Controllability of Higher Order Fractional Damped Delay Dynamical Systems with Time Varying Multiple Delays in Control

Yıl 2021, , 246 - 259, 30.06.2021

Sivabalan M , Sathiyanathan K

https://doi.org/10.31197/atnaa.685326

Cited By: 2

Öz

This paper is concerned with the controllability of higher order fractional damped delay dynamical systems with time varying multiple delays in control, which involved Caputo derivatives of any different orders. A necessary and sufficient condition for the controllability of linear fractional damped delay dynamical system is obtained by using the Grammian matrix. Sufficient conditions for controllability of the corresponding nonlinear damped delay dynamical system has established by the successive approximation technique. Examples have provided to verify the results.

Anahtar Kelimeler

Damped delay dynamical systems, Controllability, Mittag-Leffler Matrix function, Iterative technique

Kaynakça

• B. N. N. Achar, J. W. Hanneken, T. Clarke, Response characteristics of a fractional oscillator, Physica A Stat. Mech. Appl., 309 (3-4) (2002) 275-288.
• A. Al-rabtah, V. S. Erturk, S. Momani, Solutions of a fractional oscillator by using differential transform method, “Comput. Math. with Appl., 59 (3) (2010) 1356-1362.
• Aziz Khan, Thabet Abdeljawad, J.F. Gómez-Aguilar, Hasib Khan, Dynamical study of fractional order mutualism parasitism food web Module, Chaos, Solitons and Fractals, 134 (2020) 109685.
• Aziz Khan, J.F. Go´ mez-Aguilar, Thabet Abdeljawad, Hasib Khan, Stability and numerical simulation of a fractional order plant-nectar-pollinator model, Alex. Eng. J., 59 (2020) 49-59.
• R. L. Bagley, A. Torvik, A Theoretical basis for the application of fractional calculus to viscoelasticity. Journal of Rheology, 27 (3) (1983) 201-210.
• K. Balachandran, V. Govindaraj, M. Rivero, J. J. Trujillo, Controllability of fractional damped dynamical systems, Appl Math Comput., 257 (2015) 66-73.
• K. Balachandran, J. Y. Park, J. J. Trujillo, Controllability of nonlinear fractional dynamical systems, Nonlinear Anal Theory Methods Appl., 75 (4) (2012) 1919-1926.
• D. Baleanu, G. C. Wu, Y. R. Bai, F. L. Chen, Stability analysis of Caputo-like discrete fractional systems, Commun. Nonlinear Sci.Numer. Simul., 48 (2017) 520-530.
• R. Bellman, K. L. Cooke, Differential-Difference equations, Academic Press, New York, NY (1963).
• Y.Q. Chen, I.D. Petras, D. Xue, Fractional order control: A tutorial, in Proceedings of the 2009. American Control Conference (ACC'09), June 10-12, 2009, St. Louis, IEEE, Piscataway, NJ, 2009, 1397-1411.
• T.S. Chow, Fractional dynamics of interfaces between soft-nanoparticles and rough substrates, Phys. Lett. A, 342 (1-2) (2005) 148-155.
• J. Dauer, R. Gahl, Controllability of nonlinear delay systems, J. Optim. Theory Appl., 21 (1) (1977) 59-70.
• A. M. A. El-Sayed, H. M. Nour, A. Elsaid, A. E. Matouk, A. Elsonbaty, Dynamical behaviors, circuit realization, chaos control, and synchronization of a new fractional order hyperchaotic system, Appl. Math. Model., 40 (5) (2016) 3516-3534.
• G. Fernández-Anaya, G. Nava-Antonio, J. Jamous-Galante, R. Muñoz-Vega, E. G. Hernández Martínez, Asymptotic stability of distributed order nonlinear dynamical systems, Commun. Nonlinear Sci. Numer. Simul., 48 (2017) 541-549.
• J. F. Gomez-Aguilar, T. Cordova-Fraga, Thabet Abdeljawad, Aziz Khan, Hasib Khan, Analysis of fractal-fractional malaria transmission model, Fractals. 28 (8) (2020) 2040041 (25 pages).
• J. Hale, Theory of functional differential equations, Springer, New York (1977).
• J. He, Nonlinear oscillation with fractional derivative and its applications, International Conferences on Vibrating Engineering, Dalian, China, (1998) 288-291.
• B.B. He, H.C. Zhou, C.H. Kou, The controllability of fractional damped dynamical systems with control delay, Commun.Nonlinear Sci.Numer. Simul., 32 (2016) 190-198.
• S. Huang, R. Zhang, D. Chen, Stability of nonlinear fractional-order time varying systems, J. Comput. Nonlinear Dyn., 11 (3) (2016) 031007 (9 pages).
• Junpeng Liu, Suli Liu, Huilai Li, Controllability result of nonlinear higher order fractional damped dynamical system, J. Nonlinear Sci. Appl., 10 (2017) 325-337.
• A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006).
• R. L. Magin, Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng., 32 (2004) 1-104.
• R. L. Magin, Fractional calculus models of complex dynamics in biological tissues, Comput. Math. with Appl., 59 (5) (2010) 1586-1593.
• F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, A Carpinteri, Francesco Mainardi (Eds.), Fractals and Fractional calculus in Continuum Mechaics, Springer-Verlag, New York, (1997) 291-348.
• S. Manabe, The non-integer integral and its application to control systems, Journal of Institute of Electrical Engineers of Japan, 6 (3-4) (1961) 83-87.
• K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley and Sons, New York (1993).
• Z. Odibat, S. Momani, The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics, Comput. Math. with Appl., 58 (2009) 2199-2208.
• J. Sabatier, O.P. Agarwal, J.A. Tenreiro Machado, Advances in fractional calculus, Theoretical developments and applications in physics and engineering, Springer-Verlag (2007).
• M. Sivabalan, K. Sathiyanathan, Relative controllability results for nonlinear higher order fractional delay integrodifferential systems with time varying delay in control, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68 (1) (2018) 889-906.
• M. Sivabalan, K. Sathiyanathan, Controllability of higher order fractional damped delay dynamical systems, J. Phys. Conf. Ser., 1139 (2018) 12030.
• A. Soukkou, M. C. Belhour, S. Leulmi, Review, design, optimization and stability analysis of fractional-order PID controller, Int. j. intell. syst. appl., 8 (7) (2016) 73-96.
• A. Tofighi, The intrinsic damping of the fractional oscillator, Phys. A, 329 (1-2) (2003), 29-34.
• L. Wiess, On the controllability of delayed differential systems, SIAM j. control, 5 (4) (1967) 575-587.
• K. Yongyong, Z. Xiue, Some comparison of two fractional oscillators, Physica B: Physics of Condensed Matter, 405 (1) (2010) 369-373.
• Z. Zhou, W. Gong, Finite element approximation of optimal control problems governed by time fractional diffusion equation, Comput. Math. with Appl., 71 (1) (2016) 301-318.