Advanced Search
Article Contents
Article Contents

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

Abstract Full Text(HTML) Figure(2) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 49K99, 26A33; Secondary: 39A99, 90C25.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Consensus parameters with using control

    Figure 2.  Consensus parameters without control

  •   T. Abdeljawad , On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011) , 1602-1611.  doi: 10.1016/j.camwa.2011.03.036.
      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.
      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.
      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.
      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.
      M. Caponigro , M. Fornasier , B. 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.
      A. Chakraborti , Distributions of money in models of market economy, Int. J. Modern Phys. C, 13 (2002) , 1315-1321.  doi: 10.1142/S0129183102003905.
      Y.-L. Chuang , Y. R. Huang , M. 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.
      I. D. Couzin , J. Krause , N. 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.
      F. Cucker  and  S. Smale , On the mathematics of emergence, Japan. J. Math., 2 (2007) , 197-227.  doi: 10.1007/s11537-007-0647-x.
      F. Cucker  and  S. Smale , Emergent behavior in flocks, IEEE Trans. Autom. Control, 52 (2007) , 852-862.  doi: 10.1109/TAC.2007.895842.
      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.
      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.
      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. 
      S. Galam , Y. 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.
      R. Hilfer, Applications of Fractional Calculus in Physics World Scientific Publishing, River Edge, NJ, 2000. doi: 10.1142/9789812817747.
      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.
      A. Jadbabaie , J. 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.
      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.
      R. Kamocki , Pontryagin Maximum Principle for fractional ordinary optimal control problems, Math. Meth. Appl. Sci., 37 (2014) , 1668-1686.  doi: 10.1002/mma.2928.
      M. P. LazarevićAdvanced Topics on Applications of Fractional Calculus on Control Problems, System Stability and Modeling, WSEAS Press, 2014. 
      J. A. T. Machado , Discrete-time fractional-order controllers, Fract. Calc. Appl. Anal., 4 (2001) , 47-66. 
      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.
      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.
      I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198, Academic Press, San Diego, CA, 1999.
      P. D. Powell, Calculating Determinants of Block Matrices 2011, arXiv: 1112.4379.
  • 加载中



Article Metrics

HTML views(449) PDF downloads(215) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint