Skip to main content
SpringerLink
Log in

Controllability of time-varying systems with impulses, delays and nonlocal conditions

  • Published:
Afrika Matematika Aims and scope Submit manuscript

Abstract

In this paper, we prove the controllability of time-varying semilinear systems with impulses, delay, and nonlocal Conditions, where some ideas are taking from previous works for this kind of systems with impulses and nonlocal conditions only, this is done by using new techniques avoiding fixed point theorems employed by A.E. Bashirov et al. In this case the delay helps us to prove the approximate controllability of this system by pulling back the control solution to a fixed curve in a short time interval, and from this position, we are able to reach a neighborhood of the final state in time \(\tau \) by assuming that the corresponding linear control system is exactly controllable on any interval \([t_0, \tau ]\), \(0< t_0 < \tau \).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bashirov, A.E., Ghahramanlou, N.: On Partial Approximate Controllability of Semilinear Systems. Cogent Engineering 1, (2014). https://doi.org/10.1080/23311916.2014.965947

  2. Bashirov, A.E., Jneid, M.: On Partial Complete Controllability of Semilinear Systems. Abst. Appli. Anal. 2013, 521052 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Bashirov, A.E., Mahmudov, N., Semi, N., Etikan, H.: On Partial Controllability Concepts. Inernational Journal of Control 80(1), 1–7 (2007)

    Article  MathSciNet  Google Scholar 

  4. Carrasco, A., Leiva, Hugo, Sanchez, J.L., Tineo, M.A.: Approximate Controllability of the Semilinear Impulsive Beam Equation with Impulses. Transaction on IoT and Cloud Computing 2(3), 70–88 (2014)

    Google Scholar 

  5. Chukwu, E.N.: Stability and Time-Optimal Control of Hereditary Systems, Mathematics in Science and Engineering, vol. 188. Academic Press, INC. (1992)

  6. Chukwu, E.N.: Nonlinear Delay Systems Controllability. Math. Anal. Appl. 162, 564–576 (1991)

    Article  MathSciNet  Google Scholar 

  7. Chukwu, E.N.: Global Null Controllability of Nonlinear Delay Equations with Controls in a Compact Set. Optim. Theo. Appl. 53, 43–57 (1987)

    Article  MathSciNet  Google Scholar 

  8. Chukwu, E.N.: Controllability of Delay Systems with Restrained Controls. Optim. Theo. Appl. 29, 301–320 (1979)

    Article  MathSciNet  Google Scholar 

  9. Chukwu, E.N.: On the Null-Controllability of Nonlinear Delay Systems with Restrained Controls. Math. Anal. Appl. 76, 283–296 (1980)

    Article  MathSciNet  Google Scholar 

  10. Chukwu, E.N.: Null-controllability of nonlinear delay systems with restrained controls. J. Math. Anal. Appl. 76, 283–296 (1980)

    Article  MathSciNet  Google Scholar 

  11. Coron, J.-M.: Control and Nonlinearity. Mathematical Surveys and Monographs, vol. 136. American Mathematical Society, Provience, RI (2007)

  12. curtain, R.F., Pritchard, A.J.: Infinite Dimensional Linear Systems. Lecture Notes in Control and Information Sciences, 8. Springer Verlag, Berlin (1978)

  13. Curtain, R.F., Zwart, H.J.: An Introduction to Infinite Dimensional Linear Systems Theory. Text in Applied Mathematics, 21. Springer Verlag, New York (1995)

  14. Dauer, J.P.: Nonlinear Perturbation of Quasilinear Control Systems. J. Math. Anal. Appl. 54(3), 717–725 (1976)

    Article  MathSciNet  Google Scholar 

  15. Do, V.N.: Controllability of Semilinear Systems. J. Optim. Theory Appl. 65(1), 41–52 (1990)

    Article  MathSciNet  Google Scholar 

  16. Zhi-Qing, Z.H.U., Qing-Wen, L.I.N.: Exact Controllability of Semilinear Systems with Impulses. Bulletin of Math. Anal. and Appl. 4(1), 157–167 (2012)

    MathSciNet  Google Scholar 

  17. Iturriaga, E., Leiva, H.: A Characterization of Semilinear Surjective Operators and Applications to Control Problems. Applied Mathematics 1, 137–145 (2010)

    Article  Google Scholar 

  18. Lee, E.B., Markus, L.: Foundations of Optimal Control Theory. Wiley, New York (1967)

    MATH  Google Scholar 

  19. Leiva, Hugo: Hitcher Zambrano, Rank Condition for the Controllability of Linear Time Varying System. International Journal of Control 72, 920–931 (1999)

