January  2018, 23(1): 347-357. doi: 10.3934/dcdsb.2018023

Optimal control of the discrete-time fractional-order Cucker-Smale model

1. 

Faculty of Computer Science, Bialystok University of Technology, 15-351 Bia lystok, Poland

2. 

Department of Mathematics and Mathematical Economics, Warsaw School of Economics, 02-554 Warsaw, Poland

* Corresponding author: a.malinowska@pb.edu.pl

Received  September 2016 Revised  November 2016 Published  January 2018

We obtain necessary optimality conditions for the discrete-time fractional-order Cucker-Smale optimal control problem. By using fractional order differences on the left side of nonlinear system we introduce memory effects to the considered problem.

Citation: Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023
References:
[1]

T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011), 1602-1611.  doi: 10.1016/j.camwa.2011.03.036.  Google Scholar

[2]

I. Aoki, A simulation study on the schooling mechanism in fish, Bull. Japan. Soc. Sci. Fish, 48 (1982), 1081-1088.  doi: 10.2331/suisan.48.1081.  Google Scholar

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F. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc., 137 (2009), 981-989.  doi: 10.1090/S0002-9939-08-09626-3.  Google Scholar

[4]

B. Bijnan and S. Kamal, Stabilization and Control of Fractional Order Systems: A Sliding Mode Approach Springer, 2015. doi: 10.1007/978-3-319-08621-7.  Google Scholar

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L. Bourdin, Contributions au calcul des variations et au Principe du Maximum de Pontryagin en calculs time scale et fractionnaire, PhD Thesis, Université de Pau et des Pays de l'Adour, 2013. Google Scholar

[6]

M. CaponigroM. FornasierB. Piccoli and E. Trelat, Sparse stabilization and optimal control of the Cucker-Smale model, Math. Cont. Related Fields AIMS, 3 (2013), 447-466.  doi: 10.3934/mcrf.2013.3.447.  Google Scholar

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A. Chakraborti, Distributions of money in models of market economy, Int. J. Modern Phys. C, 13 (2002), 1315-1321.  doi: 10.1142/S0129183102003905.  Google Scholar

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Y.-L. ChuangY. R. HuangM. R. D'Orsogna and A. L. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials, IEEE International Conference on Robotics and Automation, (2007), 2292-2299.  doi: 10.1109/ROBOT.2007.363661.  Google Scholar

[9]

I. D. CouzinJ. KrauseN. R. Franks and S. Levin, Effective leadership and decision making in animal groups on the move, Nature, 433 (2005), 513-516.  doi: 10.1038/nature03236.  Google Scholar

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F. Cucker and S. Smale, On the mathematics of emergence, Japan. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.  Google Scholar

[11]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Autom. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[12]

P. Degond and S. Motsch, Macroscopic limit of self-driven particles with orientation interaction, C.R. Math. Acad. Sci. Paris, 345 (2007), 555-560.  doi: 10.1016/j.crma.2007.10.024.  Google Scholar

[13]

J. B. Diaz and T. J. Osler, Differences of fractional order, Math. Comp., 28 (1974), 185-202.  doi: 10.1090/S0025-5718-1974-0346352-5.  Google Scholar

[14]

A. Dzieliński and P. M. Czyronis, Fixed final time and free final state optimal control problem for fractional dynamic systems -linear quadratic discrete-time case, Bull. Pol. Acad. Sci., Tech. Sci., 61 (2013), 681-690.   Google Scholar

[15]

S. GalamY. Gefen and Y. Shapir, Sociophysics: A new approach of sociological collective behavior, J. Math. Sociology, 9 (1982), 1-13.  doi: 10.1007/978-1-4614-2032-3.  Google Scholar

[16]

R. Hilfer, Applications of Fractional Calculus in Physics World Scientific Publishing, River Edge, NJ, 2000. doi: 10.1142/9789812817747.  Google Scholar

[17]

A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems, Translated from the Russian by Karol Makowski. Studies in Mathematics and its Applications, North-Holand Pub. Co. Amsterdam, New York, Oxford, 1979.  Google Scholar

[18]

A. JadbabaieJ. Lin and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. on Autom. Control, 48 (2003), 988-1001.  doi: 10.1109/TAC.2003.812781.  Google Scholar

[19]

T. Kaczorek, Selected Problems of Fractional Systems Theory, Lecture Notes in Control and Information Sciences, vol. 411, Springer–Verlag, Berlin, 2011. doi: 10.1007/978-3-642-20502-6.  Google Scholar

[20]

R. Kamocki, Pontryagin Maximum Principle for fractional ordinary optimal control problems, Math. Meth. Appl. Sci., 37 (2014), 1668-1686.  doi: 10.1002/mma.2928.  Google Scholar

[21] M. P. Lazarević, Advanced Topics on Applications of Fractional Calculus on Control Problems, System Stability and Modeling, WSEAS Press, 2014.   Google Scholar
[22]

J. A. T. Machado, Discrete-time fractional-order controllers, Fract. Calc. Appl. Anal., 4 (2001), 47-66.   Google Scholar

[23]

K. S. Miller and B. Ross, Fractional difference calculus, Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, Japan, Ellis Horwood Ser. Math. Appl., Horwood, Chichester, (1989), 139–152.  Google Scholar

[24]

P. Ostalczyk, Discrete Fractional Calculus: Applications in Control and Image Processing Series in Computer Vision, 4. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016. doi: 10.1142/9833.  Google Scholar

