Opuscula Math. 35, no. 1 (2015), 71-83
http://dx.doi.org/10.7494/OpMath.2015.35.1.71

 
Opuscula Mathematica

Controllability of semilinear systems with fixed delay in control

Surendra Kumar
N. Sukavanam

Abstract. In this paper, different sufficient conditions for exact controllability of semilinear systems with a single constant point delay in control are established in infinite dimensional space. The existence and uniqueness of mild solution is also proved under suitable assumptions. In particular, local Lipschitz continuity of a nonlinear function is used. To illustrate the developed theory some examples are given.

Keywords: first order delay system, mild solution, fixed point, exact controllability.

Mathematics Subject Classification: 93B05.

Full text (pdf)

  1. K. Balachandran, J.P. Dauer, Controllability of nonlinear systems in Banach spaces: a survey, J. Optim. Theory Appl. 115 (2002), 7-28.
  2. R.F. Curtain, H.J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, Springer-Verlag, New York, 1995.
  3. I. Davies, P. Jackreece, Controllability and null controllability of linear systems, J. Appl. Sci. Environ. Manag. 9 (2005), 31-36.
  4. J. Klamka, Relative controllability and minimum energy control of linear systems with distributed delays in control, IEEE T. Automat. Contr. 21 (1976) 4, 594-595.
  5. J. Klamka, Controllability of linear systems with time-variable delays in control, Int. J. Control. 24 (1976) 6, 869-878.
  6. J. Klamka, On the controllability of linear systems with delays in the control, Int. J. Control. 25 (1977) 6, 875-883.
  7. J. Klamka, Schauder's fixed-point theorem in nonlinear controllability problems, Control Cybern. 29 (2000) 1, 153-165.
  8. J. Klamka, Stochastic controllability of systems with variable delay in control, Bull. Pol. Ac.: Tech. 56 (2008) 3, 279-284.
  9. J. Klamka, Stochastic controllability and minimum energy control of systems with multiple delays in control, Applied Math. Comput. 206 (2008), 704-715.
  10. J. Klamka, Stochastic controllability of systems with multiple delays in control, Int. J. Appl. Math. Comput. Sci. 19 (2009), 39-47.
  11. Y. Liu, S. Zhao, Controllability analysis of linear time-varying systems with multiple time delay and impulsive effects, Nonlinear Analysis: RWA. 13 (2012), 558-568.
  12. K. Naito, Controllability of semilinear control systems dominated by the linear part, SIAM J. Control Optim. 25 (1987), 715-722.
  13. L. Wang, Approximate controllability for integrodifferential equations with multiple delays, J. Optim. Theory Appl. 143 (2009), 185-206.
  14. L. Wang, Approximate boundary controllability for semilinear delay differential equations, J. Applied Math. 2011 (2011), Article ID 587890, 10 pp.
  • Surendra Kumar
  • University of Delhi, Department of Mathematics, Delhi - 110007, India
  • N. Sukavanam
  • Indian Institute of Technology, Roorkee, Department of Mathematics, Roorkee (Uttarakhand) - 247667, India
  • Communicated by Alexander Gomilko.
  • Received: 2013-05-29.
  • Revised: 2014-02-07.
  • Accepted: 2014-03-23.
  • Published online: 2014-11-12.
Opuscula Mathematica - cover

Cite this article as:
Surendra Kumar, N. Sukavanam, Controllability of semilinear systems with fixed delay in control, Opuscula Math. 35, no. 1 (2015), 71-83, http://dx.doi.org/10.7494/OpMath.2015.35.1.71

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.