Skip Navigation Links
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu


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) 291--304 | DOI:10.5890/JAND.2019.06.011

R. Nirmalkumar, R. Murugesu

Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore-641 020, Tamilnadu, India

Download Full Text PDF

 

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 Krasnoselskii-Schaefer-type fixed point theorem and stochastic analysis theory. An application involving nonlinear differential equation with unbounded delay is addressed.

References

  1. [1]  Caraballo, T. (1990), Asymptotic exponential stability of stochastic partial differential equations with delay, Stochastics, 33, 27-47.
  2. [2]  Da Prato, G. and Zabczyk, J. (1992), Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge.
  3. [3]  Mao, X. (1997), Stochastic Differential equations and Applications, Horwood, Chichester.
  4. [4]  Oksendal, B. (2000), Stochastic Differential Equations, An Introduction with Applications, Springer-Verlag.
  5. [5]  Sobczyk, K. (1991), Stochastic Differential Equations with Applications to Physics and Engineering, Kluwer Academic, London.
  6. [6]  Hale, J.K. and Lunel, S.M.V. (1993), Introduction to Functional-Differential Equations, in: Applied Mathematical Sciences, Vol. 99, Springer-Verlag, New York.
  7. [7]  Kolmanovskii, V.B. and Myshkis, A. (1992), Applied theory of Functional Differential Equations, Kluwer Academic, Norwell.
  8. [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), 309-319.
  9. [9]  Balasubramaniam, P. and Tamilalagan, P. (2015), Approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay by using Mainardi’s function, Applied Mathematics and Computation, 256, 232-246.
  10. [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, 373-394.
  11. [11]  Mahmudov, N.I. (2001), Controllability of linear stochastic systems in Hilbert spaces, Journal of Mathematical Analysis Applications, 259, 64-82.
  12. [12]  Mahmudov, N.I. and Denker, A. (2000, On controllability of linear stochastic systems, Int. J. Control, 73, 144-151.
  13. [13]  Ren, Y. and Sun, D.D. (2010), Second-order neutral stochastic evolution equations with infinite delay under Caratheodory conditions, J. Optim. Theory Appl, 147, 569-582.
  14. [14]  Revathi, P., Sakthivel, R., and Ren, Y. (2016), Stochastic functional differential equations of Sobolev-type with infinite delay, Statistics and Probability Letters, 109, 68-77.
  15. [15]  Sakthivel, R, Mahmudov, N.I., and Lee, S.G. (2009), Controllability of non-linear impulsive stochastic systems, Int. J. Control, 82, 801-807.
  16. [16]  Anguraj, A., Kanjanadevi, S., and Trujillo, J.J. (2017), Existence of mild solution of abstract fractional differential equations with fractional non-instantaneous impulsive conditions, Discontinuity, Nonlinearlity and Complexity, 6(2), 173-183.
  17. [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), 672-684.
  18. [18]  Chang, Y.K. (2007), Controllability of impulsive functional differential systems with infinite delay in Banach space, Chaos Solitons Fractals, 33, 1601-1609.
  19. [19]  Fen, F.T. and Karaca, I.Y. (2015), Nonlinear four-point impulsive fractional differential equations with p-Laplacian operator, Discontinuity, Nonlinearity, and Complexity, 4(4), 467-486.
  20. [20]  Shen, L.J. and Sun, J.T. (2012), Approximate controllability of stochastic impulsive functional systems with infinite delay, Automatica, 48, 2705-2709.
  21. [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), 29-51.
  22. [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), 11-24.
  23. [23]  Vijayakumar, V., Ravichandran, C., and Murugesu, R. (2013), Approximate controllability for a class of fractional neutral integro-differential inclusions with state-dependent delay, Nonlinear Studies, 20(4), 511-530.
  24. [24]  Vijayakumar, V., Selvakumar, A., and Murugesu, R. (2014), Controllability for a class of fractional neutral integro-differential equations with unbounded delay, Applied Mathematics and Computation, 232, 303-312.
  25. [25]  Joice Nirmala, R. and Balachandran, K. (2016), Controllability of Nonlinear Fractional Delay Integrodifferential Systems, Discontinuity, Nonlinearity, and Complexity, 5(1), 59-73
  26. [26]  Mahmudov, N.I., Vijayakumar, V., and Murugesu, R. (2016) , Approximate controllability of second-order evolution differential inclusions in Hilbert spaces, Mediterranean Journal of Mathematics, 13(5), 3433-3454.
  27. [27]  Vijayakumar, V. (2016), Approximate controllability results for abstract neutral integro-differential inclusions with infinite delay in Hilbert spaces, IMA Journal of Mathematical Control and Information, 1-18. doi: 10.1093/imamci/dnw049.
  28. [28]  Zhou, Y., Vijayakumar, V., and Murugesu, R. (2015), Controllability for fractional evolution inclusions without compactness, Evolution Equations and Control Theory, 4(4), 507-524.
  29. [29]  Agarwal, S. and Bahuguna, D. (2006), Existence of solutions to Sobolev-type partial neutral differential equations, Journal of Applied Mathematics and Stochastic Analysis, 1-10. Article ID 16308.
  30. [30]  Fattorini, H.O., (1985), In Second Order Linear Differential Equations in Banach Spaces, North-Holland Mathematics Studies, vol 108, North-Holland, Amsterdam.
  31. [31]  Travis, C.C. and Webb, G.F. (1978), Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hung , 32, 76-96.
  32. [32]  Burton, T, A. and Kirk, C. (1998), A fixed point theorem of Kranoselskii-Schaefer type, Math.Nachr, 189, 23-31.
  33. [33]  Wang, J., Feckan, M. and Zhou, Y. (2014), Controllability of Sobolev type fractional evolution systems, Dynamics of Partial Differential Equations, 11(1), 71-87.