Journal of Applied and Pure Mathematics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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COMPLETE CONTROLLABILITY OF SEMILINEAR STOCHASTIC INTEGRO-DIFFERENTIAL EQUATIONS WITH INFINITE DELAY AND POISSON JUMPS

  • D.N., CHALISHAJAR (Department of Applied Mathematics, Mallory Hall, Virginia Military Institute) ;
  • A., ANGURAJ (Department of Mathematics, PSG College of Arts and Science) ;
  • K., RAVIKUMAR (Department of Mathematics, PSG College of Arts and Science) ;
  • K., MALAR (Department of Mathematics, Erode Arts and Science College)
  • Received : 2021.06.24
  • Accepted : 2021.09.28
  • Published : 2022.11.30
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Abstract

This manuscript deals with the exact (complete) controllability of semilinear stochastic differential equations with infinite delay and Poisson jumps utilizing some basic and readily verified conditions. The results are obtained by using fixed-point approach and by using advance phase space definition for infinite delay part. We have used the axiomatic definition of the phase space in terms of stochastic process to consider the time delay of the system. An infinite delay along with the Poisson jump is the new investigation for the given stochastic system. An example is given to illustrate the effectiveness of the results.

Keywords

Acknowledgement

The authors wish to thank the anonymous reviewers for their valuable comments and suggestions.

