Applied Mathematics & Optimization

, Volume 77, Issue 3, pp 443–462 | Cite as

The Solvability and Optimal Controls for Fractional Stochastic Differential Equations Driven by Poisson Jumps Via Resolvent Operators

  • P. Tamilalagan
  • P. Balasubramaniam


In this manuscript, we investigate the solvability and optimal controls for fractional stochastic differential equations driven by Poisson jumps in Hilbert space by using analytic resolvent operators. Sufficient conditions are derived to prove that the system has a unique mild solution by using the classical Banach contraction mapping principle. Then, the existence of optimal control for the corresponding Lagrange optimal control problem is investigated. Finally, the derived theoretical result is validated by an illustrative example.


Contraction mapping principle Fractional calculus Optimal controls Poisson jumps Solvability 

Mathematics Subject Classification

26A33 34A12 34A08 34K50 47H10 



This work is supported by Council of Scientific and Industrial Research, Extramural Research Division, Pusa, New Delhi, India under the Grant No. 25(0217)/13/EMR-II. The authors would like to express their deep gratitude to the Editor-in-Chief, associate editor and the anonymous referees for their careful reading and valuable suggestions to improve the quality of this manuscript.


  1. 1.
    Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, 323–337 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agrawal, O.P., Defterli, O., Baleanu, D.: Fractional optimal control problems with several state and control variables. J. Vib. Control 16, 1967–1976 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Al-Hussein, A.: Necessary conditions for optimal control of stochastic evolution equations in Hilbert spaces. Appl. Math. Optim. 63, 385–400 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Applebaum, D.: Levy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bajlekova, E.G.: Fractional evolution equations in Banach spaces. Ph.D Thesis, Univ. Press Facilities, Eindhoven University of Technology (2001)Google Scholar
  6. 6.
    Balasubramaniam, P., Ntouyas, S.K.: Controllability for neutral stochastic functional differential inclusions with infinite delay in abstract space. J. Math. Anal. Appl. 324, 161–176 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Balasubramaniam, P., Tamilalagan, P.: The solvability and optimal controls for impulsive fractional stochastic integro-differential equations via resolvent operators. J. Optim. Theory Appl. (2016). doi: 10.1007/s10957-016-0865-6 zbMATHGoogle Scholar
  8. 8.
    Balasubramaniam, P., Tamilalagan, P.: Approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay by using Mainardi’s function. Appl. Math. Comput. 256, 232–246 (2015)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Balder, E.J.: Necessary and sufficient conditions for \(L_1\)-strong weak lower semicontinuity of integral functionals. Nonlinear Anal. 11, 1399–1404 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Belmekki, M., Mekhalfi, K., Ntouyas, S.K.: Existence and uniqueness for semilinear fractional differential equations with infinite delay via resolvent operators. J. Frac. Calc. Appl. 4, 267–282 (2013)Google Scholar
  11. 11.
    Dabas, J., Chauhan, A.: Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equation with infinite delay. Math. Comput. Model. 57, 754–763 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fan, Z., Mophou, G.: Existence of optimal controls for a semilinear composite fractional relaxation equation. Rep. Math. Phys. 73, 311–323 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hernandez, E., O’Regan, D., Balachandran, K.: On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal. 73, 3462–3471 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hernandez, E., O’Regan, D., Balachandran, K.: Existence results for abstract fractional differential equations with nonlocal conditions via resolvent operators. Indag. Math. 24, 68–82 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hino, Y., Naito, T., Murakami, S.: Functional Differential Equations with Infinite Delay. Lecture Notes in Mathematics, vol. 1473. Springer, Berlin (1991)Google Scholar
  16. 16.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  17. 17.
    Li, K., Peng, J., Gao, J.: On inhomogeneous fractional differential equations in Banach spaces. Numer. Funct. Anal. Optim. 34, 415–429 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Liu, X., Liu, Z., Han, J.: The solvability and optimal Controls for some fractional impulsive equation. Abstr. Appl. Anal. 2013 (2013) Article ID 914592Google Scholar
  19. 19.
    Luo, J., Taniguchi, T.: The existence and uniqueness for non-Lipschitz stochastic neutral delay evolution equations driven by Poisson jumps. Stoch. Dyn. 9, 135–152 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mao, X.: Stochastic Differential Equations and Applications. Horwood, Chichester (1997)zbMATHGoogle Scholar
  21. 21.
    Mokkedem, F.Z., Fu, X.: Approximate controllability for a semilinear stochastic evolution system with infinite delay in \(L_p\) space. Appl. Math. Optim. (2016). doi: 10.1007/s00245-016-9332-x zbMATHGoogle Scholar
  22. 22.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983)CrossRefzbMATHGoogle Scholar
  23. 23.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1998)zbMATHGoogle Scholar
  24. 24.
    Pruss, J.: Evolutionary Integral Equations and Applications. Monographs in Mathematics, 87, Birkhauser Verlag, Basel (1993)Google Scholar
  25. 25.
    Ren, Y., Chen, L.: A note on the neutral stochastic functional differential equation with infinite delay and Poisson jumps in an abstract space. J. Math. Phys. 50, 082704 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ren, Y., Sakthivel, R.: Existence, uniqueness, and stability of mild solutions for second-order neutral stochastic evolution equations with infinite delay and Poisson jumps. J. Math. Phys. 53, 073517 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ren, Y., Zhou, Q., Chen, L.: Existence, uniqueness and stability of mild solutions for time-dependent stochastic evolution equations with Poisson jumps and infinite delay. J. Optim. Theory Appl. 149, 315–331 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sakthivel, R., Revathi, P., Ren, Y.: Existence of solutions for nonlinear fractional stochastic differential equations. Nonlinear Anal. 81, 70–86 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Shukla, A., Arora, U., Sukavanam, N.: Approximate controllability of retarded semilinear stochastic system with non local conditions. J. Appl. Math. Comput. 49, 513–527 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wang, J., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal. Real World Appl. 12, 262–272 (2011)Google Scholar
  31. 31.
    Wang, J., Zhou, Y., Medved, M.: On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay. J. Optim. Theory Appl. 152, 31–50 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Wang, J., Wei, W., Zhou, Y.: Fractional finite time delay evolution systems and optimal controls in infinite-dimensional spaces. J. Dyn. Control Syst. 17, 515–535 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsThe Gandhigram Rural Institute - Deemed UniversityGandhigramIndia

Personalised recommendations