On the controllability of fractional dynamical systems

• Autorzy: Krishnan Balachandran , Jayakumar Kokila
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

controllability; fractional differential equations; Mittag-Leffler function

• Wydawca: University of Zielona Gora Press
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2012
• Tom: 22
• Numer: 3
• Strony: 523-531

Daty

wydano

2012

otrzymano

2011-06-03

poprawiono

2011-11-04

Twórcy

autor

Krishnan Balachandran

• Department of Mathematics, Bharathiar University, Coimbatore 641 046, India

autor

Jayakumar Kokila

• Department of Mathematics, Bharathiar University, Coimbatore 641 046, India

Bibliografia

• 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.
• Al Akaidi, M. (2008). Fractal Speech Processing, Cambridge University Press, Cambridge.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• Do, V.N. (1990). Controllability of semilinear systems, Journal of Optimization Theory and Applications 65(1): 41-52.
• 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.
• Herrmann, R. (2011). Fractional Calculus: An Introduction for Physicists, World Scientific Publishing, Singapore.
• 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.
• 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.
• Kexue, L. and Jigen, P. (2011). Laplace transform and fractional differential equations, Applied Mathematics Letters 24(12): 2019-2023.
• Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier, New York, NY.
• Klamka, J. (1993). Controllability of Dynamical Systems, Kluwer Academic, Dordrecht.
• Klamka, J. (2008). Constrained controllability of semilinear systems with delayed controls, Bulletin of the Polish Academy of Sciences: Technical Sciences 56(4): 333-337.
• 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.
• 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.
• 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.
• Machado, J.T. (1997). Analysis and design of fractional order digital control systems, Systems Analysis, Modelling and Simulation 27(2-3): 107-122.
• Metzler, R. and Klafter, J. (2000). The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics Reports 339(1): 1-77.
• Miller, K.S. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY.
• 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.
• Oldham, K.B., and Spanier, J. (1974). The Fractional Calculus, Academic Press, New York, NY.
• Oustaloup, A. (1991). La Commade CRONE: Commande Robuste d'Ordre Non Entier, Hermès, Paris.
• Podlubny, I. (1999a). Fractional Differential Equations, Academic Press, London.
• Podlubny, I. (1999b). Fractional-order systems and P I λ Dμ controllers, IEEE Transactions on Automatic Control 44(1): 208-214.
• Renardy, M., Hrusa, W.J. and Nohel, J.A.(1987). Mathematical Problems in Viscoelasticity, Longman Scientific and Technical, New York, NY.
• Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993). Fractional Integrals and Derivatives: Theory and Applications, Gordan and Breach, Amsterdam.
• Shamardan, A.B. and Moubarak, M.R.A. (1999). Controllability and observability for fractional control systems, Journal of Fractional Calculus 15(1): 25-34.
• 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.
• West, B.J., Bologna, M. and Grigolini, P. (2003). Physics of Fractal Operators, Springer-Verlag, Berlin.

Identyfikatory

DOI

10.2478/v10006-012-0039-0