On the controllability of fractional dynamical systems

Krishnan Balachandran; Jayakumar Kokila

International Journal of Applied Mathematics and Computer Science (2012)

  • Volume: 22, Issue: 3, page 523-531
  • ISSN: 1641-876X

Abstract

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This paper is concerned with the controllability of linear and nonlinear fractional dynamical systems in finite dimensional spaces. Sufficient conditions for controllability are obtained using Schauder's fixed point theorem and the controllability Grammian matrix which is defined by the Mittag-Leffler matrix function. Examples are given to illustrate the effectiveness of the theory.

How to cite

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Krishnan Balachandran, and Jayakumar Kokila. "On the controllability of fractional dynamical systems." International Journal of Applied Mathematics and Computer Science 22.3 (2012): 523-531. <http://eudml.org/doc/244055>.

@article{KrishnanBalachandran2012,
abstract = {This paper is concerned with the controllability of linear and nonlinear fractional dynamical systems in finite dimensional spaces. Sufficient conditions for controllability are obtained using Schauder's fixed point theorem and the controllability Grammian matrix which is defined by the Mittag-Leffler matrix function. Examples are given to illustrate the effectiveness of the theory.},
author = {Krishnan Balachandran, Jayakumar Kokila},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {controllability; fractional differential equations; Mittag-Leffler function},
language = {eng},
number = {3},
pages = {523-531},
title = {On the controllability of fractional dynamical systems},
url = {http://eudml.org/doc/244055},
volume = {22},
year = {2012},
}

TY - JOUR
AU - Krishnan Balachandran
AU - Jayakumar Kokila
TI - On the controllability of fractional dynamical systems
JO - International Journal of Applied Mathematics and Computer Science
PY - 2012
VL - 22
IS - 3
SP - 523
EP - 531
AB - This paper is concerned with the controllability of linear and nonlinear fractional dynamical systems in finite dimensional spaces. Sufficient conditions for controllability are obtained using Schauder's fixed point theorem and the controllability Grammian matrix which is defined by the Mittag-Leffler matrix function. Examples are given to illustrate the effectiveness of the theory.
LA - eng
KW - controllability; fractional differential equations; Mittag-Leffler function
UR - http://eudml.org/doc/244055
ER -

