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]

PNAS, 105 (2008), 1232-1237. doi: 10.1073/pnas.0711437105.  Google Scholar

[2]

arXiv:1202.4554, 2012. Google Scholar

[3]

J. Funct. Anal., 262 (2012), 2142-2230. doi: 10.1016/j.jfa.2011.12.012.  Google Scholar

[4]

AIMS Series on Applied Mathematics, 2, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.  Google Scholar

[5]

Reprint of the 2001 original, Princeton Studies in Complexity, Princeton University Press, Princeton, NJ, 2003.  Google Scholar

[6]

Comm. Pure Appl. Math., 59 (2006), 1207-1223. 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]

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.  Google Scholar

[9]

SIAM J. Control Optim., 50 (2012), 1735-1752. doi: 10.1137/110843216.  Google Scholar

[10]

Applications of Mathematics (New York), 17, Springer-Verlag, New York, 1983.  Google Scholar

[11]

in "IEEE International Conference on Robotics and Automation," Roma, (2007), 2292-2299. doi: 10.1109/ROBOT.2007.363661.  Google Scholar

[12]

IEEE Trans. Automat. Control, 42 (1997), 1394-1407. doi: 10.1109/9.633828.  Google Scholar

[13]

ESAIM Control Optim. Calc. Var., 17 (2011), 243-266. doi: 10.1051/cocv/2010003.  Google Scholar

[14]

Comput. Optim. Appl., 53 (2012), 155-171. doi: 10.1007/s10589-011-9444-9.  Google Scholar

[15]

SIAM J. Control Optim., 43 (2004), 549-569 (electronic). doi: 10.1137/S036301290342471X.  Google Scholar

[16]

Commun. Contemp. Math., 8 (2006), 535-567. doi: 10.1142/S0219199706002209.  Google Scholar

[17]

Proc. R. Soc. Lond. B, 270 (2002), 139-146. doi: 10.1098/rspb.2002.2210.  Google Scholar

[18]

Nature, 433 (2005), 513-516. doi: 10.1038/nature03236.  Google Scholar

[19]

J. Basic Engineering, 87 (1965), 39-57. doi: 10.1115/1.3650527.  Google Scholar

[20]

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.  Google Scholar

[21]

Multiscale Model. Simul., 9 (2011), 155-182. doi: 10.1137/100797515.  Google Scholar

[22]

IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.  Google Scholar

[23]

Jpn. J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x.  Google Scholar

[24]

Found. Comput. Math., 4 (2004), 315-343. doi: 10.1007/s10208-003-0101-2.  Google Scholar

[25]

IEEE Trans. Inform. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.  Google Scholar

[26]

Phys. Rev. E, 78 (2008), 056103, 12 pp. doi: 10.1103/PhysRevE.78.056103.  Google Scholar

[27]

IEEE Trans. Inform. Theory, 56 (2010), 505-519. doi: 10.1109/TIT.2009.2034789.  Google Scholar

[28]

SIAM J. Numer. Anal., 46 (2008), 577-613. doi: 10.1137/0606668909.  Google Scholar

[29]

chapter Compressive Sensing, Springer-Verlag, 2010. Google Scholar

[30]

IEEE Trans. Automat. Control, 55 (2010), 1679-1683. doi: 10.1109/TAC.2010.2046113.  Google Scholar

[31]

SIAM J. Control and Optimization, 50 (2012), 943-963. doi: 10.1137/100815037.  Google Scholar

[32]

Jahresber. Dtsch. Math.-Ver., 105 (2003), 103-165.  Google Scholar

[33]

Jahresber. Dtsch. Math.-Ver., 106 (2004), 51-69.  Google Scholar

[34]

[IEEE Trans. Automat. Control 48 (2003), 988-1001; MR 1986266] IEEE Trans. Automat. Control, 48 (2003), 1675.  Google Scholar

[35]

Complexity, 7 (2002), 41-54. doi: 10.1002/cplx.10030.  Google Scholar

[36]

J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[37]

Trans. Res.: Part B: Methodological, 45 (2011), 1572-1589. doi: 10.1016/j.trb.2011.07.011.  Google Scholar

[38]

Jpn. J. Math. (3), 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.  Google Scholar

[39]

Computer Graphics Forum, 31 (2012), 489-498. doi: 10.1111/j.1467-8659.2012.03028.x.  Google Scholar

[40]

in "Proc. 40th IEEE Conf. Decision Contr.," (2001), 2968-2973. Google Scholar

[41]

Third edition, Elsevier/Academic Press, Amsterdam, 2009.  Google Scholar

[42]

PLoS Computational Biology, 8 (2012), e1002442. Google Scholar

[43]

J. Theor. Biol., 171 (1994), 123-136. doi: 10.1006/jtbi.1994.1218.  Google Scholar

[44]

Science, 294 (1999), 99-101. doi: 10.1126/science.284.5411.99.  Google Scholar

[45]

