Equations in Biology,", Frontiers in Mathematics, (2007). Google Scholar

[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. doi: 10.1137/120889137. Google Scholar

[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, (1962). Google Scholar

[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, (2013). Google Scholar

[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). Google Scholar

[53]

Y. Privat, E. Trélat and E. Zuazua, Optimal observation of the one-dimensional wave equation,, J. Fourier Anal. Appl., 19 (2013), 514. doi: 10.1007/s00041-013-9267-4. Google Scholar

[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. doi: 10.1137/060674909. Google Scholar

[55]

W. Romey, Individual differences make a difference in the trajectories of simulated schools of fish,, Ecol. Model., 92 (1996), 65. doi: 10.1016/0304-3800(95)00202-2. Google Scholar

[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. doi: 10.1109/TAC.2007.898077. Google Scholar

[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. doi: 10.1142/S0218202508003029. Google Scholar

[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. doi: 10.1007/s10589-007-9150-9. Google Scholar

[59]

K. Sugawara and M. Sano, Cooperative acceleration of task performance: Foraging behavior of interacting multi-robots system,, Physica D, 100 (1997), 343. Google Scholar

[60]

N. Taleb, "The Black Swan,", Penguin, (2010). Google Scholar

[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. doi: 10.1103/PhysRevLett.75.4326. Google Scholar

[62]

E. Trélat, "Contrôle Optimal. Théorie & Applications,", Vuibert, (2005). Google Scholar

[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. doi: 10.1103/PhysRevLett.75.1226. Google Scholar

[64]

G. Vossen and H. Maurer, $L^1$ minimization in optimal control and applications to robotics,, Optimal Control Applications and Methods, 27 (2006), 301. doi: 10.1002/oca.781. Google Scholar

[65]

G. Wachsmuth and D. Wachsmuth, Convergence and regularization results for optimal control problems with sparsity functional,, ESAIM, 17 (2011), 858. doi: 10.1051/cocv/2010027. Google Scholar

[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. doi: 10.1073/pnas.0811195106. Google Scholar

[67]

M. I. Zelikin and V. F. Borisov, "Theory of Chattering Control. With Applications to Astronautics, Robotics, Economics, and Engineering,", Systems & Control: Foundations & Applications, (1994). doi: 10.1007/978-1-4612-2702-1. Google Scholar

[1]

Pia Heins, Michael Moeller, Martin Burger. Locally sparse reconstruction using the $l^{1,\infty}$-norm. Inverse Problems & Imaging, 2015, 9 (4) : 1093-1137. doi: 10.3934/ipi.2015.9.1093

[2]

Ewa Girejko, Luís Machado, Agnieszka B. Malinowska, Natália Martins. On consensus in the Cucker–Smale type model on isolated time scales. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 77-89. doi: 10.3934/dcdss.2018005

[3]

Seung-Yeal Ha, Jinwook Jung, Peter Kuchling. Emergence of anomalous flocking in the fractional Cucker-Smale model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5465-5489. doi: 10.3934/dcds.2019223

[4]

Lei Wu, Zhe Sun. A new spectral method for $l_1$-regularized minimization. Inverse Problems & Imaging, 2015, 9 (1) : 257-272. doi: 10.3934/ipi.2015.9.257

[5]

Yingying Li, Stanley Osher, Richard Tsai. Heat source identification based on $l_1$ constrained minimization. Inverse Problems & Imaging, 2014, 8 (1) : 199-221. doi: 10.3934/ipi.2014.8.199

[6]

Gero Friesecke, Felix Henneke, Karl Kunisch. Frequency-sparse optimal quantum control. Mathematical Control & Related Fields, 2018, 8 (1) : 155-176. doi: 10.3934/mcrf.2018007

[7]

Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023

[8]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034

[9]

Lidan Li, Hongwei Zhang, Liwei Zhang. Inverse quadratic programming problem with $ l_1 $ norm measure. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-13. doi: 10.3934/jimo.2019061

[10]

Zhaohui Guo, Stanley Osher. Template matching via $l_1$ minimization and its application to hyperspectral data. Inverse Problems & Imaging, 2011, 5 (1) : 19-35. doi: 10.3934/ipi.2011.5.19

[11]

Jiying Liu, Jubo Zhu, Fengxia Yan, Zenghui Zhang. Compressive sampling and $l_1$ minimization for SAR imaging with low sampling rate. Inverse Problems & Imaging, 2013, 7 (4) : 1295-1305. doi: 10.3934/ipi.2013.7.1295

[12]

Vladimir Gaitsgory, Tanya Tarnopolskaya. Threshold value of the penalty parameter in the minimization of $L_1$-penalized conditional value-at-risk. Journal of Industrial & Management Optimization, 2013, 9 (1) : 191-204. doi: 10.3934/jimo.2013.9.191

[13]

Duo Wang, Zheng-Fen Jin, Youlin Shang. A penalty decomposition method for nuclear norm minimization with l1 norm fidelity term. Evolution Equations & Control Theory, 2019, 8 (4) : 695-708. doi: 10.3934/eect.2019034

[14]

Jerzy Klamka, Helmut Maurer, Andrzej Swierniak. Local controllability and optimal control for\newline a model of combined anticancer therapy with control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 195-216. doi: 10.3934/mbe.2017013

[15]

M. Sumon Hossain, M. Monir Uddin. Iterative methods for solving large sparse Lyapunov equations and application to model reduction of index 1 differential-algebraic-equations. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 173-186. doi: 10.3934/naco.2019013

[16]

Mattia Bongini, Massimo Fornasier, Oliver Junge, Benjamin Scharf. Sparse control of alignment models in high dimension. Networks & Heterogeneous Media, 2015, 10 (3) : 647-697. doi: 10.3934/nhm.2015.10.647

[17]

Eduardo Casas, Mariano Mateos, Arnd Rösch. Finite element approximation of sparse parabolic control problems. Mathematical Control & Related Fields, 2017, 7 (3) : 393-417. doi: 10.3934/mcrf.2017014

[18]

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

[19]

Chao Zhang, Jingjing Wang, Naihua Xiu. Robust and sparse portfolio model for index tracking. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1001-1015. doi: 10.3934/jimo.2018082

[20]

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

2018 Impact Factor: 1.292

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (18)

[Back to Top]