Relative controllability results for nonlinear higher order fractional delay integrodifferential systems with time varying delay in control

This paper is concerned with the controllability of nonlinear higher order fractional delay integrodifferential equations with time varying delay in control, which involved Caputo derivatives of any different orders. A formula for the solution expression of the system is derived by using Laplace transform. A necessary and sufficient condition for the relative controllability of linear fractional delay dynamical systems with time varying delays in control is proved, and a sufficient condition for the corresponding nonlinear integrodifferential equation has obtained. Examples has given to verify the results.

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  • Bagley, R.L. and Torvik, A., A Theoretical basis for the application of fractional calculus to viscoelasticity. Journal of Rheology, 27, (1983), 201-210. Bagley, R.L. and Torvik, A., Fractional calculus in the transient analysis of viscoelastically damped structures, American Institute of Aeronautics and Astronautics, 23, (1985), 918-925.
  • Balachandran, K. and Divya, S., Relative controllability of nonlinear neutral fractional volterra integrodifferential systems with multiple delays in control, Journal of Applied Nonlinear Dynamics, 5, (2016), 147-160.
  • Balachandran, K., Kokila, J. and Trujillo, J.J., Relative controllability of fractional dyamial systems with multiple delays in control, Computers and Mathematics with Applications, 64, (2012), 3037-3045.
  • Balachandran, K., Yong Zhou and Kokila, J., Relative controllability of fractional dynamical systems with delays in control, Communications in Nonlinear Science and Numerical Simulation, 17, (2012). 3508-3520.
  • Chen, Y.Q., Ahn, H.S. and Xue, D., Robust controllability of interval fractional order linear time invariant systems, Signal Process, 86, (2006), 2794-2802.
  • Chow, T.S., Fractional dynamics of interfaces between soft-nanoparticles and rough substrates, Physics Letters A, 342, (2005), 148-155.
  • Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G., Higher transcendental functions, McGraw-Hill Book Company Inc., New York-Toronto-London 1955.
  • He, J.H., Nonlinear oscillation with fractional derivative and its applications, International Conferences on Vibrating Engineering '98, Dalian, China, 288-291 1998.
  • Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific Publisher, Singapore 2000.
  • John, C. and Loiseau, J.J., Applications of time delay systems, Springer-Verlag, Berlin Heidelberg 2007.
  • Joice Nirmala, R., Relative controllability of nonlinear fractional delay dynamical systems with time varying delay in control, Theory and applications of non integer order systems, 407, (2016), 369-379.
  • Joice Nirmala, R. and Balachandran, K., Relative controllability of nonlinear fractional delay integrodifferential systems with multiple delays in control, Kybernetika, 53, (2017), 161-178.
  • Joice Nirmala, R., Relative controllability of nonlinear fractional delay dynamical systems with time varying delay in control, Theory and Applications of Non-integer Order Systems, (2017), 369-379. DOI 10.1007/978-3-319-45474-0-33.
  • Junpeng, L., Suli, L. and Huilai, L., Controllability results of nonlinear higher order fractional damped dynamical system, Journal of Nonlinear Sciences and Applications, 10, (2017), 325-337.
  • Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam 2006.
  • Klamka, J. and Sikora, B., New controllability criteria for fractional systems with varying delays, Theory and Applications of Non integer Order systems , (2017), 333-344. DOI: 10.1007/978-3-319-4547-0-30.
  • Klamka, J., Relative controllability of nonlinear systems with distributed delay in control, International Journal of Control, 28, (1978), 307-312.
  • Magin, R.L., Fractional calculus in bioengineering, Critical Reviews in Biomedical Engineering, 32, (2004), 1-377.
  • Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, A Carpinteri, F. Mainardi (Eds.), Fractals and Fractional calculus in Continuum Mechaics, Springer-Verlag, New York, 291-348, 1997.
  • Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D. and Feliu, V., Fractional order systems and controls fundamentals and applications, London, Springer 2010).
  • Ortigueira, M.D., On the initial conditions in continuous time fractional linear systems, Signal Process, 83, (2003), 2301-2309.
  • Podlubny, I., Fractional Differential Equations, Academic Press, New York (1999).
  • Schi, J.L., The Laplace Transform, Theory and Applications, Springer, New York (1999).
  • Sikora, B., Controllability criteria for time delay fractional systems with retarded state, International Journal of Applied Computation Science, 26, (2016), 521-531.
  • Sivabalan, M. and Sathiyanathan, K., Controllability Results for Nonlinear Higher Order Fractional Delay Dynamical Systems with Distributed Delays in Control, Global Journal of Pure and Applied Mathematics, 13, (2017), 7969-7989.
  • Sivabalan, M. Sivasamy, R. and Sathiyanathan, K., Controllability results for nonlinear higher order fractional delay dynamical systems with control delay, Journal of Applied Nonlinear Dynamics, 8, (2019), (In Press).
  • Sabatier, J., Agarwal, O.P. and Tenreiro Machado, J.A., Advances in fractional calculus, Theoretical developments and applications in physics and engineering, Springer-Verlag 2007.