September  2015, 10(3): 647-697. doi: 10.3934/nhm.2015.10.647

Sparse control of alignment models in high dimension

1. 

Technische Universität München, Fakultät Mathematik, Boltzmannstraße 3, D-85748 Garching

2. 

Technische Universität München, Fakultät Mathematik, Boltzmannstrasse 3, D-85748 Garching, Germany, Germany

Received  August 2014 Revised  December 2014 Published  July 2015

For high dimensional particle systems, governed by smooth nonlinearities depending on mutual distances between particles, one can construct low-dimensional representations of the dynamical system, which allow the learning of nearly optimal control strategies in high dimension with overwhelming confidence. In this paper we present an instance of this general statement tailored to the sparse control of models of consensus emergence in high dimension, projected to lower dimensions by means of random linear maps. We show that one can steer, nearly optimally and with high probability, a high-dimensional alignment model to consensus by acting at each switching time on one agent of the system only, with a control rule chosen essentially exclusively according to information gathered from a randomly drawn low-dimensional representation of the control system.
Citation: Mattia Bongini, Massimo Fornasier, Oliver Junge, Benjamin Scharf. Sparse control of alignment models in high dimension. Networks and Heterogeneous Media, 2015, 10 (3) : 647-697. doi: 10.3934/nhm.2015.10.647
References:
[1]

S. Ahn, H.-O. Bae, S.-Y. Ha, Y. Kim and H. Lim, Application of flocking mechanism to the modeling of stochastic volatility, Math. Models Methods Appl. Sci., 23 (2013), 1603-1628. doi: 10.1142/S0218202513500176.

[2]

R. G. Baraniuk and M. B. Wakin, Random projections of smooth manifolds, Found. Comput. Math., 9 (2009), 51-77. doi: 10.1007/s10208-007-9011-z.

[3]

M. Bongini and M. Fornasier, Sparse stabilization of dynamical systems driven by attraction and avoidance forces, Netw. Heterog. Media, 9 (2014), 1-31. doi: 10.3934/nhm.2014.9.1.

[4]

M. Bongini, D. Kalise and M. Fornasier, (Un)conditional consensus emergence under perturbed and decentralized feedback controls, Discrete Contin. Dynam. Systems, 35 (2015), 4071-4094. doi: 10.3934/dcds.2015.35.4071.

[5]

J. Bouvrie and M. Maggioni, Geometric multiscale reduction for autonomous and controlled nonlinear systems, in 51st IEEE Conference on Decision and Control (CDC), 2012, 4320-4327. doi: 10.1109/CDC.2012.6425873.

[6]

M. Caponigro, M. Fornasier, B. Piccoli and E. Trelat, Sparse stabilization and control of the Cucker-Smale model, Math. Control Relat. Fields, 3 (2013), 447-466. doi: 10.3934/mcrf.2013.3.447.

[7]

M. Caponigro, M. Fornasier, B. Piccoli and E. Trelat, Sparse stabilization and control of alignment models, Math. Models Methods Appl. Sci., 25 (2015), 521-564. doi: 10.1142/S0218202515400059.

[8]

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.

[9]

R. R. Coifman and M. J. Hirn, Diffusion maps for changing data, Appl. Comput. Harmon. Anal., 36 (2014), 79-107. doi: 10.1016/j.acha.2013.03.001.

[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]

S. Dasgupta and A. Gupta, An elementary proof of a theorem of Johnson and Lindenstrauss, Random Structures Algorithms, 22 (2003), 60-65. doi: 10.1002/rsa.10073.

[13]

S. Dirksen, Dimensionality reduction with subgaussian matrices: A unified theory, arXiv:1402.3973, 2014.

[14]

M. Fornasier, J. Haškovec and J. Vybíral, Particle systems and kinetic equations modeling interacting agents in high dimension, Multiscale Model. Simul., 9 (2011), 1727-1764. doi: 10.1137/110830617.

[15]

M. Fornasier, B. Piccoli and F. Rossi, Mean-field sparse optimal control, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130400, 21PP. doi: 10.1098/rsta.2013.0400.

[16]

M. Fornasier and F. Solombrino, Mean-field optimal control, ESAIM Control Optim. Calc. Var., 20 (2014), 1123-1152. doi: 10.1051/cocv/2014009.

[17]

S.-Y. Ha, T. H. 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.

[18]

W. B. Johnson and J. Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space, in Conference in modern analysis and probability, New Haven, Conn., 1982, Contemp. Math., 26, Amer. Math. Soc., Providence, RI, 1984, 189-206. doi: 10.1090/conm/026/737400.

show all references

References:
[1]

S. Ahn, H.-O. Bae, S.-Y. Ha, Y. Kim and H. Lim, Application of flocking mechanism to the modeling of stochastic volatility, Math. Models Methods Appl. Sci., 23 (2013), 1603-1628. doi: 10.1142/S0218202513500176.

[2]

R. G. Baraniuk and M. B. Wakin, Random projections of smooth manifolds, Found. Comput. Math., 9 (2009), 51-77. doi: 10.1007/s10208-007-9011-z.

[3]

M. Bongini and M. Fornasier, Sparse stabilization of dynamical systems driven by attraction and avoidance forces, Netw. Heterog. Media, 9 (2014), 1-31. doi: 10.3934/nhm.2014.9.1.

[4]

M. Bongini, D. Kalise and M. Fornasier, (Un)conditional consensus emergence under perturbed and decentralized feedback controls, Discrete Contin. Dynam. Systems, 35 (2015), 4071-4094. doi: 10.3934/dcds.2015.35.4071.

[5]

J. Bouvrie and M. Maggioni, Geometric multiscale reduction for autonomous and controlled nonlinear systems, in 51st IEEE Conference on Decision and Control (CDC), 2012, 4320-4327. doi: 10.1109/CDC.2012.6425873.

[6]

M. Caponigro, M. Fornasier, B. Piccoli and E. Trelat, Sparse stabilization and control of the Cucker-Smale model, Math. Control Relat. Fields, 3 (2013), 447-466. doi: 10.3934/mcrf.2013.3.447.

[7]

M. Caponigro, M. Fornasier, B. Piccoli and E. Trelat, Sparse stabilization and control of alignment models, Math. Models Methods Appl. Sci., 25 (2015), 521-564. doi: 10.1142/S0218202515400059.

[8]

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.

[9]

R. R. Coifman and M. J. Hirn, Diffusion maps for changing data, Appl. Comput. Harmon. Anal., 36 (2014), 79-107. doi: 10.1016/j.acha.2013.03.001.

[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]

S. Dasgupta and A. Gupta, An elementary proof of a theorem of Johnson and Lindenstrauss, Random Structures Algorithms, 22 (2003), 60-65. doi: 10.1002/rsa.10073.

[13]

S. Dirksen, Dimensionality reduction with subgaussian matrices: A unified theory, arXiv:1402.3973, 2014.

[14]

M. Fornasier, J. Haškovec and J. Vybíral, Particle systems and kinetic equations modeling interacting agents in high dimension, Multiscale Model. Simul., 9 (2011), 1727-1764. doi: 10.1137/110830617.

[15]

M. Fornasier, B. Piccoli and F. Rossi, Mean-field sparse optimal control, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130400, 21PP. doi: 10.1098/rsta.2013.0400.

[16]

M. Fornasier and F. Solombrino, Mean-field optimal control, ESAIM Control Optim. Calc. Var., 20 (2014), 1123-1152. doi: 10.1051/cocv/2014009.

[17]

S.-Y. Ha, T. H. 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.

[18]

W. B. Johnson and J. Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space, in Conference in modern analysis and probability, New Haven, Conn., 1982, Contemp. Math., 26, Amer. Math. Soc., Providence, RI, 1984, 189-206. doi: 10.1090/conm/026/737400.

