Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

APPROXIMATE REACHABLE SETS FOR RETARDED SEMILINEAR CONTROL SYSTEMS

  • KIM, DAEWOOK (Department of Mathematics Education, Seowon University) ;
  • JEONG, JIN-MUN (Department of Applied Mathematics, Pukyong National University)
  • Received : 2020.04.20
  • Accepted : 2020.08.07
  • Published : 2020.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we consider a control system for semilinear differential equations in Hilbert spaces with Lipschitz continuous nonlinear term. Our method is to find the equivalence of approximate controllability for the given semilinear system and the linear system excluded the nonlinear term, which is based on results on regularity for the mild solution and estimates of the fundamental solution.

Keywords

References

  1. H.O. Fattorini, On complete controllability of linear systems, J. Differential Equations 3 (1967), 391-402. https://doi.org/10.1016/0022-0396(67)90039-3
  2. H.O. Fattorini, Boundary control systems, SIAM J. Control Optimization 6 (1968), 349-402. https://doi.org/10.1137/0306025
  3. R. Triggiani, Existence of rank conditions for controllability and obserbavility to Banach spaces and unbounded operators, SIAM J. Control Optim. 14 (1976), 313-338. https://doi.org/10.1137/0314022
  4. R.F. Curtain and H. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, Springer-Velag, New-York, 1995.
  5. V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press Limited, 1993.
  6. M. Yamamoto and J.Y. Park, Controllability for parabolic equations with uniformly bounded nonlinear terms, J. optim. Theory Appl. 66 (1990), 515-532. https://doi.org/10.1007/BF00940936
  7. X. Fu, Controllability of neutral functional differential systems in abstract space, Appl. Math. Comput. 141 (2003), 281-296. https://doi.org/10.1016/S0096-3003(02)00253-9
  8. M. Benchohra, L. Gorniewicz, S.K. Ntouyas and A. Ouahab, Controllability results for implusive functional differential inclusions, Rep. Math. Phys. 54 (2004), 211-228. https://doi.org/10.1016/S0034-4877(04)80015-6
  9. H.X. Zhou, Approximate controllability for a class of semilinear abstract equations, SIAM J. Control Optim. 21 (1983).
  10. 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
  11. 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
  12. J.M. Jeong, Stabilizability of retarded functional differential equation in Hilbert space, Osaka J. Math. 28 (1991), 347-365.
  13. H. Tanabe, Fundamental solutions for linear retarded functional differential equations in Banach space, Funkcial. Ekvac. 35 (1992), 149-177.
  14. H. Tanabe, Equations of Evolution, Pitman-London, 1979.
  15. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, 1978.
  16. J. Yong and L. Pan, Quasi-linear parabolic partial differential equations with delays in the highest order partial derivatives, J. Austral. Math. Soc. 54 (1993), 174-203. https://doi.org/10.1017/S1446788700037101
  17. S. Nakagiri, Structural properties of functional differential equations in Banach spaces, Osaka J. Math. 25 (1988), 353-398.
  18. P. Balachadran and J.Y. Park, Controllability of fractional integrodifferential systems with in Banach spaces, Nonlinear Anal. Hybrid Syst. 3 (2009), 363-367. https://doi.org/10.1016/j.nahs.2009.01.014