Controllability of non-autonomous nonlinear differential system with non-instantaneous impulses

  • Muslim Malik
  • Rajesh Dhayal
  • Syed AbbasEmail author
  • Avadhesh Kumar
Original Paper


In this paper, we applied the Rothe’s fixed point theorem to study the controllability of non-autonomous nonlinear differential system with non-instantaneous impulses in the space \(\mathbb {R}^{n}\). Also, we established the sufficient conditions for the controllability of the integro-differential equation as well as nonlocal problem. Finally, we have given an example to illustrate the application of these proposed results.


Controllability Non-autonomous differential system Non-instantaneous impulses Rothe’s fixed point theorem 

Mathematics Subject Classification

93B05 34K45 34G20 



We are thankful to the anonymous reviewers for their constructive comments and suggestions which help us to improve the manuscript.


  1. 1.
    Hernández, E., O’Regan, D.: On a new class of abstract impulsive differential equations. Proc. Am. Math. Soc. 141(5), 1641–1649 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Wang, J., Fečkan, M.: A general class of impulsive evolution equations. Topol. Methods Nonlinear Anal. 46(2), 915–933 (2015)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Muslim, M., Kumar, A., Fečkan, M.: Existence, uniqueness and stability of solutions to second order nonlinear differential equations with non-instantaneous impulses. J. King Saud Univ-Sci (2016). doi: 10.1016/j.jksus.2016.11.005
  4. 4.
    Sood, A., Srivastava, S.K.: On stability of differential systems with noninstantaneous impulses. Math. Probl. Eng. (2015). doi: 10.1155/2015/691687
  5. 5.
    Pierri, M., O’Regan, D., Rolnik, V.: Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses. Appl. Math. Comput. 219(12), 6743–6749 (2013)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Acharya, F.S.: Controllability of second order semilinear impulsive partial neutral functional differential equations with infinite delay. Int. J. Math. Sci. Appl. 3(1), 207–218 (2013)Google Scholar
  7. 7.
    Sakthivel, R., Mahmudov, N.I., Kim, J.H.: On controllability of second order nonlinear impulsive differential systems. Nonlinear Anal. 71, 45–52 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Shukla, A., Sukavanam, N., Pandey, D.N.: Approximate controllability of fractional semilinear stochastic system of order \(\alpha \in (1, 2]\). Nonlinear Stud. 22(1), 131–138 (2015)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Vidyasagar, M.: A controllability condition for nonlinear systems. IEEE Trans. Autom. Control 17(4), 569–570 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Leiva, H.: Rothe’s fixed point theorem and controllability of semilinear non-autonomous systems. Syst. Control Lett. 67, 14–18 (2014)CrossRefzbMATHGoogle Scholar
  11. 11.
    Leiva, H.: Controllability of semilinear impulsive non-autonomous systems. Int. J. Control 88(3), 585–592 (2015)CrossRefzbMATHGoogle Scholar
  12. 12.
    Mahmudov, N.I., Mckibben, M.A.: Approximate controllability of second order neutral stochastic evolution equations. Dyn. Contin. Discret. Impuls. Syst. Ser. B 13(5), 619–634 (2006)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Curtain, R.F., Zwart, H.: An introduction to infinite dimensional linear systems theory, vol. 21. Springer, Berlin (2012)Google Scholar
  14. 14.
    Iturriaga, E., Leiva, H.: A characterization of semilinear surjective operators and applications to control problems. Appl. Math. 1(4), 265–273 (2010)CrossRefGoogle Scholar
  15. 15.
    Isac, G.: On Rothe’s fixed point theorem in general topological vector space. Analele Stiintifice ale Universitatii Ovidius Constanta 12(2), 127–134 (2004)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162(2), 494–505 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Balachandran, K., Chandrasekaran, M.: Existence of solutions of a delay differential equation with nonlocal condition. Indian J. Pure Appl. Math. 27, 443–450 (1996)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Ezzinbi, K., Xianlong, Fu, Hilal, K.: Existence and regularity in the \(\alpha \)-norm for some neutral partial differential equations with nonlocal conditions. Nonlinear Anal.: Theory, Methods Appl. 67(5), 1613–1622 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sontag, E.D.: Mathematical control theory: deterministic finite dimensional systems, vol. 6. Springer, Berlin (2013)Google Scholar

Copyright information

© Springer-Verlag Italia S.r.l. 2017

Authors and Affiliations

  • Muslim Malik
    • 1
  • Rajesh Dhayal
    • 1
  • Syed Abbas
    • 1
    Email author
  • Avadhesh Kumar
    • 1
  1. 1.School of Basic SciencesIndian Institute of Technology MandiKamandIndia

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