[1]

V. Afraimovich, J. Schmeling, Edgardo Ugalde, Jesús Urías. Spectra of dimensions for Poincaré recurrences. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 901-914. doi: 10.3934/dcds.2000.6.901

[2]

B. Fernandez, E. Ugalde, J. Urías. Spectrum of dimensions for Poincaré recurrences of Markov maps. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 835-849. doi: 10.3934/dcds.2002.8.835

[3]

Juan Wang, Xiaodan Zhang, Yun Zhao. Dimension estimates for arbitrary subsets of limit sets of a Markov construction and related multifractal analysis. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2315-2332. doi: 10.3934/dcds.2014.34.2315

[4]

Godofredo Iommi, Bartłomiej Skorulski. Multifractal analysis for the exponential family. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 857-869. doi: 10.3934/dcds.2006.16.857

[5]

V. Afraimovich, Jean-René Chazottes, Benoît Saussol. Pointwise dimensions for Poincaré recurrences associated with maps and special flows. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 263-280. doi: 10.3934/dcds.2003.9.263

[6]

Julien Barral, Yan-Hui Qu. On the higher-dimensional multifractal analysis. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 1977-1995. doi: 10.3934/dcds.2012.32.1977

[7]

Mario Roy, Mariusz Urbański. Multifractal analysis for conformal graph directed Markov systems. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 627-650. doi: 10.3934/dcds.2009.25.627

[8]

Zhihui Yuan. Multifractal analysis of random weak Gibbs measures. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5367-5405. doi: 10.3934/dcds.2017234

[9]

Luis Barreira. Dimension theory of flows: A survey. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3345-3362. doi: 10.3934/dcdsb.2015.20.3345

[10]

Luis Barreira, César Silva. Lyapunov exponents for continuous transformations and dimension theory. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 469-490. doi: 10.3934/dcds.2005.13.469

[11]

Valentin Afraimovich, Jean-Rene Chazottes and Benoit Saussol. Local dimensions for Poincare recurrences. Electronic Research Announcements, 2000, 6: 64-74.

[12]

Yunping Wang, Ercai Chen, Xiaoyao Zhou. Mean dimension theory in symbolic dynamics for finitely generated amenable groups. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022050

[13]

Zied Douzi, Bilel Selmi. On the mutual singularity of multifractal measures. Electronic Research Archive, 2020, 28 (1) : 423-432. doi: 10.3934/era.2020024

[14]

Mirela Domijan, Markus Kirkilionis. Graph theory and qualitative analysis of reaction networks. Networks and Heterogeneous Media, 2008, 3 (2) : 295-322. doi: 10.3934/nhm.2008.3.295

[15]

Jean-Pierre Francoise, Claude Piquet. Global recurrences of multi-time scaled systems. Conference Publications, 2011, 2011 (Special) : 430-436. doi: 10.3934/proc.2011.2011.430

[16]

Balázs Bárány, Michaƚ Rams, Ruxi Shi. On the multifractal spectrum of weighted Birkhoff averages. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2461-2497. doi: 10.3934/dcds.2021199

[17]

Jerrold E. Marsden, Alexey Tret'yakov. Factor analysis of nonlinear mappings: p-regularity theory. Communications on Pure and Applied Analysis, 2003, 2 (4) : 425-445. doi: 10.3934/cpaa.2003.2.425

[18]

Lars Olsen. First return times: multifractal spectra and divergence points. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 635-656. doi: 10.3934/dcds.2004.10.635

[19]

Imen Bhouri, Houssem Tlili. On the multifractal formalism for Bernoulli products of invertible matrices. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1129-1145. doi: 10.3934/dcds.2009.24.1129

[20]

Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173

2020 Impact Factor: 1.213

Metrics

  • PDF downloads (116)
  • HTML views (0)
  • Cited by (3)

[Back to Top]