Advertisement

Controllability of Semilinear Control Systems with Fixed Delay in State

  • Abdul Haq
  • N. Sukavanam
Conference paper
  • 48 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 307)

Abstract

This work studies the controllability of a class of delay differential equations. Instead of \(C_0\)-semigroup associated with the mild solution of the system, we use the concept of fundamental solution. Approximate controllability of the system is shown using sequence method. Finally, an illustrative example has been provided.

Keywords

Delay system Fundamental solution Mild solution Approximate controllability 

References

  1. 1.
    R.F. Curtain, H. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, Texts in Applied Mathematics, vol. 21 (Springer, New York, 1995)Google Scholar
  2. 2.
    K. Naito, Controllability of semilinear control systems dominated by the linear part. SIAM J. Control Optim. 25, 715–722 (1987)MathSciNetCrossRefGoogle Scholar
  3. 3.
    K. Balachandran, J.P. Dauer, Controllability of nonlinear systems in Banach spaces: a survey. J Optim. Theory Appl. 115(1), 7–28 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    J. Klamka, Relative controllability of minimum energy control of linear systems with distributed delays in control. IEEE T. Automat. Contr. 21, 594–595 (1976)MathSciNetCrossRefGoogle Scholar
  5. 5.
    J. Klamka, Schauder’s fixed point theorem in nonlinear controllability problems. Control Cybern. 29, 153–165 (2000)MathSciNetzbMATHGoogle Scholar
  6. 6.
    N.I. Mahmudov, N. Semi, Approximate controllability of semilinear control systems in Hilbert spaces. TWMS J. App. Eng. Math. 2, 67–74 (2012)MathSciNetzbMATHGoogle Scholar
  7. 7.
    C. Wang, R. Du, Approximate controllability of a class of semilinear degenerate systems with convection term. J. Differ. Equ. 254(9), 3665–3689 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    L. Wang, Approximate controllability of integrodifferential equations with multiple delays. J. Optim Theory Appl. 143, 185–206 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    N. Sukavanam, Approximate controllability of semilinear control systems with growing nonlinearity, in Mathematical theory of control proceedings of international conference (Marcel Dekker, New York, 1993), pp. 353–357Google Scholar
  10. 10.
    J. Klamka, Stochastic controllability of systems with variable delay in control. Bull. Pol. Ac. Tech. 56, 279–284 (2008)Google Scholar
  11. 11.
    J. Klamka, Stochastic controllability and minimum energy control of systems with multiple delays in control. Appl. Math. Comput. 206, 704–715 (2008)MathSciNetzbMATHGoogle Scholar
  12. 12.
    I. Davies, P. Jackreece, Controllability and null controllability of linear systems. J. Appl. Sci. Environ. Manag. 9, 31–36 (2005)Google Scholar
  13. 13.
    A. Shukla, N. Sukavanam, D.N. Pandey, Approximate controllability of semilinear system with state delay using sequence method. J. Frankl. Inst. 352, 5380–5392 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    N. Sukavanam, S. Tafesse, Approximate controllability of a delayed semilinear control system with growing nonlinear term. Nonlinear Anal. 74, 6868–6875 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  • Abdul Haq
    • 1
  • N. Sukavanam
    • 1
  1. 1.Department of MathematicsIndian Institute of Technology RoorkeeRoorkeeIndia

Personalised recommendations