Discontinuity, Nonlinearity, and Complexity
Boundary Controllability of Fractional Order Nonlocal Semi-linear Neutral Evolution Systems with Impulsive Condition
Discontinuity, Nonlinearity, and Complexity 8(4) (2019) 419--428 | DOI:10.5890/DNC.2019.12.006
Kamalendra Kumar$^{1}$, Rakesh Kumar$^{2}$
$^{1}$ Department of Mathematics, SRMS College of Engineering & Technology, Bareilly-243001, India
$^{2}$ Department of Mathematics, Hindu College, Moradabad-244001, India
Download Full Text PDF
Abstract
Sufficient conditions of boundary controllability of fractional nonlocal semi-linear neutral evolution equations with impulsive conditions are demonstrated. To get the result, we employ fixed point theorem and strongly continuous semi-group theory. An application is discussed to explain the theory.
References
-
[1]  | Balachandran, K. and Anandhi, E.R. (2003), Neutral functional integrodifferential control systems in Banach spaces, Ky-bernetika, 39, 359-367. |
-
[2]  | Fu, X.L. (2003), Controllability of neutral functional differential systems in abstract space, Appl. Math Comput., 141, 281-296. |
-
[3]  | Balakrishnan, A.V. (1976), Applied Functional Analysis, Springer-Verlag, New York. |
-
[4]  | Washburn, D. (1979), A bound on the boundary input map for parabolic equations with application to time optimal control, SIAM J Control Optim., 17, 652-671. |
-
[5]  | Fattorini, H.O. (1968), Boundary control systems, SIAM J Control Optim., 6, 349-384. |
-
[6]  | Lasiecka, T. (1978), Boundary Control of Parabolic Systems; Regularity of Solutions, Appl. Math Optim., 4, 301-327. |
-
[7]  | Balachandran, K. and Anandhi, E.R. (2004), Boundary controllability of neutral integrodifferential systems in Banach spaces, Nihonkai Math. J., 15, 1-13. |
-
[8]  | Balachandran, K. and Leelamani, A. (2006), A note on boundary controllability of neutral integrodifferential systems in Banach spaces, Nihonkai Math. J., 17, 89-101. |
-
[9]  | Park, J.Y. and Jeong, J.U. (2011), Boundary controllability of semilinear neutral evolution systems, Bull Korean Math Soc., 48(4), 705-712. |
-
[10]  | Oustaloup, A. (1995), La Derivation Non Entiere: Theorie, Syntheseet Applications, Hermes, Paris. |
-
[11]  | Lakshmikantham, V., Bainov, D.D., and Simeonov, P.S. (1989), Theory of impulsive differential equations, Series in Modern Applied Mathematics,World Scientific Publishing, New Jersey. |
-
[12]  | Perestyuk, N.A., Plotnikov, V.A., Samoilenko, A.M., and Skripnik, N.V. (2011), Differential equation with impulse effects: Multivalued Right-hand Side with discontinuities, De Gruyter Studies in Mathematics 40, Germany. |
-
[13]  | Ahmed, H.M. (2010), Boundary controllability of nonlinear fractional integrodifferential systems, Adv Differ Equat., 2010, Article ID 279493, 9 pages. |
-
[14]  | Ahmed, H.M. (2016), Boundary controllability of impulsive nonlinear fractional delay integrodifferential system, Cogent Engineering, 3(1), 1-8. |
-
[15]  | Kumar, K. and Kumar, R. (2017), Boundary controllability of delay differential systems of fractional order with nonlocal condition, Journal of Applied Nonlinear Dynamics, 6(4), 465-472. |
-
[16]  | Zhou, Y. and Jiao, F. (2010), Existence of mild solutions for fractional neutral evolution equations, Computers and Mathematics with Applications, 59, 1063-1077. |
-
[17]  | Pazy, A. (1983), Semi-groups of linear operators and applications to partial differential equations, Springer, New York. |
-
[18]  | Podlubny, I. (1999), Fractional Differential Equations, Academic Press, San Diego. |
-
[19]  | Samko, S., Kilbas, A., andMarichev, O.L. (1993), Fractional integrals and derivatives, Gordon and Breach, New York. |
-
[20]  | Miller, K.S. and Ross, B. (1993), An introduction to the fractional calculus and fractional differential equations,Wiley, New York. |
-
[21]  | El-Borai,M.M. (2002), Some probability densities and fundamental solutions of fractional evolution equations, Chaos, Soli-tons and Fracts, 14(3), 433-440. |
-
[22]  | El-Borai, M.M. (2006), On some stochastic fractional integrodifferential equations, Advances in Dynamical Systems and Applications, 1(1), 49-57. |