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

Sparse control of alignment models in high dimension

Mattia Bongini 1, , Massimo Fornasier 1, , Oliver Junge 2, and  Benjamin Scharf 2,

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.
Keywords: Poincaré recurrences, Dimension theory, multifractal analysis..
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
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.

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