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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 and Control Theory, 2017, 6 (4) : 517-534. doi: 10.3934/eect.2017026
##### 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. [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. [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. [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. [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. [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. [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. [8] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York. 1969. [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. [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. [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. [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. [13] J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial ekvac., 21 (1978), 11-41. [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. [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. [16] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Springer-verlag, Berlin, 1991. doi: 10.1007/BFb0084432. [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. [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. [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. [20] A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224.  doi: 10.1137/0521066. [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. [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. [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. [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. [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. [26] J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204.  doi: 10.1090/qam/295683. [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. [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. [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. [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. [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. [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. [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. [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. [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. [36] H. Tanabe, Equations of Evolution, Pitman Publishing, London, 1979. [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. [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. [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. [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. [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. [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.

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. [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. [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. [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. [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. [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. [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. [8] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York. 1969. [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. [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. [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. [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. [13] J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial ekvac., 21 (1978), 11-41. [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. [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. [16] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Springer-verlag, Berlin, 1991. doi: 10.1007/BFb0084432. [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. [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. [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. [20] A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224.  doi: 10.1137/0521066. [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. [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. [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. [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. [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. [26] J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204.  doi: 10.1090/qam/295683. [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. [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. [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. [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. [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. [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. [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. [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. [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. [36] H. Tanabe, Equations of Evolution, Pitman Publishing, London, 1979. [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. [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. [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. [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. [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. [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.
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