Biol. Bull., 202 (2002), 296-305. doi: 10.2307/1543482.  Google Scholar

[46]

Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407.  Google Scholar

[47]

AIAA Journal of Guidance, Control, and Dynamics, 32 (2009), 527-537. doi: 10.2514/1.36269.  Google Scholar

[48]

Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[49]

SIAM J. Control Optim., 51 (2013), 2788-2808. doi: 10.1137/120889137.  Google Scholar

[50]

Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962.  Google Scholar

[51]

hal-00813647, version 1, 2013. Google Scholar

[52]

to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire, (2013). Google Scholar

[53]

J. Fourier Anal. Appl., 19 (2013), 514-544. doi: 10.1007/s00041-013-9267-4.  Google Scholar

[54]

SIAM J. Control and Optimization, 48 (2009), 162-186. doi: 10.1137/060674909.  Google Scholar

[55]

Ecol. Model., 92 (1996), 65-77. doi: 10.1016/0304-3800(95)00202-2.  Google Scholar

[56]

IEEE Transactions on Automatic Control, 52 (2007), 811-824. doi: 10.1109/TAC.2007.898077.  Google Scholar

[57]

Math. Models Methods Appl. Sci., 18 (2008), 1249-1267. doi: 10.1142/S0218202508003029.  Google Scholar

[58]

Comput. Optim. Appl., 44 (2009), 159-181. doi: 10.1007/s10589-007-9150-9.  Google Scholar

[59]

Physica D, 100 (1997), 343-354. Google Scholar

[60]

Penguin, 2010. Google Scholar

[61]

Phys. Rev. Lett., 75 (1995), 4326-4329. doi: 10.1103/PhysRevLett.75.4326.  Google Scholar

[62]

Vuibert, Paris, 2005.  Google Scholar

[63]

Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[64]

Optimal Control Applications and Methods, 27 (2006), 301-321. doi: 10.1002/oca.781.  Google Scholar

[65]

ESAIM, Control Optim. Calc. Var., 17 (2011), 858-886. doi: 10.1051/cocv/2010027.  Google Scholar

[66]

Proceedings of the National Academy of Sciences, 106 (2009), 5464-5469. doi: 10.1073/pnas.0811195106.  Google Scholar

[67]

Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-2702-1.  Google Scholar

show all references

References:
[1]

PNAS, 105 (2008), 1232-1237. doi: 10.1073/pnas.0711437105.  Google Scholar

[2]

arXiv:1202.4554, 2012. Google Scholar

[3]

J. Funct. Anal., 262 (2012), 2142-2230. doi: 10.1016/j.jfa.2011.12.012.  Google Scholar

[4]

AIMS Series on Applied Mathematics, 2, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.  Google Scholar

[5]

Reprint of the 2001 original, Princeton Studies in Complexity, Princeton University Press, Princeton, NJ, 2003.  Google Scholar

[6]

Comm. Pure Appl. Math., 59 (2006), 1207-1223. 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]

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.  Google Scholar

[9]

SIAM J. Control Optim., 50 (2012), 1735-1752. doi: 10.1137/110843216.  Google Scholar

[10]

Applications of Mathematics (New York), 17, Springer-Verlag, New York, 1983.  Google Scholar

[11]

in "IEEE International Conference on Robotics and Automation," Roma, (2007), 2292-2299. doi: 10.1109/ROBOT.2007.363661.  Google Scholar

[12]

IEEE Trans. Automat. Control, 42 (1997), 1394-1407. doi: 10.1109/9.633828.  Google Scholar

[13]

ESAIM Control Optim. Calc. Var., 17 (2011), 243-266. doi: 10.1051/cocv/2010003.  Google Scholar

[14]

Comput. Optim. Appl., 53 (2012), 155-171. doi: 10.1007/s10589-011-9444-9.  Google Scholar

[15]

SIAM J. Control Optim., 43 (2004), 549-569 (electronic). doi: 10.1137/S036301290342471X.  Google Scholar

[16]

Commun. Contemp. Math., 8 (2006), 535-567. doi: 10.1142/S0219199706002209.  Google Scholar

[17]

Proc. R. Soc. Lond. B, 270 (2002), 139-146. doi: 10.1098/rspb.2002.2210.  Google Scholar

[18]

Nature, 433 (2005), 513-516. doi: 10.1038/nature03236.  Google Scholar

[19]

J. Basic Engineering, 87 (1965), 39-57. doi: 10.1115/1.3650527.  Google Scholar

[20]

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.  Google Scholar

[21]

Multiscale Model. Simul., 9 (2011), 155-182. doi: 10.1137/100797515.  Google Scholar

[22]

IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842.  Google Scholar

[23]

Jpn. J. Math., 2 (2007), 197-227. doi: 10.1007/s11537-007-0647-x.  Google Scholar

[24]

