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February  2022, 11(1): 67-93. doi: 10.3934/eect.2020103

Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces

Soniya Singh 1, , Sumit Arora 1, , Manil T. Mohan 2, and  Jaydev Dabas 1,,

1. 

Department of Applied Science and Engineering, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand 247667, India
2. 

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand 247667, India

*Corresponding author: Jaydev Dabas

Received  June 2020 Revised  September 2020 Published  February 2022 Early access  November 2020

In this paper, we consider the second order semilinear impulsive differential equations with state-dependent delay. First, we consider a linear second order system and establish the approximate controllability result by using a feedback control. Then, we obtain sufficient conditions for the approximate controllability of the considered system in a separable, reflexive Banach space via properties of the resolvent operator and Schauder's fixed point theorem. Finally, we apply our results to investigate the approximate controllability of the impulsive wave equation with state-dependent delay.

Keywords: Impulsive functional differential equations, approximate controllability, Schauder's fixed point theorem, cosine family, resolvent operator.
Mathematics Subject Classification: 34K06, 34A12, 37L05, 93B05.
Citation: Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2022, 11 (1) : 67-93. doi: 10.3934/eect.2020103
References:
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O. Arino, K. Boushaba and A. Boussouar, A mathematical model of the dynamics of the phytoplankton-nutrient system. Spatial hetrogeneity in ecological models, Nonlinear Anal. Real World Appl., 1 (2000), 69-87.  doi: 10.1016/S0362-546X(99)00394-6.  Google Scholar

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S. Arora, M. T. Mohan, and J. Dabas, Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces, accepted in Math. Control. Relat. Fields, (2020). Google Scholar

[4]

S. Arora, S. Singh, J. Dabas and M. T. Mohan, Approximate controllability of semilinear impulsive functional differential system with nonlocal conditions, IMA J. Math. Control Inform., (2020). doi: 10.1093/imamci/dnz037.  Google Scholar

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U. Arora and N. Sukavanam, Approximate controllability of second order semilinear stochastic system with variable delay in control and with nonlocal conditions, Rend. Circ. Mat. Palermo (2), 65 (2016), 307-322.  doi: 10.1007/s12215-016-0235-0.  Google Scholar

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G. Arthi and K. Balachandran, Controllability of second-order impulsive evolution systems with infinite delay, Nonlinear Anal. Hybrid Syst., 11 (2014), 139-153.  doi: 10.1016/j.nahs.2013.08.001.  Google Scholar

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K. Balachandran and S. M. Anthoni, Controllability of second-order semilinear neutral functional differential systems in Banach spaces, Comput. Math. Appl., 41 (2001), 1223-1235.  doi: 10.1016/S0898-1221(01)00093-1.  Google Scholar

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K. Balachandran and J. P. Dauer, Controllability of nonlinear systems in Banach spaces: A survey, J. Optim. Theory Appl., 115 (2002), 7-28.  doi: 10.1023/A:1019668728098.  Google Scholar

[9]

V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems, Mathematics in Science and Engineering, 190, Academic Press, Inc., Boston, MA, 1993.  Google Scholar

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A. E. Bashirov and N. I. Mahmudov, On concepts of controllability for deterministic and stochastic systems, SIAM J. Control Optim., 37 (1999), 1808-1821.  doi: 10.1137/S036301299732184X.  Google Scholar

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A. Bobrowski, The Widder-Arendt theorem on inverting of the Laplace transform, and its relationships with the theory of semigroups of operators, Methods Funct. Anal. Topology, 3 (1997), 1-39.   Google Scholar

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J. Bochenek, An abstract nonlinear second order differential equation, Ann. Polon. Math., 54 (1991), 155-166.  doi: 10.4064/ap-54-2-155-166.  Google Scholar

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D. N. Chalishajar and A. Kumar, Total controllability of the second order semi-linear differential equation with infinite delay and non-instantaneous impulses, Math. Comput. Appl., 23 (2018), 13pp. doi: 10.3390/mca23030032.  Google Scholar

[14]

Y. K. Chang, J. J. Nieto and W. S. Li, Controllability of semilinear differential systems with nonlocal initial conditions in Banach spaces, J. Optim. Theory Appl., 142 (2009), 267-273.  doi: 10.1007/s10957-009-9535-2.  Google Scholar

[15]

F. Chen, D. Sun and J. Shi, Periodicity in a food-limited population model with toxicants and state dependent delays, J. Math. Anal. Appl., 288 (2003), 136-146.  doi: 10.1016/S0022-247X(03)00586-9.  Google Scholar

[16]

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H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North-Holland Mathematics Studies, 108, North-Holland Publishing Co., North-Holland, Amsterdam, 1985.  Google Scholar

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C. Gao, K. Li, E. Feng and Z. Xiu, Nonlinear impulse system of fed-batch culture in fermentative production and its properties, Chaos Solitons Fractals, 28 (2006), 271-277.  doi: 10.1016/j.chaos.2005.05.027.  Google Scholar

[23]

J. Ginibre, A. Soffer and G. Velo, The global Cauchy problem for the critical nonlinear wave equation, J. Funct. Anal., 110 (1992), 96-130.  doi: 10.1016/0022-1236(92)90044-J.  Google Scholar

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A. Grudzka and K. Rykaczewski, On approximate controllability of functional impulsive evolution inclusions in a Hilbert space, J. Optim. Theory Appl., 166 (2015), 414-439.  doi: 10.1007/s10957-014-0671-y.  Google Scholar

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E. Hernández, R. Sakthivel and S. Tanaka Aki, Existence results for impulsive evolution differential equations with state-dependent delay, Electron. J. Differential Equations, (2008), 11pp.  Google Scholar

[26]

Y. Hino, S. Murakami and T. Naito, Functional-Differential Equations with Infinite Delay, Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.  Google Scholar

[27]

L. F. Ho, Exact controllability of the one-dimensional wave equation with locally distributed control, SIAM J. Control Optim., 28 (1990), 733-748.  doi: 10.1137/0328043.  Google Scholar

[28]

J.-R. Kang, Y.-C. Kwun and J.-Y. Park, Controllability of the second-order differential inclusion in Banach spaces, J. Math. Anal. Appl., 285 (2003), 537-550.  doi: 10.1016/S0022-247X(03)00423-2.  Google Scholar

[29]

J. Kisyński, On cosine operator functions and one-parameter groups of operators, Studia Math., 44 (1972), 93-105.  doi: 10.4064/sm-44-1-93-105.  Google Scholar

[30]

A. Kumar, R. K. Vats and A. Kumar, Approximate controllability of second-order non-autonomous system with finite delay, J. Dyn. Control Syst., 26 (2020), 611-627.  doi: 10.1007/s10883-019-09475-0.  Google Scholar

[31]

S. Kumar and N. Sukavanam, Controllability of second-order systems with nonlocal conditions in Banach spaces, Numer. Funct. Anal. Optim., 35 (2014), 423-431.  doi: 10.1080/01630563.2013.814067.  Google Scholar

[32]

M. Li and M. Huang, Approximate controllability of second-order impulsive stochastic differential equations with state-dependent delay, J. Appl. Anal. Comput., 8 (2018), 598–619.x doi: 10.11948/2018.598.  Google Scholar

[33]

M. Li and J. Ma, Approximate controllability of second order impulsive functional differential systems with infinite delay in Banach space, J. Appl. Anal. Comput., 6 (2016), 492-514.  doi: 10.11948/2016036.  Google Scholar

[34]

T. Li and Y. Zhou, Cauchy problem of one-dimensional nonlinear wave equations, in Nonlinear Wave Equations, Series in Contemporary Mathematics, 2, Springer, Berlin, Heidelberg, 2017,161–181. Google Scholar

[35]

X. J. Li and J. M. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[36]

J. Liang, J. H. Liu and T.-J. Xiao, Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Math. Comput. Modelling, 49 (2009), 798-804.  doi: 10.1016/j.mcm.2008.05.046.  Google Scholar

[37]

N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim., 42 (2003), 1604-1622.  doi: 10.1137/S0363012901391688.  Google Scholar

[38]

N. I. Mahmudov, V. Vijayakumar and R. Murugesu, Approximate controllability of second-order evolution differential inclusions in Hilbert spaces, Mediterr. J. Math., 13 (2016), 3433-3454.  doi: 10.1007/s00009-016-0695-7.  Google Scholar

[39]

E. Marschall, Remarks on normal operators on Banach spaces, Rend. Circ. Mat. Palermo (2), 35 (1986), 317-329.  doi: 10.1007/BF02843901.  Google Scholar

[40]

V. Obukhovskii and J.-C. Yao, On impulsive functional differential inclusions with Hille-Yosida operators in Banach spaces, Nonlinear Anal., 73 (2010), 1715-1728.  doi: 10.1016/j.na.2010.05.009.  Google Scholar

[41]

T. W. Palmer, Unbounded normal operators on Banach spaces, Trans. Amer. Math. Soc., 133 (1968), 385-414.  doi: 10.1090/S0002-9947-1968-0231213-6.  Google Scholar

[42]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[43]

K. Ravikumar, M. T. Mohan and A. Anguraj, Approximate controllability of a non-autonomous evolution equation in Banach spaces, Numer. Algebra Control Optim., (2020). doi: 10.3934/naco.2020038.  Google Scholar

[44]

R. Sakthivel, E. R. Anandhi and N. I. Mahmudov, Approximate controllability of second-order systems with state-dependent delay, Numer. Funct. Anal. Optim., 29 (2008), 1347-1362.  doi: 10.1080/01630560802580901.  Google Scholar

[45]

R. Sakthivel, N. I. Mahmudov and J. H. Kim, Approximate controllability of nonlinear impulsive differential systems, Rep. Math. Phys., 60 (2007), 85-96.  doi: 10.1016/S0034-4877(07)80100-5.  Google Scholar

[46]

R. Sakthivel, J. J. Nieto and N. I. Mahmudov, Approximate controllability of nonlinear deterministic and stochastic systems with unbounded delay, Taiwanese J. Math., 14 (2010), 1777-1797.  doi: 10.11650/twjm/1500406016.  Google Scholar

[47]

A. M. Samo${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$lenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 14, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. doi: 10.1142/9789812798664.  Google Scholar

[48]

S. Tang and L. Chen, Density-dependent birth rate, birth pulses and their population dynamic consequences, J. Math. Biol., 44 (2002), 185-199.  doi: 10.1007/s002850100121.  Google Scholar

[49]

C. C. Travis and G. F. Webb, Compactness, regularity, and uniform continuity properties of strongly continuous cosine families, Houston J. Math., 3 (1977), 555-567.   Google Scholar

[50]

C. C. Travis and G. F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hungar., 32 (1978), 75-96.  doi: 10.1007/BF01902205.  Google Scholar

[51]

C. C. Travis and G. F. Webb, Second order differential equations in Banach space, in Nonlinear Equations in Abstract Spaces, Academic Press, New York 1978,331–361.  Google Scholar

[52]

R. Triggiani, A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optim., 15 (1977), 407-411.  doi: 10.1137/0315028.  Google Scholar

show all references

References:
[1]

O. Arino, K. Boushaba and A. Boussouar, A mathematical model of the dynamics of the phytoplankton-nutrient system. Spatial hetrogeneity in ecological models, Nonlinear Anal. Real World Appl., 1 (2000), 69-87.  doi: 10.1016/S0362-546X(99)00394-6.  Google Scholar

[2]

S. Arora, M. T. Mohan and J. Dabas, Approximate controllability of the non-autonomous impulsive evolution equation with state-dependent delay in Banach space, accepted in Nonlinear Anal. Hybrid System. Google Scholar

[3]

S. Arora, M. T. Mohan, and J. Dabas, Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces, accepted in Math. Control. Relat. Fields, (2020). Google Scholar

[4]

S. Arora, S. Singh, J. Dabas and M. T. Mohan, Approximate controllability of semilinear impulsive functional differential system with nonlocal conditions, IMA J. Math. Control Inform., (2020). doi: 10.1093/imamci/dnz037.  Google Scholar

[5]

U. Arora and N. Sukavanam, Approximate controllability of second order semilinear stochastic system with variable delay in control and with nonlocal conditions, Rend. Circ. Mat. Palermo (2), 65 (2016), 307-322.  doi: 10.1007/s12215-016-0235-0.  Google Scholar

[6]

G. Arthi and K. Balachandran, Controllability of second-order impulsive evolution systems with infinite delay, Nonlinear Anal. Hybrid Syst., 11 (2014), 139-153.  doi: 10.1016/j.nahs.2013.08.001.  Google Scholar

[7]

K. Balachandran and S. M. Anthoni, Controllability of second-order semilinear neutral functional differential systems in Banach spaces, Comput. Math. Appl., 41 (2001), 1223-1235.  doi: 10.1016/S0898-1221(01)00093-1.  Google Scholar

[8]

K. Balachandran and J. P. Dauer, Controllability of nonlinear systems in Banach spaces: A survey, J. Optim. Theory Appl., 115 (2002), 7-28.  doi: 10.1023/A:1019668728098.  Google Scholar

[9]

V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems, Mathematics in Science and Engineering, 190, Academic Press, Inc., Boston, MA, 1993.  Google Scholar

[10]

A. E. Bashirov and N. I. Mahmudov, On concepts of controllability for deterministic and stochastic systems, SIAM J. Control Optim., 37 (1999), 1808-1821.  doi: 10.1137/S036301299732184X.  Google Scholar