References

  1. A. Anguraj and K. Ramkumar, Approximate controllability of semilinear stochastic interodifferential system with nonlocal conditions, Fractal Fract 2 (2018), 29.
  2. S. Barnett, Introduction to mathematical control theory, Clarendon Press, Oxford, 1975.
  3. K. Balachandran and J.P. Dauer, Controllability of nonlinear systems in Banach spaces, Journal of Optimization Theory and Applications 115 (2002), 7-28. https://doi.org/10.1023/A:1019668728098
  4. A.E. Bashirov and K.R. Kerimov, On controllability conception for stochastic systems, SIAM journal on control and optimization 35 (1997), 384-398. https://doi.org/10.1137/S0363012994260970
  5. D.N. Chalishajar, Controllability of second order impulsive neutral functional differential inclusions with infinite delay, Journal of Optimization Theory and Applications 154 (2012), 672-684. https://doi.org/10.1007/s10957-012-0025-6
  6. D.N. Chalishajar, K. Malar and K. Karthikeyan, Approximate controllability of abstract impulsive fractional neutral evolution equations with infinite delay in Banach spaces, Electron. J. Differ. Equ. 275 (2013), 1-21.
  7. R.F. Curtain and H. Zwart, An introduction to infinite-dimensional linear systems theory, Springer, New York, 1995.
  8. M.A. Diop, K. Ezzinbi and M.M. Zene, Existence and stability results for a partial impulsive stochastic integro-differential equation with infinite delay, SeMA Journal 73 (2016), 17-30. https://doi.org/10.1007/s40324-015-0053-x
  9. G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, 1992.
  10. R.C. Grimmer, Resolvent operator for integral equations in Banach space, Transactions of the American Mathematical Society 273 (1982), 333-349. https://doi.org/10.1090/S0002-9947-1982-0664046-4
  11. J.K. Hale and J. Kato, Phase Space for Retarded Equations with Infinite Delay, Funkcial. Ekvac. 21 (1978), 11-41.
  12. Y. Hino, S. Murakami and T. Naito, Functional differential equations with infinite delay, Lecture Notes in Mathematics, Springer, New York, 1473, 1991.
  13. R.E. Kalman, Y.C. Ho, and K.S. Narendra, Controllability of linear systems, Contributions to Differential Equations 1 (1963), 189-213.
  14. J. Klamka, Stochastic controllability of linear systems with state delays, International Journal of Applied Mathematics and Computer Science 17 (2007), 5-13.
  15. J. Klamka, Stochastic controllability of linear systems with state delays, International Journal of Applied Mathematics and Computer Science 17 (2007), 5.
  16. J. Klamka and L. Socha, Some remarks about stochastic controllability, IEEE Transactions on Automatic Control 22 (1977), 880-881. https://doi.org/10.1109/TAC.1977.1101615
  17. J. Klamka and L. Socha, Some remarks about stochastic controllability for delayed linear systems, International Journal of Control 32 (1980), 561-566. https://doi.org/10.1080/00207178008922872
  18. H. Khan, A. Khan, F. Jarad and A. Shah, Existence and data dependence theorems for solutions of an ABC-fractional order impulsive system, Chaos, Solitons & Fractals 131 (2020), 109477. DOI: 10.1016/j.chaos.2019.109477
  19. H. Long, J. Hu and Y. Li, Approximate controllability of stochastic PDE with infinite delays driven by Possion jumps, In 2012 IEEE International Conference on Information Science and Technology, 2012, 194-199.
  20. N.I. Mahumudov, Controllability of linear stochastic systems, IEEE Trans. Automatic Control 46 (2001), 724-731.
  21. N.I. Mahumudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM journal on control and optimization 42 (2003), 1604-1622. https://doi.org/10.1137/S0363012901391688
  22. N.I. Mahmudov and S. Zorlu, Controllability of non-linear stochastic systems, International Journal of Control 76 (2003), 95-104. https://doi.org/10.1080/0020717031000065648
  23. N.I. Mahmudov, On controllability of semilinear stochastic systems in Hilbert spaces, IMA Journal of Mathematical Control and Information 19 (2002), 363-376. https://doi.org/10.1093/imamci/19.4.363
  24. N.I. Mahmudov, Controllability of linear stochastic systems in Hilbert spaces, Journal of mathematical Analysis and Applications 259 (2001), 64-82. https://doi.org/10.1006/jmaa.2000.7386
  25. N.I. Mahmudov and N. Semi, Approximate controllability of semilinear control systems in Hilbert spaces, TWMS J. App. Eng. Math. 2 (2012), 67-74.
  26. N.I. Mahmudov and A. Denker, On controllability of linear stochastic systems, International Journal of Control 73 (2000), 144-151. https://doi.org/10.1080/002071700219849
  27. P. Muthukumar and C. Rajivganthi, Approximate controllability of fractional order stochastic variational inequalities driven by Poisson jumps, Taiwanese Journal of Mathematics 18 (2014), 1721-1738. https://doi.org/10.11650/tjm.18.2014.3885
  28. K. Naito, Controllability of semilinear control systems dominated by the linear part, SIAM Journal on control and Optimization 25 (1987), 715-722. https://doi.org/10.1137/0325040
  29. C. Rajivganthi, K. Thiagu, P. Muthukumar and P. Balasubramaniam, Existence of solutions and approximate controllability of impulsive fractional stochastic differential systems with infinite delay and Poisson jumps, Applications of Mathematics 60 (2015), 395-419. https://doi.org/10.1007/s10492-015-0103-9
  30. R. Sakthivel and Y. Ren, Complete controllability of stochastic evolution equations with jumps, Reports on Mathematical Physics 68 (2011), 163-174. https://doi.org/10.1016/S0034-4877(12)60003-2
  31. A. Shukla, N. Sukavanam and D.N. Pandey, Complete controllability of semi-linear stochastic system with delay, Rend. Circ. Mat. Palermo. 64 (2015), 209-220. https://doi.org/10.1007/s12215-015-0191-0
  32. L. Shen and J. Sun, Relative controllability of stochastic nonlinear systems with delay in control, Nonlinear Analysis: Real World Applications 13 (2012), 2880-2887. https://doi.org/10.1016/j.nonrwa.2012.04.017
  33. R. Sakthivel, J.H. Kim and N.I. Mahumudov, On controllability of nonlinear stochastic systems, Reports on Mathematical Physics 58 (2006), 433-443. https://doi.org/10.1016/S0034-4877(06)80963-8
  34. N. Sukavanam and M. Kumar, S-controllability of an abstract first order semilinear control system, Numerical Functional Analysis and Optimization 31 (2010), 1023-1034. https://doi.org/10.1080/01630563.2010.498598