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A necessary condition of Pontryagin type for fuzzy fractional optimal control problems
On consensus in the Cucker–Smale type model on isolated time scales
1. | Faculty of Computer Science, Bialystok University of Technology, 15-351 Bia lystok, Poland |
2. | Department of Mathematics, UTAD, 5001-801 Vila Real, Portugal |
3. | Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal |
This article addresses a consensus phenomenon in a Cucker-Smale model where the magnitude of the step size is not necessarily a constant but it is a function of time. In the considered model the weights of mutual influences in the group of agents do not change. A sufficient condition under which the proposed model tends to a consensus is obtained. This condition strikingly demonstrates the importance of the graininess function in a consensus phenomenon. The results are illustrated by numerical simulations.
References:
[1] |
F. M. Atici, D. C. Biles and A. Lebedinsky,
An application of time scales to economics, Math. Comput. Modelling, 43 (2006), 718-726.
doi: 10.1016/j.mcm.2005.08.014. |
[2] |
B. Aulbach and S. Hilger, A unified approach to continuous and discrete dynamics, in Qualitative Theory of Differential Equations (Szeged, 1988), Colloq. Math. Soc. J. Bolyai, 53,
North-Holland, Amsterdam, 1990, 37–56. |
[3] |
Z. Bartosiewicz, N. Martins and D. F. M. Torres,
The second Euler–Lagrange equation of variational calculus on time scales, European Journal of Control, 17 (2011), 9-18.
doi: 10.3166/ejc.17.9-18. |
[4] |
Z. Bartosiewicz,
Linear positive control systems on time scales: Controllability, Mathematics of Control, Signals and Systems, 25 (2013), 327-343.
doi: 10.1007/s00498-013-0106-6. |
[5] |
J. Belikov, U. Kotta and M. Tonso,
Realization of nonlinear MIMO system on homogeneous time scales, European Journal of Control, 23 (2015), 48-54.
doi: 10.1016/j.ejcon.2015.01.006. |
[6] |
V. D. Blondel, J. M. Hendrickx and J. N. Tsitsiklis,
On Krause's multi-agent consensus model with state-dependent connectivity, IEEE Transactions on Automatics Control, 54 (2009), 2586-2597.
doi: 10.1109/TAC.2009.2031211. |
[7] |
V. D. Blondel, J. M. Hendrickx and J. N. Tsitsiklis,
Continuous-time average-preserving opinion dynamics with opinion-dependent communications, SIAM J. Control Optim., 48 (2010), 5214-5240.
doi: 10.1137/090766188. |
[8] |
M. Bohner, M. Fan and J. Zhang,
Periodicity of scalar dynamic equations on time scales and applications to population models, J. Math. Anal. Appl., 330 (2007), 1-9.
doi: 10.1016/j.jmaa.2006.04.084. |
[9] |
M. Bohner and A. Peterson,
Dynamic Equations on Time Scales Birkhäuser Boston, Boston, MA, 2001.
doi: 10.1007/978-1-4612-0201-1. |
[10] |
M. Bohner and A. Peterson,
Advances in Dynamic Equations on Time Scales Birkhäuser Boston, Boston, MA, 2003.
doi: 10.1007/978-1-4612-0201-1. |
[11] |
M. Bohner and H. Warth,
The Beverton-Holt dynamic equation, Appl. Anal., 86 (2007), 1007-1015.
doi: 10.1080/00036810701474140. |
[12] |
M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat,
Sparse stabilization and optimal control of the Cucker-Smale model, Mathematics of Control and Related Fields, 3 (2013), 447-466.
doi: 10.3934/mcrf.2013.3.447. |
[13] |
D. Casagrande, U. Kotta, M. Tonso and M. Wyrwas,
Transfer equivalence and realization of nonlinear input-output delta-differential equations on homogeneous time scales, IEEE Transactions on Automatic Control, 55 (2010), 2601-2606.
doi: 10.1109/TAC.2010.2060251. |
[14] |
S. Chatterjee and E. Seneta,
Towards consensus: Some convergence theorems on repeated averaging, J. Appl. Prob., 14 (1977), 89-97.
doi: 10.1017/S0021900200104681. |
[15] |
Cucker Smale and S. Smale,
Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[16] |
Cucker Smale and S. Smale,
On the mathematics of emergence, Japanese Journal of Mathematics, 2 (2007), 197-227.
doi: 10.1007/s11537-007-0647-x. |
[17] |
M. H. DeGroot,
Reaching a consensus, Journal of the American Statistical Association, 69 (1974), 118-121.
doi: 10.1080/01621459.1974.10480137. |
[18] |
J. French,
A formal theory of social power, Social Networks, A Developing Paradigm, (1977), 35-48.
doi: 10.1016/B978-0-12-442450-0.50010-9. |
[19] |
G. Fu, W. Zhang and Z. Li,
Opinion dynamics of modified Hegselmann–Krause model in a group-based population with heterogeneous bounded confidence, Physica A, 419 (2015), 558-565.