[11]

A. Bobrowski, The Widder-Arendt theorem on inverting of the Laplace transform, and its relationships with the theory of semigroups of operators, Methods Funct. Anal. Topology, 3 (1997), 1-39.   Google Scholar

[12]

J. Bochenek, An abstract nonlinear second order differential equation, Ann. Polon. Math., 54 (1991), 155-166.  doi: 10.4064/ap-54-2-155-166.  Google Scholar

[13]

D. N. Chalishajar and A. Kumar, Total controllability of the second order semi-linear differential equation with infinite delay and non-instantaneous impulses, Math. Comput. Appl., 23 (2018), 13pp. doi: 10.3390/mca23030032.  Google Scholar

[14]

Y. K. Chang, J. J. Nieto and W. S. Li, Controllability of semilinear differential systems with nonlocal initial conditions in Banach spaces, J. Optim. Theory Appl., 142 (2009), 267-273.  doi: 10.1007/s10957-009-9535-2.  Google Scholar

[15]

F. Chen, D. Sun and J. Shi, Periodicity in a food-limited population model with toxicants and state dependent delays, J. Math. Anal. Appl., 288 (2003), 136-146.  doi: 10.1016/S0022-247X(03)00586-9.  Google Scholar

[16]

R. F. Curtain and H. Zwart,, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar

[17]

S. Y. Dobrokhotov, V. E. Nazaikinskii and B. Tirozzi, Asymptotic solution of the one-dimensional wave equation with localized initial data and with degenerating velocity: I, Russ. J. Math. Phys., 17 (2010), 434-447.  doi: 10.1134/S1061920810040059.  Google Scholar

[18]

I. Ekeland and T. Turnbull, Infinite-Dimensional Optimization and Convexity, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1983.  Google Scholar

[19]

H. O. Fattorini, Ordinary differential equations in linear topological spaces. I, J. Differential Equations, 5 (1969), 72-105.  doi: 10.1016/0022-0396(69)90105-3.  Google Scholar

[20]

H.O. Fattorini, Ordinary differential equations in linear topological spaces. II, J. Differential Equations, 6 (1969), 50-70.  doi: 10.1016/0022-0396(69)90117-X.  Google Scholar

[21]

H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North-Holland Mathematics Studies, 108, North-Holland Publishing Co., North-Holland, Amsterdam, 1985.  Google Scholar

[22]

C. Gao, K. Li, E. Feng and Z. Xiu, Nonlinear impulse system of fed-batch culture in fermentative production and its properties, Chaos Solitons Fractals, 28 (2006), 271-277.  doi: 10.1016/j.chaos.2005.05.027.  Google Scholar

[23]

J. Ginibre, A. Soffer and G. Velo, The global Cauchy problem for the critical nonlinear wave equation, J. Funct. Anal., 110 (1992), 96-130.  doi: 10.1016/0022-1236(92)90044-J.  Google Scholar

[24]

A. Grudzka and K. Rykaczewski, On approximate controllability of functional impulsive evolution inclusions in a Hilbert space, J. Optim. Theory Appl., 166 (2015), 414-439.  doi: 10.1007/s10957-014-0671-y.  Google Scholar

[25]

E. Hernández, R. Sakthivel and S. Tanaka Aki, Existence results for impulsive evolution differential equations with state-dependent delay, Electron. J. Differential Equations, (2008), 11pp.  Google Scholar

[26]

Y. Hino, S. Murakami and T. Naito, Functional-Differential Equations with Infinite Delay, Lecture Notes in Mathematics, 1473, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0084432.  Google Scholar

[27]

L. F. Ho, Exact controllability of the one-dimensional wave equation with locally distributed control, SIAM J. Control Optim., 28 (1990), 733-748.  doi: 10.1137/0328043.  Google Scholar

[28]

J.-R. Kang, Y.-C. Kwun and J.-Y. Park, Controllability of the second-order differential inclusion in Banach spaces, J. Math. Anal. Appl., 285 (2003), 537-550.  doi: 10.1016/S0022-247X(03)00423-2.  Google Scholar

[29]

J. Kisyński, On cosine operator functions and one-parameter groups of operators, Studia Math., 44 (1972), 93-105.  doi: 10.4064/sm-44-1-93-105.  Google Scholar

[30]

A. Kumar, R. K. Vats and A. Kumar, Approximate controllability of second-order non-autonomous system with finite delay, J. Dyn. Control Syst., 26 (2020), 611-627.  doi: 10.1007/s10883-019-09475-0.  Google Scholar

