Non-Linear Fractional Integrodifferential Systems with Delays: A Controllability Study

E. Onyeocha¹ , C.A. Nse² , I.J. Njoku³

Section:Research Paper, Product Type: Journal-Paper
Vol.9 , Issue.2 , pp.43-48, Apr-2022

Online published on Apr 30, 2022

Copyright © E. Onyeocha, C.A. Nse, I.J. Njoku . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

View this paper at   Google Scholar | DPI Digital Library

XML View     PDF Download

How to Cite this Paper

Abstract :
A dynamical system which can be acted on using suitable controls is said to be a control system. This work present results for the controllability of a class of non-linear implicit neutral fractional integro-differential equations using a fixed point method. This research aims to present novel results on the controllability of nonlinear fractional integro-differential systems. A neutral system in this paper refers to a dynamical system, in which there is a delay/relay while the system runs, i.e., the time and system functionality are not constant. The Schauder’s fixed point theory is used to establish the controllability results for the system. Furthermore, in order to establish non-compactness, the Arzela-Ascoli theorem was used and with that the controllability conditions for the dynamical system with delay was established. The study could be extended to consider systems without delay using the Schauder fixed point theory and also, other methods such as the algebraic and functional analytic method could be applied to this problem. The results of this study could find applications in Engineering.

Key-Words / Index Term :
Controllability; Neutral fractional integro-differential systems; Schauder fixed point theorem; Fixed point; Arzela-Ascoli theorem

