Controllability of nonlinear implicit fractional integrodifferential systems

Balachandran, Krishnan; Divya, Shanmugam

doi:10.2478/amcs-2014-0052

Abstract

In this paper, we study the controllability of nonlinear fractional integrodifferential systems with implicit fractional derivative. Sufficient conditions for controllability results are obtained through the notion of the measure of noncompactness of a set and Darbo’s fixed point theorem. Examples are included to verify the result.

References

Anichini, G., Conti, G. and Zecca, P. (1986). A note on controllability of certain nonlinear system, Note di Mathematica 6(2): 99-111.
Search in Google Scholar
Balachandran, K. (1988). Controllability of nonlinear systems with implicit derivatives, IMA Journal of Mathematical Control and Information 5(2): 77-83.10.1093/imamci/5.2.77
Search in Google Scholar
Balachandran, K. and Dauer, J.P. (1987). Controllability of nonlinear systems via fixed point theorems, Journal of Optimization Theory and Applications 53(3): 345-352.10.1007/BF00938943
Search in Google Scholar
Balachandran, K. and Balasubramaniam, P. (1992). A note on controllability of nonlinear Volterra integrodifferential systems, Kybernetika 28(4): 284-291.
Search in Google Scholar
Balachandran, K. and Balasubramaniam, P. (1994).
Search in Google Scholar
Controllability of nonlinear neutral Volterra integrodifferential systems, Journal of the Australian Mathematical Society 36(1): 107-116.10.1017/S0334270000010274
Search in Google Scholar
Balachandran, K. and Kokila, J. (2012a). On the controllability of fractional dynamical systems, International Journal of Applied Mathematics and Computer Science 22(3): 523-531, DOI: 10.2478/v10006-012-0039-0.10.2478/v10006-012-0039-0
Search in Google Scholar
Balachandran, K. and Kokila, J. (2013a). Constrained controllability of fractional dynamical systems, Numerical Functional Analysis and Optimization 34(11): 1187-1205.10.1080/01630563.2013.778868
Search in Google Scholar
Balachandran, K. and Kokila, J. (2013b). Controllability of nonlinear implicit fractional dynamical systems, IMA Journal of Applied Mathematics 79(3): 562-570.10.1093/imamat/hxt003
Search in Google Scholar
Balachandran, K., Kokila, J. and Trujillo, J.J. (2012b).
Search in Google Scholar
Relative controllability of fractional dynamical systems with multiple delays in control, Computers and Mathematics with Applications 64(10): 3037-3045.10.1016/j.camwa.2012.01.071
Search in Google Scholar
Balachandran, K., Park, J.Y. and Trujillo, J.J. (2012c).
Search in Google Scholar
Controllability of nonlinear fractional dynamical systems, Nonlinear Analysis: Theory, Methods and Applications 75(4): 1919-1926.10.1016/j.na.2011.09.042
Search in Google Scholar
Balachandran, K., Zhou, Y. and J. Kokila, J. (2012d). Relative controllability of fractional dynamical systems with delays in control, Communications in Nonlinear Science and Numerical Simulation 17(9): 3508-3520.10.1016/j.cnsns.2011.12.018
Search in Google Scholar
Burton, T.A. (1983). Volterra Integral and Differential Equations, Academic Press, New York, NY.
Search in Google Scholar
Caputo, M. (1967). Linear model of dissipation whose Q is almost frequency independent, Part II, Geophysical Journal of Royal Astronomical Society 13(5): 529-539.10.1111/j.1365-246X.1967.tb02303.x
Search in Google Scholar
Dacka, C. (1980). On the controllability of a class of nonlinear systems, IEEE Transaction on Automatic Control 25(2): 263-266.10.1109/TAC.1980.1102287
Search in Google Scholar
Kaczorek, K. (2011). Selected Problems of Fractional Systems Theory, Springer, Berlin.10.1007/978-3-642-20502-6
Search in Google Scholar
Kexue, L. and Jigen, P. (2011). Laplace transform and fractional differential equations, Applied Mathematics Letters 24(12): 2019-2013.10.1016/j.aml.2011.05.035
Search in Google Scholar
Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.
Search in Google Scholar
Klamka, J. (1975a). On the global controllability of perturbed nonlinear systems, IEEE Transactions on Automatic Control AC-20(1): 170-172.10.1109/TAC.1975.1100870
Search in Google Scholar
Klamka, J. (1975b). On the local controllability of perturbed nonlinear systems, IEEE Transactions on Automatic Control AC-20(2): 289-291.10.1109/TAC.1975.1100929
Search in Google Scholar
Klamka, J. (1975c). Controllability of nonlinear systems with delays in control, IEEE Transactions on Automatic Control AC-20(5): 702-704.10.1109/TAC.1975.1101046
Search in Google Scholar
Klamka, J. (1993). Controllability of Dynamical Systems, Kluwer Academic, Dordrecht.
Search in Google Scholar
Klamka, J. (1999). Constrained controllability of dynamic systems, International Journal of Applied Mathematics and Computer Science 9(2): 231-244.
Search in Google Scholar
Klamka, J. (2000). Schauder’s fixed-point theorem in nonlinear controllability problems, Control and Cybernetics 29(1): 153-165.
Search in Google Scholar
Klamka, J. (2001). Constrained controllability of semilinear delayed systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 49(3): 505-515.
Search in Google Scholar
Klamka, J. (2008). Constrained controllability of semilinear systems with delayed controls, Bulletin of the Polish Academy of Sciences: Technical Sciences 56(4): 333-337.
Search in Google Scholar
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. 503-509.10.1007/978-90-481-3293-5_45
Search in Google Scholar
Miller, K.S. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY.
Search in Google Scholar
Mittal, R.C. and Nigam, R. (2008). Solution of fractional integrodifferential equations by Adomian decomposition method, International Journal of AppliedMathematics and Mechanics 4(2): 87-94.
Search in Google Scholar
Oldham, K.B and Spanier, J. (1974). The Fractional Calculus, Academic Press, London.
Search in Google Scholar
Olmstead, W.E. and Handelsman, R.A. (1976). Diffusion in a semi-infinite region with nonlinear surface dissipation, SIAM Review 18(2): 275-291.10.1137/1018044
Search in Google Scholar
Podlubny, I. (1999). Fractional Differential Equations, Academic Press, New York, NY.
Search in Google Scholar

Controllability of nonlinear implicit fractional integrodifferential systems

Abstract

Paradigm

My account