[31]

S. Kumar and N. Sukavanam, Controllability of second-order systems with nonlocal conditions in Banach spaces, Numer. Funct. Anal. Optim., 35 (2014), 423-431.  doi: 10.1080/01630563.2013.814067.  Google Scholar

[32]

M. Li and M. Huang, Approximate controllability of second-order impulsive stochastic differential equations with state-dependent delay, J. Appl. Anal. Comput., 8 (2018), 598–619.x doi: 10.11948/2018.598.  Google Scholar

[33]

M. Li and J. Ma, Approximate controllability of second order impulsive functional differential systems with infinite delay in Banach space, J. Appl. Anal. Comput., 6 (2016), 492-514.  doi: 10.11948/2016036.  Google Scholar

[34]

T. Li and Y. Zhou, Cauchy problem of one-dimensional nonlinear wave equations, in Nonlinear Wave Equations, Series in Contemporary Mathematics, 2, Springer, Berlin, Heidelberg, 2017,161–181. Google Scholar

[35]

X. J. Li and J. M. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[36]

J. Liang, J. H. Liu and T.-J. Xiao, Nonlocal impulsive problems for nonlinear differential equations in Banach spaces, Math. Comput. Modelling, 49 (2009), 798-804.  doi: 10.1016/j.mcm.2008.05.046.  Google Scholar

[37]

N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim., 42 (2003), 1604-1622.  doi: 10.1137/S0363012901391688.  Google Scholar

[38]

N. I. Mahmudov, V. Vijayakumar and R. Murugesu, Approximate controllability of second-order evolution differential inclusions in Hilbert spaces, Mediterr. J. Math., 13 (2016), 3433-3454.  doi: 10.1007/s00009-016-0695-7.  Google Scholar

[39]

E. Marschall, Remarks on normal operators on Banach spaces, Rend. Circ. Mat. Palermo (2), 35 (1986), 317-329.  doi: 10.1007/BF02843901.  Google Scholar

[40]

V. Obukhovskii and J.-C. Yao, On impulsive functional differential inclusions with Hille-Yosida operators in Banach spaces, Nonlinear Anal., 73 (2010), 1715-1728.  doi: 10.1016/j.na.2010.05.009.  Google Scholar

[41]

T. W. Palmer, Unbounded normal operators on Banach spaces, Trans. Amer. Math. Soc., 133 (1968), 385-414.  doi: 10.1090/S0002-9947-1968-0231213-6.  Google Scholar

[42]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[43]

K. Ravikumar, M. T. Mohan and A. Anguraj, Approximate controllability of a non-autonomous evolution equation in Banach spaces, Numer. Algebra Control Optim., (2020). doi: 10.3934/naco.2020038.  Google Scholar

[44]

R. Sakthivel, E. R. Anandhi and N. I. Mahmudov, Approximate controllability of second-order systems with state-dependent delay, Numer. Funct. Anal. Optim., 29 (2008), 1347-1362.  doi: 10.1080/01630560802580901.  Google Scholar

[45]

R. Sakthivel, N. I. Mahmudov and J. H. Kim, Approximate controllability of nonlinear impulsive differential systems, Rep. Math. Phys., 60 (2007), 85-96.  doi: 10.1016/S0034-4877(07)80100-5.  Google Scholar

[46]

R. Sakthivel, J. J. Nieto and N. I. Mahmudov, Approximate controllability of nonlinear deterministic and stochastic systems with unbounded delay, Taiwanese J. Math., 14 (2010), 1777-1797.  doi: 10.11650/twjm/1500406016.  Google Scholar

[47]

A. M. Samo${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$lenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 14, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. doi: 10.1142/9789812798664.  Google Scholar

[48]

S. Tang and L. Chen, Density-dependent birth rate, birth pulses and their population dynamic consequences, J. Math. Biol., 44 (2002), 185-199.  doi: 10.1007/s002850100121.  Google Scholar

[49]

C. C. Travis and G. F. Webb, Compactness, regularity, and uniform continuity properties of strongly continuous cosine families, Houston J. Math., 3 (1977), 555-567.   Google Scholar

[50]

C. C. Travis and G. F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hungar., 32 (1978), 75-96.  doi: 10.1007/BF01902205.  Google Scholar

[51]

C. C. Travis and G. F. Webb, Second order differential equations in Banach space, in Nonlinear Equations in Abstract Spaces, Academic Press, New York 1978,331–361.  Google Scholar

[52]

R. Triggiani, A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optim., 15 (1977), 407-411.  doi: 10.1137/0315028.  Google Scholar

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