doi: 10.3934/mcrf.2020049

Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces

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  April 2020 Revised  July 2020 Published  November 2020

In this paper, we investigate the approximate controllability problems of certain Sobolev type differential equations. Here, we obtain sufficient conditions for the approximate controllability of a semilinear Sobolev type evolution system in Banach spaces. In order to establish the approximate controllability results of such a system, we have employed the resolvent operator condition and Schauder's fixed point theorem. Finally, we discuss a concrete example to illustrate the efficiency of the results obtained.

Citation: Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2020049
References:
[1]

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K. Balachandran and N. Annapoorani, Existence results for impulsive neutral evolution integrodifferential equations with infinite delay, Nonlinear Anal. Hybrid Syst., 3 (2009), 674-684.  doi: 10.1016/j.nahs.2009.06.004.  Google Scholar

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K. Balachandran and J. Y. Park, Sobolev type integrodifferential equation with nonlocal condition in Banach spaces, Taiwanese J. Math., 7 (2003), 155-163.  doi: 10.11650/twjm/1500407525.  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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W. M. Bian, Approximate controllability of semilinear systems, Acta Math. Hungar., 81 (1998), 41-57.  doi: 10.1023/A:1006510809870.  Google Scholar

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W. M. Bian, Controllability of nonlinear evolution systems with preassigned responses, J. Optim. Theory Appl., 100 (1999), 265-285.  doi: 10.1023/A:1021726017996.  Google Scholar

[11]

J. M. Borwein and J. Vanderwerff, Fréchet-Legendre functions and reflexive Banach spaces, J. Convex Anal., 17 (2010), 915-924.   Google Scholar

[12]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.  Google Scholar

[13]

H. Brill, A semilinear Sobolev evolution equation in Banach space, J. Differential Equations, 24 (1977), 412-425.  doi: 10.1016/0022-0396(77)90009-2.  Google Scholar

[14]

Y.-K. ChangA. Pereira and R. Ponce, Approximate controllability for fractional differential equations of Sobolev type via properties on resolvent operators, Fract. Calc. Appl. Anal., 20 (2017), 963-987.  doi: 10.1515/fca-2017-0050.  Google Scholar

[15]

P. ChenX. Zhang and Y. Li, Approximate controllability of non-autonomous evolution system with nonlocal conditions, J. Dyn. Control Syst., 26 (2020), 1-16.  doi: 10.1007/s10883-018-9423-x.  Google Scholar

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S. GaoL. ChenJ. J. Nieto and A. Torres, Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, 24 (2006), 6037-6045.  doi: 10.1016/j.vaccine.2006.05.018.  Google Scholar

[22]

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[23]

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

[24]

E. Hernández, R. Sakthivel and S. Tanaka Aki, Existence results for impulsive evolution differential equations with state-dependent delay, Electron. J. Differential Equations, 2008 (2008), 28, 1–11. doi: EJDE-2008/28.  Google Scholar

[25]

J.-M. Jeong and H.-H. Roh, Approximate controllability for semilinear retarded systems, J. Appl. Math. Anal. Appl., 321 (2006), 961-975.  doi: 10.1016/j.jmaa.2005.09.005.  Google Scholar

[26]

M. Kerboua, A. Debbouche and D. Baleanu, Approximate controllability of Sobolev type nonlocal fractional stochastic dynamic systems in Hilbert spaces, Abstr. Appl. Anal., 2013 (2013), 262191, 10pp. doi: 10.1155/2013/262191.  Google Scholar

[27]

J. Klamka, Constrained controllability of semilinear systems with delays, Nonlinear Dyn., 56 (2009), 169-177.  doi: 10.1007/s11071-008-9389-4.  Google Scholar

[28]

J. Klamka, Schauder's fixed point theorem in nonlinear controllability problems, Control Cybernet., 29 (2000), 153-165.   Google Scholar

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J. Klamka, Controllability and Minimum Energy Control, in Series Studies in Systems, Decision and Control, Springer-Verlag, New York, 2019. doi: 10.1007/978-3-319-92540-0.  Google Scholar

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J. KlamkaA. Babiarz and M. Niezabitowski, Banach fixed-point theorem in semilinear controllability problems–a survey, Bull. Polish Acad. Sci. Tech. Sci., 64 (2016), 21-35.  doi: 10.1515/bpasts-2016-0004.  Google Scholar

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J. KlamkaA. Babiarz and M. Niezabitowski, Schauder's fixed point theorem in approximate controllability problems, Int. J. Appl. Math. Comput. Sci., 26 (2016), 263-275.  doi: 10.1515/amcs-2016-0018.  Google Scholar

[32]

K. D. Kucche and M. B. Dhakne, Sobolev typen Volterra-Fredholmfunctional integrodifferential equations in Banach spaces, Bol. Soc. Parana. Mat., 32 (2014), 239-255.  doi: 10.5269/bspm.v32i1.19901.  Google Scholar

[33]

H. Leiva and P. Sundar, Approximate controllability of the Burgers equation with impulses and delay, Far East J. Math. Sci., 102 (2017), 2291-2306.  doi: 10.17654/MS102102291.  Google Scholar

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[36]

A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224.  doi: 10.1137/0521066.  Google Scholar

[37]

N. I. Mahmudov, Approximate controllability of fractional Sobolev type evolution equations in Banach Spaces, Abstr. Appl. Anal., 2013 (2013), 502839, 1–9. doi: 10.1155/2013/502839.  Google Scholar

[38]

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

[39]

