Acknowledgement
This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (2019R1F1A1048077).
References
- V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press Limited, Cambridge, (1993).
- M. Benchohra and A. Ouahab, Controllability results for functional semilinear differential inclusion in Frechet spaces, Nonlinear Anal, 61(3)(2005), 405-423. https://doi.org/10.1016/j.na.2004.12.002
- J. P. Dauer and K. I. Mahmudov, Exact null controllability of semilinear integrodifferential systems in Hilbert spaces, J. Math. Anal. Appl., 299(2)(2004), 322-333. https://doi.org/10.1016/j.jmaa.2004.01.050
- G. Di Blasio and K. Kunisch and E. Sinestrari, L2-regularity for parabolic partial integrodifferential equations with delay in the highest-order derivatives, J. Math. Anal. Appl., 102(1)(1984), 38-57. https://doi.org/10.1016/0022-247X(84)90200-2
- X. Fu, Controllability of neutral functional differential systems in abstract space, Appl. Math. Comput., 141(2-3)(2003), 281-296. https://doi.org/10.1016/S0096-3003(02)00253-9
- X. Fu and J. Lu and Y. You, Approximate controllability of a semilinear neutral evolution systems with delay, Inter. J. Contro., 87(4)(2014), 665-681. https://doi.org/10.1080/00207179.2013.852254
- L. Gorniewicz and S. K. Ntouyas and D. O'Reran, Controllability of semilinear differential equations and inclusions via semigroup theory in Banach spaces, Rep. Math. Phys., 56(3)(2005), 437-470. https://doi.org/10.1016/S0034-4877(05)80096-5
- E. Hernandez and M. Mckibben, On state-dependent delay partial neutral functional differential equations, Appl. Math. Comput., 186(1)(2007), 294-301. https://doi.org/10.1016/j.amc.2006.07.103
- E. Hernandez and M. Mckibben and H. Henrrquez, Existence results for partial neutral functional differential equations with state-dependent delay, Math. Comput. Modell., 49(5-6)(2009), 1260-1267. https://doi.org/10.1016/j.mcm.2008.07.011
- J. M. Jeong and Y. H. Kang, Controllability for trajectories of semilinear functional differential equations, Bull. Korean Math. Soc., 55(1)(2018), 63-79. https://doi.org/10.4134/BKMS.b160848
- J. M. Jeong and Y. C. Kwun and J. Y. Park Approximate controllability for semilinear retarded functional differential equations., J. Dyn. Control Syst., 5(3)(1995), 329-346. https://doi.org/10.1023/A:1021714500075
- Y. H. Kang and J. M. Jeong, Control problems for semi-linear retarded integrodifferential equations by the Fredholm theory, Inter. J. Control., 92(1)(2019), 56-64. https://doi.org/10.1080/00207179.2017.1390260
- Y. Kobayashi, T. Matsumoto and N. Tanaka, Semigroups of locally Lipschitz operators associated with semilinear evolution equations, J. Math. Anal. Appl., 330(2)(2007), 1042-1067. https://doi.org/10.1016/j.jmaa.2006.08.028
- Y. C. Kwun, S. H. Park, D. G. Park and S. J. Park, Controllability of semilinear neutral functional differential evolution equations with nonlocal conditions, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math., 15(3)(2008), 245-257.
- F. Z. Mokkedem and X. Fu, Approximate controllability of a semi-linear neutral evolution systems with infinite delay, Internat. J. Robust Nonlinear Control, 27(2017), 1122-1146. https://doi.org/10.1002/rnc.3619
- K. Naito, Controllability of semilinear control systems dominated by the linear part, SIAM J. Control Optim., 25(3)(1987), 715-722. https://doi.org/10.1137/0325040
- A. Pazy, Semigroups of Linear Operators and Applications to partial Differential Equations, Springer-Verlag Newyork, Inc, Newyork, (1983).
- B. Radhakrishnan and K. Balachandran, Controllability of neutral evolution integrodifferential systems with state dependent delay, J. Optim. Theory Appl., 153(1)(2012), 85-97. https://doi.org/10.1007/s10957-011-9934-z
- B. Radhakrishnan and K. Balachandran, Controllability of impulsive neutral functional evolution integrodifferential systems with infinite delay, Nonlinear Anal. Hybrid Syst., 5(4)(2011), 655-670. https://doi.org/10.1016/j.nahs.2011.05.001
- N. Sukavanam and N. K. Tomar, Approximate controllability of semilinear delay control system, Nonlinear Func. Anal. Appl., 12(1)(2007), 53-59.
- H. Tanabe, Equations of Evolution, Pitman, London, (1979).
- H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland publ., North-Holland, (1978).
- L. Wang, Approximate controllability and approximate null controllability of semilinear systems, Commun. Pure Appl. Anal., 5(4)(2006), 953-962.
- L. Wang, Approximate controllability for integrodifferential equations and multiple delays, J. Optim. Theory Appl., 143(1)(2009), 185-206. https://doi.org/10.1007/s10957-009-9545-0
- H. X. Zhou, Approximate controllability for a class of semilinear abstract equations, SIAM J. Control Optim., 21(4)(1983),551-565. https://doi.org/10.1137/0321033