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Nonautonomous Dynamical Systems

formerly Nonautonomous and Stochastic Dynamical Systems

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Controllability of the Semilinear Heat Equation with Impulses and Delay on the State

Hugo Leiva
  • Leiva: Louisiana State University, College of Science, Math. Department, Baton Rouge, LA 70803-USA
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2015-10-16 | DOI: https://doi.org/10.1515/msds-2015-0004


In this paper we prove the interior approximate controllability of the following Semilinear Heat Equation with Impulses and Delay

where Ω is a bounded domain in RN(N ≥ 1), φ : [−r, 0] × Ω → ℝ is a continuous function, ! is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ! and the distributed control u be- longs to L2([0, τ]; L2(Ω; )). Here r ≥ 0 is the delay and the nonlinear functions f , Ik : [0, τ] × ℝ × ℝ → ℝ are smooth enough, such that

Under this condition we prove the following statement: For all open nonempty subset ! of Ω the system is approximately controllable on [0, τ], for all τ > 0.

Keywords: interior approximate controllability; semilinear heat equation with impulses and delay; strongly continuous semigroup


  • Google Scholar

  • [1] N. Abada, M. Benchohra and H. Hammouche, Existence Results for Semilinear Differential Evolution Equations with Impulses and Delay,CUBO A Mathematical Journal Vol. 02, (1-17). june 2010. Google Scholar

  • [2] H. Akca, a. Boucherif and V. Covachev, Impulsive Functional-Differential Equations with Nonlocal Conditions, IJMMs 29:5 (2002) 251-256. Google Scholar

  • [3] A.E. Bashirov and Noushin Ghahramanlou, On Partial Approximate Controllability of Semilinear Systems. COGENTENG- Engeneering, Vol. 1, 1-13 (2014), doi: 10.1080/23311916.2014.965947 CrossrefGoogle Scholar

  • [4] A.E. Bashirov and Noushin Ghahramanlou, On Partial Complete Controllability of Semilinear Systems. Abstract and Applied Analysis, Vol. 2013, Article ID 52105, 8 pages. Web of ScienceGoogle Scholar

  • [5] A.E. Bashirov, N. Mahmudov, N. Semi and H. Etikan, No On Partial Controllability Concepts. Inernational Journal of Control, Vol. 80, No. 1, January 2007, 1-7. Google Scholar

  • [6] D. N. Chalishajar, Controllability of Impulsive Partial Neutral Funcional Differential Equation with Infinite Delay. Int. Journal of Math. Analysis, Vol. 5, 2011, No. 8, 369-380. Google Scholar

  • [7] D.Barcenas, H. Leiva and Z. Sivoli, A Broad Class of Evolution Equations are Approximately Controllable, but Never Exactly Controllable. IMA J. Math. Control Inform. 22, no. 3 (2005), 310–320. Google Scholar

  • [8] K. Balachandran and J. H. Kim, Remarks on the Paper Controllability of Second Order Differential Inclusion in Banach Spaces, (J. Math. Anal. Appl.) J. Math. Anal. Appli. 324 (2006), 746-749. CrossrefGoogle Scholar

  • [9] Lizhen Chen and Gang Li, Approximate Controllability of Impulsive Differential Equations with Nonlocal Conditions. Inter- national Journal of Nonlinear Science, Vol.10(2010), No. 4, pp. 438-446. Google Scholar

  • [10] R.F. Curtain, A.J. Pritchard, Infinite Dimensional Linear Systems. Lecture Notes in Control and Information Sciences, 8. Springer Verlag, Berlin (1978). Google Scholar

  • [11] R.F. Curtain, H.J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory. Text in Applied Mathematics, 21. Springer Verlag, New York (1995). Google Scholar

  • [12] V. Lakshmikantham, D. D. Bainov and P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. Google Scholar

  • [13] H. Leiva, N. Merentes and J. Sanchez "A Characterization of Semilinear Dense Range Operators and Applications", to appear in Abstract and Applied Analysis. Google Scholar

  • [14] H. Leiva, N. Merentes and J. Sanchez “Interior Controllability of the Benjamin-Bona-Mahony Equation”. Journal of Mathe- matis and Applications, No. 33,pp. 51-59 (2010). Google Scholar

  • [15] H. Leiva, N. Merentes and J. Sanchez “Interior Controllability of the Semilinear Benjamin-Bona-Mahony Equation”. Journal of Mathematis and Applications, No. 35,pp. 97-109 (2012). Google Scholar

  • [16] H. Leiva, N. Merentes and J. Sanchez “Approximate Controllability of a Semilinear Heat Equation”. Interntional Journal of Partial Differential Equations Mathematis,Vol. 2013, Art. ID 424309, 7 pages. Google Scholar

  • [17] Hugo Leiva, Approximate Controllability of Semilinear Impulsive Evolution Equations, Abstract and Applied Analysis, Vol. 2015, Article ID 797439, 7 pages Web of ScienceGoogle Scholar

  • [18] A. M. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations. World Scientific Series on Nonlinear Science Series A, Vol. 14, 1995. Google Scholar

  • [19] A. Carrasco, Hugo Leiva, J.L. Sanchez and A. Tineo M, Approximate Controllability of the Semilinear Impulsive BeamEquation with Impulses. Transaction on IoT and Cloud Computing 2(3) 70-88, 2014. Google Scholar

  • [20] Hugo Leiva, Controllability of Semilinear Impulsive Nonautonomous Systems, International Journal of Control, 2014, http/dx.doi.org/10.1080/00207179.2014.966759. CrossrefWeb of ScienceGoogle Scholar

  • [21] Hugo Leiva and N. Merentes,Approximate Controllability of the Impulsive Semilinear Heat Equation. Journal of Mathematics and Applications, No. 38, pp 85-104 (2015) Google Scholar

  • [22] S. Selvi and M. Mallika Arjunan, Controllability Results for Impulsive Differential Systems with Finite Delay J. Nonlinear Sci. Appl. 5 (2012), 206-219. Google Scholar

  • [23] R. Shikharchand Jain and M. Baburao Dhakne, On Mild Solutions of Nonlocal Semilinear Impulsive Functional Integro- Differential Equations, Applied Mathematics E-Notes, 13(2014), 109-119. Google Scholar

About the article

Received: 2015-05-20

Accepted: 2015-08-15

Published Online: 2015-10-16

Citation Information: Nonautonomous Dynamical Systems, Volume 2, Issue 1, ISSN (Online) 2353-0626, DOI: https://doi.org/10.1515/msds-2015-0004.

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©2015 Hugo Leiva. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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