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 and 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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[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. 

[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.

[16]

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

[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.

[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.

[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.

[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.

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

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

[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.

[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.

[25]

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

[26]

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

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[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. 

[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.

[16]

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

[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.

[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.

[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.

[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.

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

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

[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.

[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.

[25]

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

[26]

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

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