Advertisement

Journal of Dynamical and Control Systems

, Volume 25, Issue 1, pp 45–78 | Cite as

Pseudo Almost Periodic Solutions to Impulsive Non-autonomous Stochastic Differential Equations with Unbounded Delay and its Optimal Control

  • Zuomao YanEmail author
  • Fangxia Lu
Article
  • 95 Downloads

Abstract

This paper is concerned with the pseudo almost periodic in distribution mild solutions for impulsive non-autonomous stochastic differential equations with unbounded delay and optimal controls in Hilbert spaces. Firstly, a suitable pseudo almost periodic in distribution mild solutions is introduced. The existence of pseudo almost periodic in distribution mild solutions are proved by means of a fixed-point theorem for condensing maps combined with stochastic analysis theory and evolution family. Secondly, the existence of optimal pairs of system governed by impulsive non-autonomous stochastic differential equations is also presented. Finally, an example is given for demonstration.

Keywords

Impulsive non-autonomous stochastic differential equations Pseudo almost periodic in distribution functions Optimal control Unbounded delay Fixed point 

Mathematics Subject Classification (2010)

34A37 60H15 35B15 34H05 34F05 

Notes

Acknowledgments

The authors would like to thank the referee for his careful reading of the paper.

Funding information

This work is supported by the National Natural Science Foundation of China (11461019), the President Fund of Scientific Research Innovation and Application of Hexi University (xz2013-10), and the Scientific Research Fund of Young Teacher of Hexi University (QN2015-01).

