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December  2017, 6(4): 517-534. doi: 10.3934/eect.2017026

## Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay

 Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China

* Corresponding author: Xianlong Fu

Received  December 2016 Revised  July 2017 Published  September 2017

Fund Project: This work is supported by NSF of China (Nos. 11671142 and 11371087), STCSM (No. 13dz2260400) and Shanghai Leading Academic Discipline Project (No. B407)

The controllability of non-autonomous evolution systems is an important and difficult topic in control theory. In this paper, we study the approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay. The theory of linear evolution operators is used instead of $C_0-$semigroup to discuss the problem. Some sufficient conditions of approximate controllability are formulated and proved here by using the resolvent operator condition. Finally, two examples are provided to illustrate the applications of the obtained results.

Citation: Xianlong Fu. Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay. Evolution Equations & Control Theory, 2017, 6 (4) : 517-534. doi: 10.3934/eect.2017026
##### References:
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Control Syst., 15 (2009), 425-443. doi: 10.1007/s10883-009-9068-x. Google Scholar [11] X. Fu and J. Zhang, Approximate controllability of neutral functional differential systems with state-dependent delay, Chinese Ann. Math. (B), 37 (2016), 291-308. doi: 10.1007/s11401-016-0934-z. Google Scholar [12] R. K. Georgr, Approximate controllability of non-autonomous semiliear systems, Nonl. Anal. (TMA), 24 (1995), 1377-1393. doi: 10.1016/0362-546X(94)E0082-R. Google Scholar [13] J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial ekvac., 21 (1978), 11-41. Google Scholar [14] E. Hernández and D. O'Regan, $C^α-$Hölder classical solutionss for neutral differential euations, Discr. Cont. Dyn. Syst. (A), 29 (2011), 241-260. Google Scholar [15] E. Hernández, A. Prokopczyk and L. Ladeira, A note on partial functional differential equations with state-dependent delay, Nonl. Anal. (RWA), 7 (2006), 510-519. doi: 10.1016/j.nonrwa.2005.03.014. Google Scholar [16] Y. 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Mackey, Hematopoietic model with moving boundary condition and state dependent delay: Applications in erythropoiesis, J. Theory Bio., 190 (1998), 135-146. Google Scholar [22] N.I. Mahmudov and S. Zorlu, On the approximate controllability of fractional evolution equations with compact analytic semigroup, J. Comp. Appl. Math., 259 (2014), 194-204. doi: 10.1016/j.cam.2013.06.015. Google Scholar [23] F. Z. Mokkedem and X. Fu, Approximate controllability for a semilinear evolut ion system with infinite delay, J. Dyn. Control Sys., 22 (2016), 71-89. doi: 10.1007/s10883-014-9252-5. Google Scholar [24] K. Naito, Controllability of semilinear control systems dominated by the linear part, SIAM J. Control Optim., 25 (1987), 715-722. doi: 10.1137/0325040. Google Scholar [25] K. Naito, Approximate controllability for trajectories of a delay Voltera control system, J. Optim. Theory Appl., 61 (1989), 271-279. doi: 10.1007/BF00962800. Google Scholar [26] J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204. doi: 10.1090/qam/295683. Google Scholar [27] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar [28] S. M. Rankin III, Existence and asymptotic behavior of a functional differential equation in Banach space, J. Math. Anal. Appl., 88 (1982), 531-542. doi: 10.1016/0022-247X(82)90211-6. Google Scholar [29] R. Sakthivel and E. R. Ananndhi, Approximate controllability of impulsive differential equations with state-dependent delay, Int. J. control., 83 (2010), 387-393. doi: 10.1080/00207170903171348. Google Scholar [30] R. Sakthivel, R. Ganesh and S. M. Anthoni, Approximate controllability of fractional nonlinear differential inclusions, Appl. Math. Comp., 225 (2013), 708-717. doi: 10.1016/j.amc.2013.09.068. Google Scholar [31] R. Sakthivel and Y. Ren, Approximate controllability of fractional differential equations with state-dependent delay, Results Math., 63 (2013), 949-963. doi: 10.1007/s00025-012-0245-y. Google Scholar [32] R. Sakthivel, S. Suganya and S. M. Anthoni, Approximate controllability of fractional stochastic evolution equations, Comp. Math. Appl., 63 (2012), 660-668. doi: 10.1016/j.camwa.2011.11.024. Google Scholar [33] J. P. C. dos Santos, On state-dependent delay partial neutral functional integro-differential equations, Appl. Math. Comp., 216 (2010), 1637-1644. doi: 10.1016/j.amc.2010.03.019. Google Scholar [34] L. Shen and J. Sun, Approximate controllability of stochastic impulsive functional systems with infinite delay, Automatica, 48 (2012), 2705-2709. doi: 10.1016/j.automatica.2012.06.098. Google Scholar [35] N. Sukavanam and S. Kumar, Approximate controllability of fractional order semilinear delay systems, J. Optim. Theory Appl., 151 (2011), 373-384. doi: 10.1007/s10957-011-9905-4. Google Scholar [36] H. Tanabe, Equations of Evolution, Pitman Publishing, London, 1979. Google Scholar [37] C. C. Travis and G. F. Webb, Partial differential equations with deviating arguments in the time variable, J. Math. Anal. Appl., 56 (1976), 397-409. doi: 10.1016/0022-247X(76)90052-4. Google Scholar [38] C. C. Travis and G. F. Webb, Existence, stability and compactness in the $α-$norm for partial functional differential equations, Trans. Amer. Math. Soc., 240 (1978), 129-143. doi: 10.2307/1998809. Google Scholar [39] L. Wang, Approximate controllability for integrodifferential equations with multiple delays, J. Optim. Theory Appl., 143 (2009), 185-206. doi: 10.1007/s10957-009-9545-0. Google Scholar [40] L. Wang, Approximate controllability results of semilinear integrodifferential equations with infinite delays, Sci. China Ser. F-Inf. Sci., 52 (2009), 1095-1102. doi: 10.1007/s11432-009-0127-4. Google Scholar [41] Z. Yan, Approximate controllability of partial neutral functional differential systems of fractional order with state-dependent delay, Int. J. Contr., 85 (2012), 1051-1062. doi: 10.1080/00207179.2012.675518. Google Scholar [42] Z. Yan, Approximate controllability of fractional neutral integro-differential inclusions with state-dependent delay in Hilbert spaces, IMA J. Math. Control Inform., 30 (2013), 443-462. doi: 10.1093/imamci/dns033. Google Scholar

