Journal of Applied Nonlinear Dynamics
Approximate Controllability Results for Second Order Neutral Impulsive Stochastic Evolution Equations of Sobolev Type with Unbounded Delay
Journal of Environmental Accounting and Management 8(2) (2019) 291304  DOI:10.5890/JAND.2019.06.011
R. Nirmalkumar, R. Murugesu
Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore641 020, Tamilnadu, India
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Abstract
In this paper, we discuss the approximate controllability of the second order neutral impulsive stochastic evolution equations of Sobolev type with unbounded delay in Hilbert Spaces. A set of sufficient conditions are established for the existence and approximate controllability of the mild solutions using KrasnoselskiiSchaefertype fixed point theorem and stochastic analysis theory. An application involving nonlinear differential equation with unbounded delay is addressed.
References

[1]  Caraballo, T. (1990), Asymptotic exponential stability of stochastic partial differential equations with delay, Stochastics, 33, 2747. 

[2]  Da Prato, G. and Zabczyk, J. (1992), Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge. 

[3]  Mao, X. (1997), Stochastic Differential equations and Applications, Horwood, Chichester. 

[4]  Oksendal, B. (2000), Stochastic Differential Equations, An Introduction with Applications, SpringerVerlag. 

[5]  Sobczyk, K. (1991), Stochastic Differential Equations with Applications to Physics and Engineering, Kluwer Academic, London. 

[6]  Hale, J.K. and Lunel, S.M.V. (1993), Introduction to FunctionalDifferential Equations, in: Applied Mathematical Sciences, Vol. 99, SpringerVerlag, New York. 

[7]  Kolmanovskii, V.B. and Myshkis, A. (1992), Applied theory of Functional Differential Equations, Kluwer Academic, Norwell. 

[8]  Balachandran, K., Kiruthika, S., Rivero, M., and Trujillo, J.J. (2012), Existence of Solutions for Fractional Delay Integrodifferential Equations, Journal of Applied Nonlinear Dynamics, 1(4), 309319. 

[9]  Balasubramaniam, P. and Tamilalagan, P. (2015), Approximate controllability of a class of fractional neutral stochastic integrodifferential inclusions with infinite delay by using Mainardi’s function, Applied Mathematics and Computation, 256, 232246. 

[10]  Dauer, J.P. and Mahmudov, N.I. (2004), Controllability of stochastic semilinear functional differential equations in Hilbert spaces, Journal of Mathematical Analysis Applications, 290, 373394. 

[11]  Mahmudov, N.I. (2001), Controllability of linear stochastic systems in Hilbert spaces, Journal of Mathematical Analysis Applications, 259, 6482. 

[12]  Mahmudov, N.I. and Denker, A. (2000, On controllability of linear stochastic systems, Int. J. Control, 73, 144151. 

[13]  Ren, Y. and Sun, D.D. (2010), Secondorder neutral stochastic evolution equations with infinite delay under Caratheodory conditions, J. Optim. Theory Appl, 147, 569582. 

[14]  Revathi, P., Sakthivel, R., and Ren, Y. (2016), Stochastic functional differential equations of Sobolevtype with infinite delay, Statistics and Probability Letters, 109, 6877. 

[15]  Sakthivel, R, Mahmudov, N.I., and Lee, S.G. (2009), Controllability of nonlinear impulsive stochastic systems, Int. J. Control, 82, 801807. 

[16]  Anguraj, A., Kanjanadevi, S., and Trujillo, J.J. (2017), Existence of mild solution of abstract fractional differential equations with fractional noninstantaneous impulsive conditions, Discontinuity, Nonlinearlity and Complexity, 6(2), 173183. 

[17]  Chalishajar, D.N. (2012), Controllability of second order impulsive neutral functional differential inclusions with infinite delay, Journal of Optimization Theory and Applications, 154(2), 672684. 

[18]  Chang, Y.K. (2007), Controllability of impulsive functional differential systems with infinite delay in Banach space, Chaos Solitons Fractals, 33, 16011609. 

[19]  Fen, F.T. and Karaca, I.Y. (2015), Nonlinear fourpoint impulsive fractional differential equations with pLaplacian operator, Discontinuity, Nonlinearity, and Complexity, 4(4), 467486. 

[20]  Shen, L.J. and Sun, J.T. (2012), Approximate controllability of stochastic impulsive functional systems with infinite delay, Automatica, 48, 27052709. 

[21]  Vijayakumar, V., Murugesu, R., Poongodi, R., and Dhanalakshmi, S. (2017), Controllability of second order impulsive nonlocal Cauchy problem via measure of noncompactness, Mediterranean Journal of Mathematics, 14(1), 2951. 

[22]  Chang, Y.K. and Li, W.T. (2006), Controllability of Sobolev type semilinear functional differential and integrodifferential inclusions with an unbounded delay, Georgian Mathematical Journal, 13(1), 1124. 

[23]  Vijayakumar, V., Ravichandran, C., and Murugesu, R. (2013), Approximate controllability for a class of fractional neutral integrodifferential inclusions with statedependent delay, Nonlinear Studies, 20(4), 511530. 

[24]  Vijayakumar, V., Selvakumar, A., and Murugesu, R. (2014), Controllability for a class of fractional neutral integrodifferential equations with unbounded delay, Applied Mathematics and Computation, 232, 303312. 

[25]  Joice Nirmala, R. and Balachandran, K. (2016), Controllability of Nonlinear Fractional Delay Integrodifferential Systems, Discontinuity, Nonlinearity, and Complexity, 5(1), 5973 

[26]  Mahmudov, N.I., Vijayakumar, V., and Murugesu, R. (2016) , Approximate controllability of secondorder evolution differential inclusions in Hilbert spaces, Mediterranean Journal of Mathematics, 13(5), 34333454. 

[27]  Vijayakumar, V. (2016), Approximate controllability results for abstract neutral integrodifferential inclusions with infinite delay in Hilbert spaces, IMA Journal of Mathematical Control and Information, 118. doi: 10.1093/imamci/dnw049. 

[28]  Zhou, Y., Vijayakumar, V., and Murugesu, R. (2015), Controllability for fractional evolution inclusions without compactness, Evolution Equations and Control Theory, 4(4), 507524. 

[29]  Agarwal, S. and Bahuguna, D. (2006), Existence of solutions to Sobolevtype partial neutral differential equations, Journal of Applied Mathematics and Stochastic Analysis, 110. Article ID 16308. 

[30]  Fattorini, H.O., (1985), In Second Order Linear Differential Equations in Banach Spaces, NorthHolland Mathematics Studies, vol 108, NorthHolland, Amsterdam. 

[31]  Travis, C.C. and Webb, G.F. (1978), Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hung , 32, 7696. 

[32]  Burton, T, A. and Kirk, C. (1998), A fixed point theorem of KranoselskiiSchaefer type, Math.Nachr, 189, 2331. 

[33]  Wang, J., Feckan, M. and Zhou, Y. (2014), Controllability of Sobolev type fractional evolution systems, Dynamics of Partial Differential Equations, 11(1), 7187. 