Advances in Difference Equations (Oct 2021)

Results on exact controllability of second-order semilinear control system in Hilbert spaces

  • Urvashi Arora,
  • V. Vijayakumar,
  • Anurag Shukla,
  • Kottakkaran Sooppy Nisar,
  • Shahram Rezapour,
  • Wasim Jamshed

DOI
https://doi.org/10.1186/s13662-021-03620-5
Journal volume & issue
Vol. 2021, no. 1
pp. 1 – 13

Abstract

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Abstract In our manuscript, we extend the controllability outcomes given by Bashirov (Math. Methods Appl. Sci. 44(9):7455–7462, 2021) for a family of second-order semilinear control system by formulating a sequence of piecewise controls. This approach does not involve large estimations which are required to apply fixed point theorems. Therefore, we avoid the use of fixed point theory and the contraction mapping principle. We establish that a second-order semilinear system drives any starting position to the required final position from the domain of the system. To achieve the required results, we suppose that the linear system is exactly controllable at every non-initial time period, the norm of the inverse of the controllability Grammian operator increases as the time approaches zero with the slower rate in comparison to the reciprocal of the square function, and the nonlinear term is bounded. Finally, an example has been presented to validate the results.

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