show all references

##### References:
 [1] O. Arino, M. Habid and R. Bravo de la Parra, A mathematical model of growth of population of fish in the larval stage: density-dependence effects, Math. Biosc., 150 (1998), 1-20. doi: 10.1016/S0025-5564(98)00008-X. Google Scholar [2] Z. Balanov, Q. Hu and W. Krawcewicz, Global Hopf bifurcation of differential equations with threshold type state-dependent delay, J. Diff. Equ., 257 (2014), 2622-2670. doi: 10.1016/j.jde.2014.05.053. Google Scholar [3] A. E. Bashirov and N. I. Mahmudov, On concepts of controllability for linear deterministic and stochastic systems, SIAM J. Control Optim., 37 (1999), 1808-1821. doi: 10.1137/S036301299732184X. Google Scholar [4] M. Belmekki, M. Benchohra and K. Ezzinbi, Existence results for some partial functional differential equations with state-dependent delay, Appl. Math. Lett., 24 (2011), 1810-1816. doi: 10.1016/j.aml.2011.04.039. Google Scholar [5] R. Curtain and H. J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6. Google Scholar [6] J. P. Dauer and N. I. Mahmudov, Approximate controllability of semilinear functional equations in Hilbert spaces, J. Math. Anal. Appl., 273 (2002), 310-327. doi: 10.1016/S0022-247X(02)00225-1. Google Scholar [7] W. Fitzgibbon, Semilinear functional differential equations in Banach space, J. Diff. Equ., 29 (1978), 1-14. doi: 10.1016/0022-0396(78)90037-2. Google Scholar [8] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York. 1969. Google Scholar [9] X. Fu and X. Liu, Existence of periodic solutions for abstract neutral non-autonomous equations with infinite delay, J. Math. Anal. Appl., 325 (2007), 249-267. doi: 10.1016/j.jmaa.2006.01.048. Google Scholar [10] X. Fu and K. Mei, Approximate controllability of semilinear partial functional differential systems, J. Dyn. Control Syst., 15 (2009), 425-443. doi: 10.1007/s10883-009-9068-x. Google Scholar [11] X. Fu and J. Zhang, Approximate controllability of neutral functional differential systems with state-dependent delay, Chinese Ann. Math. (B), 37 (2016), 291-308. doi: 10.1007/s11401-016-0934-z. Google Scholar [12] R. K. Georgr, Approximate controllability of non-autonomous semiliear systems, Nonl. Anal. (TMA), 24 (1995), 1377-1393. doi: 10.1016/0362-546X(94)E0082-R. Google Scholar [13] J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial ekvac., 21 (1978), 11-41. Google Scholar [14] E. Hernández and D. O'Regan, $C^α-$Hölder classical solutionss for neutral differential euations, Discr. Cont. Dyn. Syst. (A), 29 (2011), 241-260. Google Scholar [15] E. Hernández, A. Prokopczyk and L. Ladeira, A note on partial functional differential equations with state-dependent delay, Nonl. Anal. (RWA), 7 (2006), 510-519. doi: 10.1016/j.nonrwa.2005.03.014. Google Scholar [16] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Springer-verlag, Berlin, 1991. doi: 10.1007/BFb0084432. Google Scholar [17] J. Jeong, Y. Kwun and J. Park, Approximate controllability for semilinear retarded functional differential equations, J. Dyn. Contr. Syst., 5 (1999), 329-346. doi: 10.1023/A:1021714500075. Google Scholar [18] J. Jeong and H. Roh, Approximate controllability for semilinear retarded systems, J. Math. Anal. Appl., 321 (2006), 961-975. doi: 10.1016/j.jmaa.2005.09.005. Google Scholar [19] V. Keyantuo and C. Lizama, Hölder continuous solutions for integro-differential equations and maximal regularity, J. Diff. Equ., 230 (2006), 634-660. doi: 10.1016/j.jde.2006.07.018. Google Scholar [20] A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224. doi: 10.1137/0521066. Google Scholar [21] J. M. Mahaffy, J. Belair and M. C. Mackey, Hematopoietic model with moving boundary condition and state dependent delay: Applications in erythropoiesis, J. Theory Bio., 190 (1998), 135-146. Google Scholar [22] N.I. Mahmudov and S. Zorlu, On the approximate controllability of fractional evolution equations with