Sparse control of alignment models in high dimension

Mattia Bongini; Massimo Fornasier; Oliver  Junge; Benjamin  Scharf; Mattia Bongini; Massimo Fornasier; Oliver  Junge; Benjamin  Scharf

doi:10.3934/nhm.2015.10.647

Networks and Heterogeneous Media

2015, Volume 10, Issue 3: 647-697. doi: 10.3934/nhm.2015.10.647

Previous Article Next Article

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

Received: 01 August 2014 Revised: 01 December 2014
Primary: 58F15, 58F17; Secondary: 53C35.

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.
- Dimension theory,
- Poincaré recurrences,
- multifractal analysis
Citation: Mattia Bongini, Massimo Fornasier, Oliver Junge, Benjamin Scharf. Sparse control of alignment models in high dimension[J]. Networks and Heterogeneous Media, 2015, 10(3): 647-697. doi: 10.3934/nhm.2015.10.647

Related Papers:

Abstract

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.

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

Reader Comments

Your name:*

Email:*
© 2015 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)