Found. Comput. Math., 4 (2004), 315-343. doi: 10.1007/s10208-003-0101-2.  Google Scholar

[25]

IEEE Trans. Inform. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.  Google Scholar

[26]

Phys. Rev. E, 78 (2008), 056103, 12 pp. doi: 10.1103/PhysRevE.78.056103.  Google Scholar

[27]

IEEE Trans. Inform. Theory, 56 (2010), 505-519. doi: 10.1109/TIT.2009.2034789.  Google Scholar

[28]

SIAM J. Numer. Anal., 46 (2008), 577-613. doi: 10.1137/0606668909.  Google Scholar

[29]

chapter Compressive Sensing, Springer-Verlag, 2010. Google Scholar

[30]

IEEE Trans. Automat. Control, 55 (2010), 1679-1683. doi: 10.1109/TAC.2010.2046113.  Google Scholar

[31]

SIAM J. Control and Optimization, 50 (2012), 943-963. doi: 10.1137/100815037.  Google Scholar

[32]

Jahresber. Dtsch. Math.-Ver., 105 (2003), 103-165.  Google Scholar

[33]

Jahresber. Dtsch. Math.-Ver., 106 (2004), 51-69.  Google Scholar

[34]

[IEEE Trans. Automat. Control 48 (2003), 988-1001; MR 1986266] IEEE Trans. Automat. Control, 48 (2003), 1675.  Google Scholar

[35]

Complexity, 7 (2002), 41-54. doi: 10.1002/cplx.10030.  Google Scholar

[36]

J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[37]

Trans. Res.: Part B: Methodological, 45 (2011), 1572-1589. doi: 10.1016/j.trb.2011.07.011.  Google Scholar

[38]

Jpn. J. Math. (3), 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.  Google Scholar

[39]

Computer Graphics Forum, 31 (2012), 489-498. doi: 10.1111/j.1467-8659.2012.03028.x.  Google Scholar

[40]

in "Proc. 40th IEEE Conf. Decision Contr.," (2001), 2968-2973. Google Scholar

[41]

Third edition, Elsevier/Academic Press, Amsterdam, 2009.  Google Scholar

[42]

PLoS Computational Biology, 8 (2012), e1002442. Google Scholar

[43]

J. Theor. Biol., 171 (1994), 123-136. doi: 10.1006/jtbi.1994.1218.  Google Scholar

[44]

Science, 294 (1999), 99-101. doi: 10.1126/science.284.5411.99.  Google Scholar

[45]

Biol. Bull., 202 (2002), 296-305. doi: 10.2307/1543482.  Google Scholar

[46]

Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407.  Google Scholar

[47]

AIAA Journal of Guidance, Control, and Dynamics, 32 (2009), 527-537. doi: 10.2514/1.36269.  Google Scholar

[48]

Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[49]

SIAM J. Control Optim., 51 (2013), 2788-2808. doi: 10.1137/120889137.  Google Scholar

[50]

Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962.  Google Scholar

[51]

hal-00813647, version 1, 2013. Google Scholar

[52]

to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire, (2013). Google Scholar

[53]

J. Fourier Anal. Appl., 19 (2013), 514-544. doi: 10.1007/s00041-013-9267-4.  Google Scholar

[54]

SIAM J. Control and Optimization, 48 (2009), 162-186. doi: 10.1137/060674909.  Google Scholar

[55]

Ecol. Model., 92 (1996), 65-77. doi: 10.1016/0304-3800(95)00202-2.  Google Scholar

[56]

IEEE Transactions on Automatic Control, 52 (2007), 811-824. doi: 10.1109/TAC.2007.898077.  Google Scholar

[57]

Math. Models Methods Appl. Sci., 18 (2008), 1249-1267. doi: 10.1142/S0218202508003029.  Google Scholar

[58]

Comput. Optim. Appl., 44 (2009), 159-181. doi: 10.1007/s10589-007-9150-9.  Google Scholar

[59]

Physica D, 100 (1997), 343-354. Google Scholar

[60]

Penguin, 2010. Google Scholar

[61]

Phys. Rev. Lett., 75 (1995), 4326-4329. doi: 10.1103/PhysRevLett.75.4326.  Google Scholar

[62]

Vuibert, Paris, 2005.  Google Scholar

[63]

Phys. Rev. Lett., 75 (1995), 1226-1229. doi: 10.1103/PhysRevLett.75.1226.  Google Scholar

[64]

Optimal Control Applications and Methods, 27 (2006), 301-321. doi: 10.1002/oca.781.  Google Scholar

[65]

ESAIM, Control Optim. Calc. Var., 17 (2011), 858-886. doi: 10.1051/cocv/2010027.  Google Scholar

[66]

Proceedings of the National Academy of Sciences, 106 (2009), 5464-5469. doi: 10.1073/pnas.0811195106.  Google Scholar

[67]

Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-2702-1.  Google Scholar

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