compact analytic semigroup, J. Comp. Appl. Math., 259 (2014), 194-204. doi: 10.1016/j.cam.2013.06.015. Google Scholar [23] F. Z. Mokkedem and X. Fu, Approximate controllability for a semilinear evolut ion system with infinite delay, J. Dyn. Control Sys., 22 (2016), 71-89. doi: 10.1007/s10883-014-9252-5. Google Scholar [24] K. Naito, Controllability of semilinear control systems dominated by the linear part, SIAM J. Control Optim., 25 (1987), 715-722. doi: 10.1137/0325040. Google Scholar [25] K. Naito, Approximate controllability for trajectories of a delay Voltera control system, J. Optim. Theory Appl., 61 (1989), 271-279. doi: 10.1007/BF00962800. Google Scholar [26] J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204. doi: 10.1090/qam/295683. Google Scholar [27] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar [28] S. M. Rankin III, Existence and asymptotic behavior of a functional differential equation in Banach space, J. Math. Anal. Appl., 88 (1982), 531-542. doi: 10.1016/0022-247X(82)90211-6. Google Scholar [29] R. Sakthivel and E. R. Ananndhi, Approximate controllability of impulsive differential equations with state-dependent delay, Int. J. control., 83 (2010), 387-393. doi: 10.1080/00207170903171348. Google Scholar [30] R. Sakthivel, R. Ganesh and S. M. Anthoni, Approximate controllability of fractional nonlinear differential inclusions, Appl. Math. Comp., 225 (2013), 708-717. doi: 10.1016/j.amc.2013.09.068. Google Scholar [31] R. Sakthivel and Y. Ren, Approximate controllability of fractional differential equations with state-dependent delay, Results Math., 63 (2013), 949-963. doi: 10.1007/s00025-012-0245-y. Google Scholar [32] R. Sakthivel, S. Suganya and S. M. Anthoni, Approximate controllability of fractional stochastic evolution equations, Comp. Math. Appl., 63 (2012), 660-668. doi: 10.1016/j.camwa.2011.11.024. Google Scholar [33] J. P. C. dos Santos, On state-dependent delay partial neutral functional integro-differential equations, Appl. Math. Comp., 216 (2010), 1637-1644. doi: 10.1016/j.amc.2010.03.019. Google Scholar [34] L. Shen and J. Sun, Approximate controllability of stochastic impulsive functional systems with infinite delay, Automatica, 48 (2012), 2705-2709. doi: 10.1016/j.automatica.2012.06.098. Google Scholar [35] N. Sukavanam and S. Kumar, Approximate controllability of fractional order semilinear delay systems, J. Optim. Theory Appl., 151 (2011), 373-384. doi: 10.1007/s10957-011-9905-4. Google Scholar [36] H. Tanabe, Equations of Evolution, Pitman Publishing, London, 1979. Google Scholar [37] C. C. Travis and G. F. Webb, Partial differential equations with deviating arguments in the time variable, J. Math. Anal. Appl., 56 (1976), 397-409. doi: 10.1016/0022-247X(76)90052-4. Google Scholar [38] C. C. Travis and G. F. Webb, Existence, stability and compactness in the $α-$norm for partial functional differential equations, Trans. Amer. Math. Soc., 240 (1978), 129-143. doi: 10.2307/1998809. Google Scholar [39] L. Wang, Approximate controllability for integrodifferential equations with multiple delays, J. Optim. Theory Appl., 143 (2009), 185-206. doi: 10.1007/s10957-009-9545-0. Google Scholar [40] L. Wang, Approximate controllability results of semilinear integrodifferential equations with infinite delays, Sci. China Ser. F-Inf. Sci., 52 (2009), 1095-1102. doi: 10.1007/s11432-009-0127-4. Google Scholar [41] Z. Yan, Approximate controllability of partial neutral functional differential systems of fractional order with state-dependent delay, Int. J. Contr., 85 (2012), 1051-1062. doi: 10.1080/00207179.2012.675518. Google Scholar [42] Z. Yan, Approximate controllability of fractional neutral integro-differential inclusions with state-dependent delay in Hilbert spaces, IMA J. Math. Control Inform., 30 (2013), 443-462. doi: 10.1093/imamci/dns033. Google Scholar
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