References :
[1] J-M , Coron, “Control and non-linearlity, Mathematical surveys and monographs”, ISSN 0076-5376, vol. 136, 2007.
[2] A.E. Bashirov, K.R. Kerimov, “On controllability conception for stochastic systems”, SIAM Journal on Control and Optimization, Vol. 35, Issue 2, pp. 384–398, 1997.
[3] A.E. Bashirov, N.I. Mahmudov, On concepts of controllability for deterministic and stochastic systems”, SIAM Journal on Control and Optimization, Vol. 37, Issue 6, pp. 1808–1821, 1999.
[4] M. Benchohra, A. Ouahab, “Controllability results for functional semilinear differential inclusions in Frechet spaces”, Nonlinear Analysis: Theory, Methods & Applications, Vol. 61, Issue 3, pp. 405–423, 2005.
[5] L. Gorniewicz, S. Ntouyas, D. O’Regan, “Controllability of semilinear differential equations and inclusions via semigroup theory in Banach spaces”, Reports on Mathematical Physics, Vol. 56, Issue 3, pp. 437–470, 2005.
[6] A. Babiarz, J. Klamka, M. Niezabitowski, “Schauder’s fixed–point theorem in approximate controllability problems”, Int. J. Appl. Math. Comput. Sci., Vol. 26, Issue 2, pp. 263–275, 2016.
[7] M. Nawaz, J. Wei, J. Sheng, A. Ullah, K. Niazi, L.C. Yang, “On the controllability of nonlinear fractional system with control delay”, Hacet. J. Math. Stat., pp. 1 – 9, 2018.
[8] P.C. Jackreece, C.A. Godspower, H.C. Ugorji, “Controllability of Fractional-order Dynamical System with Time Varying-Delays”, General Letters in Mathematics, Vol. 5, Issue 3, pp. 148 -156, 2018.
[9] K. Balachandran, S. Divya, “Controllability of Nonlinear Implicit Fractional Integrodifferential Systems”, Int. J. Appl. Math. Comput. Sci., Vol. 24, Issue 4, pp. 713–722, 2014.
[10] K. Mourad, E. Fateh, D. Baleanu, “Stochastic fractional perturbed control systems with fractional Brownian motion and Sobolev stochastic non-local conditions”, Collect. Math. Vol. 69, pp. 283–296, 2018.
[11] M. Caputo, “Linear model of dissipation whose Q is almost frequency independent, Part II”, Geophys. J. Royal Astronomical Soc., Vol. 13, Issue 5, pp. 529-539, 1967.
[12] W.E. Olmstead, R.A. Handelsaman, “Diffusion in a semi-infinite region with nonlinear surface dissipation”, SIAM Review, Vol. 18, Issue 2, pp. 275-291, 1976.
[13] J. Sabatier, O.P. Agarwal, J.A. Tenreiro Machado, (Eds.),. “Advances in fractional calculus: Theoretical Developments and Applications in Physics and Engineering”, Springer-Verlag, New York, NY, 2007.
[14] R.C. Mittal, R. Nigan, “Solution of fractional integrodifferential equations by Adomian Decomposition method”, Int. J. Appl. Math. Mech., Vol. 4, Issue 2, pp. 87-94, 2008.
[15] E.A. Rawashdeh, “Legendre wavelets method for fractional integrodifferential equations”, Appl. Math. Sci., Vol. 5, Issue 2, pp. 2467-2474, 2011.
[16] D.J.N. Xiao-Li, “Controllability of Nonlinear Fractional Delay Dynamical Systems with prescribed controls”, Nonlinear Analysis: Modelling and Control, Vol. 23, Issue 1, pp. 1-18, 2018.
[17] H. Zhang, J. Cao, W. Jiang, “Controllability criteria for Linear Fractional Differential Systems with State Delay and Impulse”, Journal of Applied Mathematics, (2013).
[18] J. Klamka, “On the Global Controllability of Perturbed nonlinear Systems”, IEEE Transactions on Automatic Control, Vol. 20, Issue 1, pp. 170-172, 1975.
[19] J. Klamka, “On the Local Controllability of Perturbed Nonlinear Control systems”, IEEE Transactions on Automatic control, Vol. 20, Issue 2, pp. 289-291, 1975.
[20] J. Klamka, “Constrained Controllability of Semilinear Systems with Delayed Controls”, Bulletin of the Polish Academy of Science: Technical Science, Vol. 56, Issue 4, pp. 333-337, 2008.
[21] J. Klamka, “Constrained Controllability of Dynamical systems”, International Journal of Applied Mathematics and Computer science, Vol. 9, Issue 2, pp. 231-244, 1999.
[22] K. Balachandran, J.P. Dauer, “Controllability of Nonlinear Systems via Fixed point Theorem”, Journal of Optimization Theory and Applications, Vol. 5, Issue 3, pp. 345- 352, 1987.
[23] K. Balachandran, P. Balasubramaniam, “A Note on Controllability of Nonlinear Volterra Integrodifferential Systems”, Kybernetika, Vol. 28, Issue 4, pp. 284-291, 1992.
[24] K. Balachandran, P. Balasubramaniam, “Controllability of Nonlinear Neutral Volterra Integrodifferential Systems”, Journal of Australian Mathematical Society, Vol. 36, Issue 1, pp. 107-116, 1994.
[25] K. Balachandran, “Controllability of Nonlinear Systems with Implicit Derivatives”, IMA Journal of Mathematical Control and Information, Vol. 5, Issue 2, pp. 77-83, 1988.
[26] C. Dacka, “On the Controllability of a Class of Nonlinear Systems”, IEEE Transactions on Automatic Control, Vol. 25, Issue 2, pp. 263-266, 1980.
[27] T.L. Guo, “Controllability and Observability of Impulsive Fractional Linear Time- Invariant Systems”, Computers and Mathematics with Applications, Vol. 64, Issue 10, pp. 3171-3182, 2012.
[28] R.E. Kalman, Y.C. Ho, K.S. Narendra, “Controllability of Linear Dynamical Systems”, Contributions to Differential Equations, Vol. 1, Issue 2, pp. 189-213, 1963.
[29] V. Govindaraj, R.K. George, “Controllability of fractional dynamical systems: A functional analytic approach”, Mathematical Control and Related Fields, Vol. 7, Issue 4, pp. 537–562.
[30] A. Dabbouchi, D. Baleanu, “Exact Null Controllability for Fractional Nonlinear Integrodifferential equations via Implicit Evolution systems”, Journal of Applied Mathematics, Article ID 931975, 1-17, 2012.
[31] N.I. Mahmudov, “Approximate Controllability of Fractional Sobolev-type Evolution Equation in Banach Spaces”, Abstract and Applied analysis, pp. 1-9, 2013.
[32] J. Schauder, “Der Fixpunktsatz in Functionalraumen”, Studia Math., Vol. 2, pp. 171-180, 1930.
[33] F.E. Browder, “A further generalization of the Schauder fixed point theorem”, Duke Math. J.. Vol. 32, pp. 575–578, 1965.
[34] F.E. Browder, “Asymptotic fixed point theorems”, Math. Ann. Vol. 185, pp. 38–60, 1970.
[35] D. Bugajewski, “Weak solutions of integral equations with weakly singular kernel in Banach spaces”, Comm. Math. Vol. 34, pp. 49–58, 1994.
[36] K. Deimling, “Nonlinear Functional Analysis”, Springer-Verlag, Berlin-Heidelberg, 1985.
[37] J. Dugundji, A. Granas, “Fixed Point Theory”, Polish Scientific Publishers, Warszawa, 1982.
[38] L. Górniewicz, D. Rozp?och-Nowakowska, “On the Schauder Fixed Point Theorem”, Topology in Nonlinear Analysis, Banach Center Publications, vol. 35, Polish Academy of Sciences, Warszawa, 1996.
[39] R.P. Agarwal, S. Arshad, D. O’Regan, V., Lupulescu, “A Schauder fixed point theorem in semilinear spaces and applications”, Fixed Point Theory Appl., Vol. 2013, Issue 306, 2013.
[40] J. Dauer, N., Mahmudov, “Approximate controllability of semilinear functional equations in Hilbert spaces”, Journal of Mathematical Analysis and Applications, Vol. 273, Issue 2, pp. 310–327, 2002.
[41] J. Dauer, N., Mahmudov, M., Matar, “Approximate controllability of backward stochastic evolution equations in Hilbert spaces”, Journal of Mathematical Analysis and Applications Vol. 323, Issue 1, pp. 42–56, 2006.
[42] L. Kexue, P. Jigen, “Laplace transform and fractional differential equations”, Applied Mathematics Letters, Vol. 24, Issue 12, pp. 2013–2019, 2011.
[43] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, “Theory and Applications of Fractional Differential Equations”, Elsevier, Amsterdam, 2006.
[44] K.S. Miller, B. Ross, “An Introduction to the Fractional Calculus and Fractional Differential Equations”, Wiley, New York, NY, 1993.
[45] K.B. Oldham, J. Spanier, “The Fractional Calculus”, Academic Press, London, 1974.
[46] I. Podlubny, “Fractional Differential Equations”, Academic Press, New York, NY, 1999.
[47] S.G. Samko, A.A., Kilbas, O.I., Marichev, “Fractional Integrals and Derivatives; Theory and Applications”, Gordon and Breach, Amsterdam, 1993.
[48] K. Kaczorek, “Selected Problems of Fractional Systems Theory”, Springer, Berlin, 2011.
[49] K. Balachandran, J. Kokila, “Constrained controllability of fractional dynamical systems”, Numerical Functional Analysis and Optimization, Vol. 34, Issue 11, pp. 1187–1205, 2016.