[25]

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198, Academic Press, San Diego, CA, 1999.  Google Scholar

[26]

P. D. Powell, Calculating Determinants of Block Matrices 2011, arXiv: 1112.4379. Google Scholar

show all references

References:
[1]

T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011), 1602-1611.  doi: 10.1016/j.camwa.2011.03.036.  Google Scholar

[2]

I. Aoki, A simulation study on the schooling mechanism in fish, Bull. Japan. Soc. Sci. Fish, 48 (1982), 1081-1088.  doi: 10.2331/suisan.48.1081.  Google Scholar

[3]

F. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc., 137 (2009), 981-989.  doi: 10.1090/S0002-9939-08-09626-3.  Google Scholar

[4]

B. Bijnan and S. Kamal, Stabilization and Control of Fractional Order Systems: A Sliding Mode Approach Springer, 2015. doi: 10.1007/978-3-319-08621-7.  Google Scholar

[5]

L. Bourdin, Contributions au calcul des variations et au Principe du Maximum de Pontryagin en calculs time scale et fractionnaire, PhD Thesis, Université de Pau et des Pays de l'Adour, 2013. Google Scholar

[6]

M. CaponigroM. FornasierB. Piccoli and E. Trelat, Sparse stabilization and optimal control of the Cucker-Smale model, Math. Cont. Related Fields AIMS, 3 (2013), 447-466.  doi: 10.3934/mcrf.2013.3.447.  Google Scholar

[7]

A. Chakraborti, Distributions of money in models of market economy, Int. J. Modern Phys. C, 13 (2002), 1315-1321.  doi: 10.1142/S0129183102003905.  Google Scholar

[8]

Y.-L. ChuangY. R. HuangM. R. D'Orsogna and A. L. Bertozzi, Multi-vehicle flocking: Scalability of cooperative control algorithms using pairwise potentials, IEEE International Conference on Robotics and Automation, (2007), 2292-2299.  doi: 10.1109/ROBOT.2007.363661.  Google Scholar

[9]

I. D. CouzinJ. KrauseN. R. Franks and S. Levin, Effective leadership and decision making in animal groups on the move, Nature, 433 (2005), 513-516.  doi: 10.1038/nature03236.  Google Scholar

[10]

F. Cucker and S. Smale, On the mathematics of emergence, Japan. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x.  Google Scholar

[11]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Autom. Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.  Google Scholar

[12]

P. Degond and S. Motsch, Macroscopic limit of self-driven particles with orientation interaction, C.R. Math. Acad. Sci. Paris, 345 (2007), 555-560.  doi: 10.1016/j.crma.2007.10.024.  Google Scholar

[13]

J. B. Diaz and T. J. Osler, Differences of fractional order, Math. Comp., 28 (1974), 185-202.  doi: 10.1090/S0025-5718-1974-0346352-5.  Google Scholar

[14]

A. Dzieliński and P. M. Czyronis, Fixed final time and free final state optimal control problem for fractional dynamic systems -linear quadratic discrete-time case, Bull. Pol. Acad. Sci., Tech. Sci., 61 (2013), 681-690.   Google Scholar

[15]

S. GalamY. Gefen and Y. Shapir, Sociophysics: A new approach of sociological collective behavior, J. Math. Sociology, 9 (1982), 1-13.  doi: 10.1007/978-1-4614-2032-3.  Google Scholar

[16]

R. Hilfer, Applications of Fractional Calculus in Physics World Scientific Publishing, River Edge, NJ, 2000. doi: 10.1142/9789812817747.  Google Scholar

[17]

A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems, Translated from the Russian by Karol Makowski. Studies in Mathematics and its Applications, North-Holand Pub. Co. Amsterdam, New York, Oxford, 1979.  Google Scholar

[18]

A. JadbabaieJ. Lin and A. S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. on Autom. Control, 48 (2003), 988-1001.  doi: 10.1109/TAC.2003.812781.  Google Scholar

[19]

T. Kaczorek, Selected Problems of Fractional Systems Theory, Lecture Notes in Control and Information Sciences, vol. 411, Springer–Verlag, Berlin, 2011. doi: 10.1007/978-3-642-20502-6.  Google Scholar

[20]

R. Kamocki, Pontryagin Maximum Principle for fractional ordinary optimal control problems, Math. Meth. Appl. Sci., 37 (2014), 1668-1686.  doi: 10.1002/mma.2928.  Google Scholar

[21] M. P. Lazarević, Advanced Topics on Applications of Fractional Calculus on Control Problems, System Stability and Modeling, WSEAS Press, 2014.   Google Scholar
[22]

J. A. T. Machado, Discrete-time fractional-order controllers, Fract. Calc. Appl. Anal., 4 (2001), 47-66.   Google Scholar

[23]

K. S. Miller and B. Ross, Fractional difference calculus, Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, Japan, Ellis Horwood Ser. Math. Appl., Horwood, Chichester, (1989), 139–152.  Google Scholar

[24]

P. Ostalczyk, Discrete Fractional Calculus: Applications in Control and Image Processing Series in Computer Vision, 4. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2016. doi: 10.1142/9833.  Google Scholar

[25]

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198, Academic Press, San Diego, CA, 1999.  Google Scholar

[26]

P. D. Powell, Calculating Determinants of Block Matrices 2011, arXiv: 1112.4379. Google Scholar

Figure 1.  Consensus parameters with using control
Figure 2.  Consensus parameters without control
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