    Article  Google Scholar 

  20. Leiva, Hugo: Rothe’s Fixed Point Theorem and Controllability of Semilinear Nonautonomous Systems. System and Control Letters 67, 14–18 (2014)

    Article  MathSciNet  Google Scholar 

  21. Leiva, Hugo: Controllability of Semilinear Impulsive Nonautonomous Systems, International Journal of Control, https://doi.org/10.1080/00207179.2014.966759 (2014)

  22. Hugo Leiva, N.: Merentes, Approximate Controllability of the Impulsive Semilinear Heat Equation. Journal of Mathematics and Applications N0(38), 85–104 (2015)

    MATH  Google Scholar 

  23. Leiva, Hugo: Approximate Controllability of Semilinear Impulsive Evolution Equations, Abstract and Applied Analysis, 2015, 797439. https://doi.org/10.1155/2015/797439

  24. Leiva, H., Cabada, D., Gallo, R.: Roughness of the Controllability for Time Varying Systems Under the Influence of Impulses, Delay, and Nonlocal Conditions. Nonautonomous Dynamical Systems 7(1), 126–139 (2020)

    Article  MathSciNet  Google Scholar 

  25. Leiva, Hugo, Rojas, Raul: Controllability of Semilinear Nonautonomous Systems with Impulses and Nonlocal Conditions. Equilibrium: Journal of Natural Sciences, Volumen 1, abril ISSN: 2470-1998 (2016)

  26. Leiva, H., Sundar, P: Existence of solutions for a class of semilinear evolution equations whit impulses and delays, Journal of Nonlinear Evolution Equations and Applications ISSN 2161-3680. 2017(7) 95-108 (2017)

  27. Leiva, Hugo: Karakostas Fixed Point Theorem and the Existence of Solutions for Impulsive Semilinear Evolution Equations with Delays and Nonlocal Conditions. Communications inMathematicalAnalysis, Volume 21, Number 2, pp. 68-91, ISSN 1938-9787 (2018)

  28. LukesS, D.L.: Global Controllability of Nonlinear Systems. SIAM J. Control Optim. 10 (1972), 1, 112-126, Erratum 11, 1, 186 (1973)

  29. Mirza, K.B., Womack, B.F.: On the Controllability of Nonlinear Time-Delay Systems. IEEE Trans. Auto. Cont. Short Papers 17(6), 812–814 (1972)

    Article  MathSciNet  Google Scholar 

  30. Sinha, A.S.C.: Null-Controllability of Non-Linear Infinite Delay Systems with Restrained Controls. Int. J. Cont. 42, 735–741 (1985)

    Article  Google Scholar 

  31. Nieto, J.J., Tisdell, C.: On exact controllability of first-order impulsive differential equations. Adv. Differ. Equ. 2010(1), 136504 (2010). https://doi.org/10.1155/2010/136504

    Article  MathSciNet  MATH  Google Scholar 

  32. Sinha, A. S. C.: Yokomoto, C. F.: Null Controllability of a Nonlinear System with Variable Time Delay, IEEE Trans. Auto. Cont. AC-25 1234-1236 (1980)

  33. Selvi, S., Arjunan, M. Mallika: Controllability results for Impulsive Differential Systems with finite Delay. Journal of Nonlinear Science and Applications. 5, 206-219(2012)

  34. Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd edn. Springer, New York, NY, USA (1998)

    Book  Google Scholar 

  35. Vidyasager, M.: A Controllability Condition for Nonlinear Systems. IEEE Trans. Automat. Control. AC-17, 5, 569-570 (1972)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, School of Mathematical Sciences and Information Technology, Yachay Tech, 100119, San Miguel de Urcuqui, Imbabura, Ecuador

    Hugo Leiva, Dalia Cabada & Rodolfo Gallo

  2. Facultad de Ciencias, Departamento de Matemática, Universidad de los Andes, Mérida, 5101, Venezuela

    Hugo Leiva, Dalia Cabada & Rodolfo Gallo

Authors
  1. Hugo Leiva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dalia Cabada
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rodolfo Gallo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hugo Leiva.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work has been supported by Yachay Tech and ULA.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Leiva, H., Cabada, D. & Gallo, R. Controllability of time-varying systems with impulses, delays and nonlocal conditions. Afr. Mat. 32, 959–967 (2021). https://doi.org/10.1007/s13370-021-00872-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13370-021-00872-y

Keywords

Mathematics Subject Classification

Search

Navigation