References

  1. 1.
    Zhang C Y. Pseudo almost periodic solutions of some differential equations. J Math Anal Appl 1994;151:62–76.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Diagana T, Hernández E M. Existence and uniqueness of pseudo almost periodic solutions to some abstract partial neutral functional-differential equations and applications. J Math Anal Appl 2007;327:776–791.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Hernández E M, Henríquez H R. Pseudo almost periodic solutions for non-autonomous neutral differential equations with unbounded delay. Nonlinear Anal RWA 2008;9:430–437.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Zhang L, Xu Y. Existence of pseudo almost periodic solutions of functional differential equations with infinite delay. Appl Anal 2009;88:1713–1726.MathSciNetCrossRefGoogle Scholar
  5. 5.
    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 2012;53:073517.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Sakthivel R, Ren Y, Debbouche A, Mahmudov N I. Approximate controllability of fractional stochastic differential inclusions with nonlocal conditions. Appl Anal 2016;95:2361–2382.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gu Y, Ren Y, Sakthivel R. Square-mean pseudo almost automorphic mild solutions for stochastic evolution equations driven by G-Brownian motion. Stoch Anal Appl 2016;34:528–545.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Tudor C A, Tudor M. Pseudo almost periodic solutions of some stochastic differential equations. Math Rep (Bucur) 1999;1:305–314.MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chérif F. Quadratic-mean pseudo almost periodic solutions to some stochastic differential equations in a Hilbert space. J Appl Math Comput 2012;40:427–443.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bezandry P H, Diagana T. Almost periodic stochastic processes. New York Inc.: Springer-Verlag; 2011.CrossRefGoogle Scholar
  11. 11.
    Bezandry P H, Diagana T. P-th mean pseudo almost automorphic mild solutions to some nonautonomous stochastic differential equations. Afr Diaspora J Math 2011;1: 60–79.MathSciNetzbMATHGoogle Scholar
  12. 12.
    Yan Z, Zhang H. Existence of Stepanov-like square-mean pseudo almost periodic solutions to partial stochastic neutral differential equations. Ann Funct Anal 2015;6: 116–138.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Diop M A, Ezzinbi K, Mbaye M M. Existence and global attractiveness of a pseudo almost periodic solution in p-th mean sense for stochastic evolution equation driven by a fractional Brownian motion. Stochastics 2015;87:1061–1093.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mellah O, de Fitte P R. Counterexamples to mean square almost periodicity of the solutions of some SDEs with almost periodic coefficients. Electron J Diff Equa 2013; 2013:1–7.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Da Prato G, Tudor C. Periodic and almost periodic solutions for semilinear stochastic equations. Stoch Anal Appl 1995;13:13–33.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kamenskii M, Mellah O, de Fitte P R. Weak averaging of semilinear stochastic differential equations with almost periodic coefficients. J Math Anal Appl 2012;427: 336–364.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Bedouhene F, Challali N, Mellah O, de Fitte P R, Smaali M. Almost automorphy various extensions for stochastic processes. J Math Anal Appl 2015; 429:1113–1152.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Samoilenko A M, Perestyuk N A. Impulsive differential equations. Singapore: World Scientific; 1995.CrossRefGoogle Scholar
  19. 19.
    Stamov G T, Alzabut J O. Almost periodic solutions for abstract impulsive differential equations. Nonlinear Anal 2010;72:2457–2464.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Liu J, Zhang C. Composition of piecewise pseudo almost periodic functions and applications to abstract impulsive differential equations. Adv Differ Equ 2013;2013: 1–21.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Xia Z. Pseudo almost periodic mild solution of nonautonomous impulsive integro-differential equations. Mediterr J Math 2016;13:1065–1086.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Sakthivel R, Luo J. Asymptotic stability of impulsive stochastic partial differential equations with infinite delays. J Math Anal Appl 2009;356:1–6.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Hu L, Ren Y. Existence results for impulsive neutral stochastic functional integro-differential equations with infinite delays. Acta Appl Math 2010;111:303–317.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Yan Z, Yan X. Existence of solutions for impulsive partial stochastic neutral integrodifferential equations with state-dependent delay. Collect Math 2013;64:235–250.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ren Y, Sakthivel X J i a R. The p-th moment stability of solutions to impulsive stochastic differential equations driven by G-Brownian motion. Appl Anal 2017;96: 988–1003.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Yan Z, Lu F. Existence and exponential stability of pseudo almost periodic solutions for impulsive nonautonomous partial stochastic evolution equations. Adv Differ Equ 2016;2016:1–37.MathSciNetCrossRefGoogle Scholar
  27. 27.
    Mao X. Stochastic Differential Equations and Applications. Chichester: Horwood; 1997.zbMATHGoogle Scholar
  28. 28.
    Sakthivel R, Revathi P, Ren Y. Existence of solutions for nonlinear fractional stochastic differential equations. Nonlinear Anal 2013;81:70–86.MathSciNetCrossRefGoogle Scholar
  29. 29.
    Yan Z, Lu F. Existence results for a new class of fractional impulsive partial neutral stochastic integro-differential equations with infinite delay. J Appl Anal Comput 2015;5: 329–346.MathSciNetGoogle Scholar
  30. 30.
    Yan Z, Lu F. The optimal control of a new class of impulsive stochastic neutral evolution integro-differential equations with infinite delay. Internat J Control 2016;89: 1592–1612.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Yan Z, Jia X. On a fractional impulsive partial stochastic integro-differential equation with state-dependent delay and optimal controls. Stochastics 2016;88:1115–1146.MathSciNetCrossRefGoogle Scholar
  32. 32.
    Fuhrman B M, Tessitore G. Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces. Ann Probab 2004;32:607–660.MathSciNetCrossRefGoogle Scholar
  33. 33.
    Agram N, Øksendal B. Infinite horizon optimal control of forward-backward stochastic differential equations with delay. J Comput Appl Math 2014;259:336–349.MathSciNetCrossRefGoogle Scholar
  34. 34.
    Hale J K, Kato J. Phase spaces for retarded equations with infinite delay. Funkcial Ekvac 1978;21:11–41.MathSciNetzbMATHGoogle Scholar
  35. 35.
    Hino Y, Murakami S, Naito T. Functional-differential equations with infinite delay. Lecture notes in mathematics. Berlin: Springer-Verlag; 1991.Google Scholar
  36. 36.
    Acquistapace P. Evolution operators and strong solution of abstract linear parabolic equations. Diff Integral Equ 1988;1:433–457.MathSciNetzbMATHGoogle Scholar
  37. 37.
    Acquistapace P, Flandoli F, Terreni B. Initial boundary value problems and optimal control for nonautonomous parabolic systems. SIAM J Control Optim 1991;29: 89–118.MathSciNetCrossRefGoogle Scholar
  38. 38.
    Maniar L, Roland S. Almost periodicity of inhomogeneous parabolic evolution equations. Lecture notes in pure and applied mathematics. New York: Dekker; 2003. p. 299–318.Google Scholar
  39. 39.
    Sadovskii B N. On a fixed-point principle. Funct Anal Appl 1967;1:74–76.MathSciNetzbMATHGoogle Scholar
  40. 40.
    Ichikawa A. Stability of semilinear stochastic evolution equations. J Math Anal Appl 1982;90:12–44.MathSciNetCrossRefGoogle Scholar
  41. 41.
    Balder E. Necessary and sufficient conditions for L 1-strong-weak lower semicontinuity of integral functional. Nonlinear Anal RWA 1987;11:1399–1404.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHexi UniversityZhangyePeople’s Republic of China

Personalised recommendations