doi: 10.1016/j.physa.2014.10.045. |
[20] |
E. Girejko, L. Machado, A. B. Malinowska and N. Martins,
Krause's model of opinion dynamics on isolated time scales, Mathematical Methods in the Applied Sciences, 39 (2016), 5302-5314.
doi: 10.1002/mma.3916. |
[21] |
E. Girejko, A. B. Malinowska and D. F. M. Torres,
The contingent epiderivative and the calculus of variations on time scales, Optimization Letters, 61 (2012), 251-264.
doi: 10.1080/02331934.2010.506615. |
[22] |
E. Girejko and D. F. M. Torres,
The existence of solutions for dynamic inclusions on time scales via duality, Applied Mathematic Letters, 25 (2012), 1632-1637.
doi: 10.1016/j.aml.2012.01.026. |
[23] |
R. Hegselmann and U. Krause,
Opinion dynamics and bounded confidence: Models, analysis, and simulation, J. Artificial Societies and Social Simulations, 5 (2002), 1-33.
|
[24] |
S. Hilger,
Ein Maßkettenkalkül mit Anwendung auf Zentrumsmanningfaltigkeiten Ph. D thesis, Universität Würzburg, 1988. |
[25] |
U. Krause, A discrete nonlinear and non-autonomous model of consensus formation, Comunications in Difference Equations (eds. S. Elaydi, G. Ladas, J. Popenda and J. Rakowski),
Gordon and Breach Publ. , Amsterdam, 2000,227–236. |
[26] |
K. Lehrer and C. Wagner,
Rational Consensus in Science and Society D. Reidel Publishing Company, Dordrecht, Holland, 1981. |
[27] |
A. B. Malinowska and D. F. M. Torres,
Natural boundary conditions in the calculus of variations, Math. Methods Appl. Sci., 33 (2010), 1712-1722.
doi: 10.1002/mma.1289. |
[28] |
A. B. Malinowska, N. Martins and D. F. M. Torres,
Transversality conditions for infinite horizon variational problems on time scales, Optim. Lett., 5 (2011), 41-53.
doi: 10.1007/s11590-010-0189-7. |
[29] |
N. Martins and D. F. M. Torres,
Calculus of variations on time scales with nabla derivatives, Nonlinear Anal., 71 (2009), e763-e773.
doi: 10.1016/j.na.2008.11.035. |
[30] |
T. Vicsek, A. Czirók, E. Ben-Jacob and O. Shochet,
Novel type of phase transition in a system of self-driven particles, Phys. Rev. Letters, 75 (1995), 1226-1229.
doi: 10.1103/PhysRevLett.75.1226. |
[31] |
H. Wang and L. Shang,
Opinion dynamics in networks with common-neighbors-based connections, Physica A, 421 (2015), 180-186.
doi: 10.1016/j.physa.2014.10.090. |
show all references
References:
[1] |
F. M. Atici, D. C. Biles and A. Lebedinsky,
An application of time scales to economics, Math. Comput. Modelling, 43 (2006), 718-726.
doi: 10.1016/j.mcm.2005.08.014. |
[2] |
B. Aulbach and S. Hilger, A unified approach to continuous and discrete dynamics, in Qualitative Theory of Differential Equations (Szeged, 1988), Colloq. Math. Soc. J. Bolyai, 53,
North-Holland, Amsterdam, 1990, 37–56. |
[3] |
Z. Bartosiewicz, N. Martins and D. F. M. Torres,
The second Euler–Lagrange equation of variational calculus on time scales, European Journal of Control, 17 (2011), 9-18.
doi: 10.3166/ejc.17.9-18. |
[4] |
Z. Bartosiewicz,
Linear positive control systems on time scales: Controllability, Mathematics of Control, Signals and Systems, 25 (2013), 327-343.
doi: 10.1007/s00498-013-0106-6. |
[5] |
J. Belikov, U. Kotta and M. Tonso,
Realization of nonlinear MIMO system on homogeneous time scales, European Journal of Control, 23 (2015), 48-54.
doi: 10.1016/j.ejcon.2015.01.006. |
[6] |
V. D. Blondel, J. M. Hendrickx and J. N. Tsitsiklis,
On Krause's multi-agent consensus model with state-dependent connectivity, IEEE Transactions on Automatics Control, 54 (2009), 2586-2597.
doi: 10.1109/TAC.2009.2031211. |
[7] |
V. D. Blondel, J. M. Hendrickx and J. N. Tsitsiklis,
Continuous-time average-preserving opinion dynamics with opinion-dependent communications, SIAM J. Control Optim., 48 (2010), 5214-5240.
doi: 10.1137/090766188. |
[8] |
M. Bohner, M. Fan and J. Zhang,
Periodicity of scalar dynamic equations on time scales and applications to population models, J. Math. Anal. Appl., 330 (2007), 1-9.
doi: 10.1016/j.jmaa.2006.04.084. |
[9] |
M. Bohner and A. Peterson,
Dynamic Equations on Time Scales Birkhäuser Boston, Boston, MA, 2001.
