• Jeong, Jin-Mun (Department of Applied Mathematics Pukyong National University) ;
  • Kang, Yong Han (Institute of Liberal Education Catholic University of Daegu)
  • Received : 2016.10.21
  • Accepted : 2017.02.24
  • Published : 2018.01.31


In this paper, we first consider the existence and regularity of solutions of the semilinear control system under natural assumptions such as the local Lipschtiz continuity of nonlinear term. Thereafter, we will also establish the approximate controllability for the equation when the corresponding linear system is approximately controllable.


Supported by : National research Foundation of Korea(NRF)


  1. J. P. Aubin, Un theoreme de compacite, C. R. Acad. Sci. 256 (1963), 5042-5044.
  2. V. Barbu, Analysis and Control of Nonlinear In nite Dimensional Systems, Academic Press Limited, 1993.
  3. A. E. Bashirov and N. I. Mahmudov, On concepts of controllability for deterministic and stochastic systems, SIAM J. Control Optim. 37 (1999), no. 6, 1808-1821.
  4. M. Benchohra, L. Gorniewicz, S. K. Ntouyas, and A. Ouahab, Controllability results for implusive functional differential inclusions, Rep. Math. Phys. 54 (2004), no. 2, 211-228.
  5. M. Benchohra and A. Ouahab, Controllability results for functional semilinear differ-ential inclusion in Frechet spaces, Nonlinear Anal. 61 (2005), no. 3, 405-423.
  6. R. F. Curtain and H. Zwart, An Introduction to In nite Dimensional Linear Systems Theory, Springer-Velag, New-York, 1995.
  7. J. P. Dauer and N. I. Mahmudov, Exact null controllability of semilinear integrodiffer-ential systems in Hilbert spaces, J. Math. Anal. Appl. 299 (2004), no. 2, 322-333.
  8. G. Di Blasio, K. Kunisch, and E. Sinestrari, $L^2$-regularity for parabolic partial integrod-ifferential equations with delay in the highest-order derivatives, J. Math. Anal. Appl. 102 (1984), no. 1, 38-57.
  9. L. Gorniewicz, S. K. Ntouyas, and D. O'Reran, Controllability of semilinear differential equations and inclusions via semigroup theory in Banach spaces, Rep. Math. Phys. 56 (2005), no. 3, 437-470.
  10. M. L. Heard, An abstract parabolic Volterra integro-differential equation, J. Appl. Math. 17 (1981), 175-202.
  11. J. M. Jeong, Y. C. Kwun, and J. Y. Park, Approximate controllability for semilinear retarded functional differential equations, J. Dynam. Control Systems 5 (1999), no. 3, 329-346.
  12. Y. Kobayashi, T. Matsumoto, and N. Tanaka, Semigroups of locally Lipschitz operators associated with semilinear evolution equations, J. Math. Anal. Appl. 330 (2007), no. 2, 1042-1067.
  13. N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim. 42 (2006), 175-181.
  14. K. Naito, Controllability of semilinear control systems dominated by the linear part, SIAM J. Control Optim. 25 (1987), no. 3, 715-722.
  15. S. Nakagiri, Controllability and identi ability for linear time-delay systems in Hilbert space, Control theory of distributed parameter systems and applications, Lecture Notes in Control and Inform. Sci., 159, Springer, Berlin, 1991.
  16. A. Pazy, Semigroups of Linear Operators and Applications to partial Differential Equations, Springer-Verlag, New-York, Inc. 1983.
  17. N. Sukavanam and N. K. Tomar, Approximate controllability of semilinear delay control system, Nonlinear Funct. Anal. Appl. 12 (2007), no. 1, 53-59.
  18. H. Tanabe, Equations of Evolution, Pitman-London, 1979.
  19. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, 1978.
  20. L. Wang, Approximate controllability and approximate null controllability of semilinear systems, Commun. Pure Appl. Anal. 5 (2006), no. 4, 953-962.
  21. L. Wang, Approximate controllability for integrodifferential equations and multiple delays, J. Optim. Theory Appl. 143 (2009), no. 1, 185-206.
  22. G. Webb, Continuous nonlinear perturbations of linear accretive operator in Banach spaces, J. Funct. Anal. 10 (1972), 191-203.
  23. M. Yamamoto and J. Y. Park, Controllability for parabolic equations with uniformly bounded nonlinear terms, J. Optim. Theory Appl. 66 (1990), no. 3, 515-532.
  24. H. X. Zhou, Approximate controllability for a class of semilinear abstract equations, SIAM J. Control Optim. 21 (1983), no. 4, 551-565.