East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
권호 표지
DOI QR코드

DOI QR Code

REMARK ON THE CONTROLLABILITY FOR SEMILINEAR EVOLUTION EQUATIONS

  • Jeong, Jin-Mun (Department of Applied Mathematics, Pukyong National University)
  • Received : 2013.05.08
  • Accepted : 2013.10.25
  • Published : 2013.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we deal with approximate controllability for semilinear system in a Hilbert space. In order to obtain the controllability, we assume that the system of the generalized eigenspaces of the principal operator is complete in the state space, which has a simple form and can be applied to many examples. Because of its simple form, some examples of controllability of the systems governed by the semilinear equations will be given.

Keywords

References

  1. G. Di Blasio, K. Kunisch and E. Sinestrari, $L^2$-regularity for parabolic partial integrodifferential equations with delay in the highest-order derivatives, J. Math. Anal. Appl. 102 (1984), 38-57. https://doi.org/10.1016/0022-247X(84)90200-2
  2. J. M. Jeong, Retarded functional differential equations with $L^1$-valued controller, Funkcial. Ekvac. 36 (1993), 71-93.
  3. K. Naito, Controllability of semilinear control systems dominated by the linear part, SIAM J. Control Optim. 25 (1987), 715-722. https://doi.org/10.1137/0325040
  4. S. Nakagiri, Structural properties of functional differential equations in banach space, Osaka J. Math. 25 (1988), 353-398.
  5. J. M. Jeong, Y. C. Kwun and J. Y. Park, Approximate controllability for semilinear retarded functional differential equations, J. Dynamics and Control Systems, 5 (1999), 329-346. https://doi.org/10.1023/A:1021714500075
  6. H. Tanabe, Functional analysis II, Jikko Suppan Publ. Co., Tokyo, 1981[in Japanese].
  7. H. Tanabe, Fundamental solution of differential equation with time delay in Banach space, Funkcial. Ekvac. 35 (1992), 149-177.
  8. K. Yosida, Functional Analysis, 3rd ed., Springer-Verlag Berlin Heidelberg New York, 1980.
  9. H. X. Zhou, Approximate controllability for a class of semilinear abstract equations, SIAM J. Control Optim. 21 (1983), 551-565. https://doi.org/10.1137/0321033
  10. H. X. Zhou, Controllability properties of linear and semilinear abstract control systems, SIAM J. Control Optim. 22 (1984), 405-422. https://doi.org/10.1137/0322026
  11. J. M. Jeong and H. H. Rho, Approximate controllability for semilinear retarded systems, J. Math. Anal. Appl. 321 (2006), 961-975. https://doi.org/10.1016/j.jmaa.2005.09.005