doi: 10.1007/978-1-4612-0201-1. |
[10] |
M. Bohner and A. Peterson,
Advances in Dynamic Equations on Time Scales Birkhäuser Boston, Boston, MA, 2003.
doi: 10.1007/978-1-4612-0201-1. |
[11] |
M. Bohner and H. Warth,
The Beverton-Holt dynamic equation, Appl. Anal., 86 (2007), 1007-1015.
doi: 10.1080/00036810701474140. |
[12] |
M. Caponigro, M. Fornasier, B. Piccoli and E. Trélat,
Sparse stabilization and optimal control of the Cucker-Smale model, Mathematics of Control and Related Fields, 3 (2013), 447-466.
doi: 10.3934/mcrf.2013.3.447. |
[13] |
D. Casagrande, U. Kotta, M. Tonso and M. Wyrwas,
Transfer equivalence and realization of nonlinear input-output delta-differential equations on homogeneous time scales, IEEE Transactions on Automatic Control, 55 (2010), 2601-2606.
doi: 10.1109/TAC.2010.2060251. |
[14] |
S. Chatterjee and E. Seneta,
Towards consensus: Some convergence theorems on repeated averaging, J. Appl. Prob., 14 (1977), 89-97.
doi: 10.1017/S0021900200104681. |
[15] |
Cucker Smale and S. Smale,
Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[16] |
Cucker Smale and S. Smale,
On the mathematics of emergence, Japanese Journal of Mathematics, 2 (2007), 197-227.
doi: 10.1007/s11537-007-0647-x. |
[17] |
M. H. DeGroot,
Reaching a consensus, Journal of the American Statistical Association, 69 (1974), 118-121.
doi: 10.1080/01621459.1974.10480137. |
[18] |
J. French,
A formal theory of social power, Social Networks, A Developing Paradigm, (1977), 35-48.
doi: 10.1016/B978-0-12-442450-0.50010-9. |
[19] |
G. Fu, W. Zhang and Z. Li,
Opinion dynamics of modified Hegselmann–Krause model in a group-based population with heterogeneous bounded confidence, Physica A, 419 (2015), 558-565.
doi: 10.1016/j.physa.2014.10.045. |
[20] |
E. Girejko, L. Machado, A. B. Malinowska and N. Martins,
Krause's model of opinion dynamics on isolated time scales, Mathematical Methods in the Applied Sciences, 39 (2016), 5302-5314.
doi: 10.1002/mma.3916. |
[21] |
E. Girejko, A. B. Malinowska and D. F. M. Torres,
The contingent epiderivative and the calculus of variations on time scales, Optimization Letters, 61 (2012), 251-264.
doi: 10.1080/02331934.2010.506615. |
[22] |
E. Girejko and D. F. M. Torres,
The existence of solutions for dynamic inclusions on time scales via duality, Applied Mathematic Letters, 25 (2012), 1632-1637.
doi: 10.1016/j.aml.2012.01.026. |
[23] |
R. Hegselmann and U. Krause,
Opinion dynamics and bounded confidence: Models, analysis, and simulation, J. Artificial Societies and Social Simulations, 5 (2002), 1-33.
|
[24] |
S. Hilger,
Ein Maßkettenkalkül mit Anwendung auf Zentrumsmanningfaltigkeiten Ph. D thesis, Universität Würzburg, 1988. |
[25] |
U. Krause, A discrete nonlinear and non-autonomous model of consensus formation, Comunications in Difference Equations (eds. S. Elaydi, G. Ladas, J. Popenda and J. Rakowski),
Gordon and Breach Publ. , Amsterdam, 2000,227–236. |
[26] |
K. Lehrer and C. Wagner,
Rational Consensus in Science and Society D. Reidel Publishing Company, Dordrecht, Holland, 1981. |
[27] |
A. B. Malinowska and D. F. M. Torres,
Natural boundary conditions in the calculus of variations, Math. Methods Appl. Sci., 33 (2010), 1712-1722.
doi: 10.1002/mma.1289. |
[28] |
A. B. Malinowska, N. Martins and D. F. M. Torres,
Transversality conditions for infinite horizon variational problems on time scales, Optim. Lett., 5 (2011), 41-53.
doi: 10.1007/s11590-010-0189-7. |
[29] |
N. Martins and D. F. M. Torres,
Calculus of variations on time scales with nabla derivatives, Nonlinear Anal., 71 (2009), e763-e773.
doi: 10.1016/j.na.2008.11.035. |
[30] |
T. Vicsek, A. Czirók, E. Ben-Jacob and O. Shochet,
Novel type of phase transition in a system of self-driven particles, Phys. Rev. Letters, 75 (1995), 1226-1229.
doi: 10.1103/PhysRevLett.75.1226. |
[31] |
H. Wang and L. Shang,
Opinion dynamics in networks with common-neighbors-based connections, Physica A, 421 (2015), 180-186.
doi: 10.1016/j.physa.2014.10.090. |









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