References

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  1. Adams, J.L. and Hartley, T.T. (2008). Finite time controllability of fractional order systems, Journal of Computational and Nonlinear Dynamics 3(2): 021402-1-021402-5. 
  2. Al Akaidi, M. (2008). Fractal Speech Processing, Cambridge University Press, Cambridge. Zbl1082.94003
  3. Arena, P., Caponetta, R., Fortuna L. and Porto, D. (2008). Nonlinear Noninteger Order Circuits and Systems: An Introduction, World Scientific Series on Nonlinear Science, Vol. 38, World Scientific, Singapore. 
  4. Balachandran, K. and Dauer, J.P. (1987). Controllability of nonlinear systems via fixed point theorems, Journal of Optimization Theory and Applications 53(3): 345-352. Zbl0596.93010
  5. Balachandran, K. and Kiruthika, S. (2010). Existence of solutions of abstract fractional impulsive semilinear evolution equations, Electronic Journal of Qualitative Theory of Differential Equations 4: 1-12. Zbl1201.34091
  6. Balachandran, K. and Trujillo, J.J. (2010). The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces, Nonlinear Analysis: Theory, Methods and Applications 72(12): 4587-4593. Zbl1196.34007
  7. Balachandran, K., Kiruthika, S. and Trujillo, J.J. (2011). Existence results for fractional impulsive integrodifferential equations in Banach spaces, Communication in Nonlinear Science and Numerical Simulation 16(4): 1970-1977. Zbl1221.34215
  8. Balachandran, K., Park J.Y. and Trujillo, J.J. (2012). Controllability of nonlinear fractional dynamical systems, Nonlinear Analysis: Theory, Methods & Applications 75(4): 1919-1926. Zbl1277.34006
  9. Benson, D.A., Wheatcraft, S.W. and Meerschaert, M.M. (2000). Application of a fractional advection-dispersion equation, Water Resources Research 36(6): 1403-1412. 
  10. Bettayeb, M. and Djennoune, S. (2008). New results on the controllability and observability of fractional dynamical systems, Journal of Vibrating and Control 14(9-10): 1531-1541. Zbl1229.93018
  11. Bonilla, B., Rivero, M. and Trujillo, J.J. (2007). On systems of linear fractional differential equations with constant coefficients, Applied Mathematics and Computation 187(1): 68-78. Zbl1121.34006
  12. Caputo, M. (1967 ). Linear model of dissipation whose Q is almost frequency independent, Part II, Geophysical Journal of Royal Astronomical Society 13(5): 529-539. 
  13. Chen, Y.Q., Ahn, H.S. and Xue, D. (2006). Robust controllability of interval fractional order linear time invariant systems, Signal Processing 86(10): 2794-2802. Zbl1172.94386
  14. Chikrii, A.A. and Matichin, I.I. (2008). Presentation of solutions of linear systems with fractional derivatives in the sense of Riemann-Liouville, Caputo and Miller-Ross, Journal of Automation and Information Sciences 40(6): 1-11. 
  15. Chikrii, A. and Matichin, I.I. (2010). Game problems for fractional order systems, in D. Baleanu, Z.B. Guvenc and J.A.T. Machado (Eds.), New Trends in Nanotechnology and Fractional Calculus, Springer-Verlag, New York, NY, pp. 233-241. Zbl1303.91042
  16. Do, V.N. (1990). Controllability of semilinear systems, Journal of Optimization Theory and Applications 65(1): 41-52. Zbl0674.93006
  17. Guermah, S.A., Djennoune, S.A. and Bettayeb, M.A.(2008). Controllability and observability of linear discrete time fractional order systems, International Journal of Applied Mathematics and Computer Science 18(2): 213-222, DOI: 10.2478/v10006-008-0019-6. Zbl1234.93014
  18. Herrmann, R. (2011). Fractional Calculus: An Introduction for Physicists, World Scientific Publishing, Singapore. Zbl1232.26006
  19. Ichise, M., Nagayanagi, Y. and Kojima, T. (1971). Analog simulation of non integer order transfer functions for analysis of electrode processes, Journal of Electroanalytical Chemistry 33(2): 253-265. 
  20. Karthikeyan, S. and Balachandran, K. (2011). Constrained controllability of nonlinear stochastic impulsive systems, International Journal of Applied Mathematics and Computer Science 21(2): 307-316, DOI: 10.2478/v10006-011-0023-0. Zbl1282.93053
  21. Kexue, L. and Jigen, P. (2011). Laplace transform and fractional differential equations, Applied Mathematics Letters 24(12): 2019-2023. Zbl1238.34013
  22. Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier, New York, NY. Zbl1092.45003
  23. Klamka, J. (1993). Controllability of Dynamical Systems, Kluwer Academic, Dordrecht. Zbl0797.93004
  24. Klamka, J. (2008). Constrained controllability of semilinear systems with delayed controls, Bulletin of the Polish Academy of Sciences: Technical Sciences 56(4): 333-337. 
  25. Klamka, J. (2010). Controllability and minimum energy control problem of fractional discrete time systems, in D. Baleanu, Z.B. Guvenc and J.A.T. Machado (Eds.), New Trends in Nanotechnology and Fractional Calculus, Springer-Verlag, New York, NY, pp. 503-509. Zbl1222.93030
  26. Liu, F., Anh, V.V., Turner, I. and Zhuang, P. (2003). Time fractional advection-dispersion equation, Journal of Applied Mathematics and Computing 13(1-2): 233-245. Zbl1068.26006
  27. Machado, J.T., Kiryakova, V. and Mainardi, F. (2011). Recent history of fractional calculus, Communications in Nonlinear Science and Numerical Simulations 16(3): 1140-1153. Zbl1221.26002
  28. Machado, J.T. (1997). Analysis and design of fractional order digital control systems, Systems Analysis, Modelling and Simulation 27(2-3): 107-122. Zbl0875.93154
  29. Metzler, R. and Klafter, J. (2000). The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics Reports 339(1): 1-77. Zbl0984.82032
  30. Miller, K.S. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY. Zbl0789.26002
  31. Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D. and Feliu, V. (2010). Fractional-order Systems and Controls: Fundamentals and Applications, Springer, London. Zbl1211.93002
  32. Oldham, K.B., and Spanier, J. (1974). The Fractional Calculus, Academic Press, New York, NY. Zbl0292.26011
  33. Oustaloup, A. (1991). La Commade CRONE: Commande Robuste d'Ordre Non Entier, Hermès, Paris. 
  34. Podlubny, I. (1999a). Fractional Differential Equations, Academic Press, London. Zbl0924.34008
  35. Podlubny, I. (1999b). Fractional-order systems and P I λ Dμ controllers, IEEE Transactions on Automatic Control 44(1): 208-214. Zbl1056.93542
  36. Renardy, M., Hrusa, W.J. and Nohel, J.A.(1987). Mathematical Problems in Viscoelasticity, Longman Scientific and Technical, New York, NY. Zbl0719.73013
  37. Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993). Fractional Integrals and Derivatives: Theory and Applications, Gordan and Breach, Amsterdam. Zbl0818.26003
  38. Shamardan, A.B. and Moubarak, M.R.A. (1999). Controllability and observability for fractional control systems, Journal of Fractional Calculus 15(1): 25-34. Zbl0964.93013
  39. Valerio, D. and Sa da Costa, J. (2004). Non integer order control of a flexible robot, Proceedings of the IFAC Workshop on Fractional Differentiation and its Applications, Bordeaux, France, pp. 520-525. 
  40. West, B.J., Bologna, M. and Grigolini, P. (2003). Physics of Fractal Operators, Springer-Verlag, Berlin. 

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