N. I. Mahmudov, Existence and approximate controllability of Sobolev type fractional stochastic evolution equations, Bull. Polish Acad. Sci. Tech. Sci., 62 (2014), 205-215.  doi: 10.2478/bpasts-2014-0020.  Google Scholar

[40]

M. McKibben, A note on the approximate controllability of a class of abstract semilinear evolution equations, Far East J. Math. Sci., 5 (2002), 113-133.   Google Scholar

[41]

M. T. Mohan, On the three dimensional Kelvin-Voigt fluids: Global solvability, exponential stability and exact controllability of Galerkin approximations, Evol. Equ. Control Theory, 9 (2020), 301-339.  doi: 10.3934/eect.2020007.  Google Scholar

[42]

K. Naito, Controllability of semilinear control systems dominated by the linear part, SIAM J. Math. Anal., 25 (1987), 715-722.  doi: 10.1137/0325040.  Google Scholar

[43]

K. Naito, Approximate controllability for a semilinear control system, J. Optim. Theory Appl., 60 (1989), 57-65.  doi: 10.1007/BF00938799.  Google Scholar

[44]

J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204.  doi: 10.1090/qam/295683.  Google Scholar

[45]

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

[46]

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

[47]

R. Sakthivel and E. R. Anandhi, Approximate controllability of impulsive differential equations with state-dependent delay, Internat. J. Control, 83 (2010), 387-393.  doi: 10.1080/00207170903171348.  Google Scholar

[48]

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

[49]

A. M. Samoilenko, N. A. Perestyuk and Y. Chapovsky, Impulsive Differential Equations, World Scientific, Singapore, 1995. doi: 10.1142/9789812798664.  Google Scholar

[50]

R. E. Showalter, Existence and representation theorem for a semilinear Sobolev equation in Banach space, SIAM J. Math. Anal., 3 (1972), 527-543.  doi: 10.1137/0503051.  Google Scholar

[51]

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

[52]

R. Triggiani, Addendum:A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optim., 18 (1980), 98-99.  doi: 10.1137/0318007.  Google Scholar

[53]

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

[54]

V. Vijayakumar, Approximate controllability results for impulsive neutral differential inclusions of Sobolev type with infinite delay, Internat. J. Control, 91 (2018), 2366-2386.  doi: 10.1080/00207179.2017.1346300.  Google Scholar

[55]

L. Wang, Approximate controllability of delayed semilinear control systems, J. Appl. Math. Stoch. Anal., 2005 (2005), 67-76.  doi: 10.1155/JAMSA.2005.67.  Google Scholar

[56]

J. WangM. Fečkan and Y. Zhou, Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions, Evol. Equ. Control Theory, 6 (2017), 471-486.  doi: 10.3934/eect.2017024.  Google Scholar

[57]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, Handbook of Differential Equations: Evolutionary Equations, 3 (2007), 527-621.  doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

show all references

References:
[1]

S. Agarwal and D. Bahuguna, Existence of solutions to Sobolev type partial neutral differential equations, J. Appl. Math. Stoch. Anal., 2006 (2006), 16308, 1–10. doi: 10.1155/JAMSA/2006/16308.  Google Scholar

[2]

O. ArinoM. L. Habid and R. Bravo de la Parra, A mathematical model of growth of population of fish in the larval stage: Density dependence effects, Math. Biosci., 150 (1998), 1-20.  doi: 10.1016/S0025-5564(98)00008-X.  Google Scholar

[3]

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

[4]

K. Balachandran and N. Annapoorani, Existence results for impulsive neutral evolution integrodifferential equations with infinite delay, Nonlinear Anal. Hybrid Syst., 3 (2009), 674-684.  doi: 10.1016/j.nahs.2009.06.004.  Google Scholar

[5]

K. Balachandran and T. N. Gopal, Approximate controllability of nonlinear evolution systems with time varying delays, IMA J. Math. Control Inform., 23 (2006), 499-513.  doi: 10.1093/imamci/dnl002.  Google Scholar

[6]

K. Balachandran and J. Y. Park, Sobolev type integrodifferential equation with nonlocal condition in Banach spaces, Taiwanese J. Math., 7 (2003), 155-163.  doi: 10.11650/twjm/1500407525.  Google Scholar

[7] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, New York, 1993.   Google Scholar
[8]

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

[9]

W. M. Bian, Approximate controllability of semilinear systems, Acta Math. Hungar., 81 (1998), 41-57.  doi: 10.1023/A:1006510809870.  Google Scholar

[10]

W. M. Bian, Controllability of nonlinear evolution systems with preassigned responses, J. Optim. Theory Appl., 100 (1999), 265-285.  doi: 10.1023/A:1021726017996.  Google Scholar

[11]

J. M. Borwein and J. Vanderwerff, Fréchet-Legendre functions and reflexive Banach spaces, J. Convex Anal., 17 (2010), 915-924.   Google Scholar

[12]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.  Google Scholar

[13]

H. Brill, A semilinear Sobolev evolution equation in Banach space, J. Differential Equations, 24 (1977), 412-425.  doi: 10.1016/0022-0396(77)90009-2.  Google Scholar

[14]

Y.-K. ChangA. Pereira and R. Ponce, Approximate controllability for fractional differential equations of Sobolev type via properties on resolvent operators, Fract. Calc. Appl. Anal., 20 (2017), 963-987.  doi: 10.1515/fca-2017-0050.  Google Scholar

[15]

P. ChenX. Zhang and Y. Li, Approximate controllability of non-autonomous evolution system with nonlocal conditions, J. Dyn. Control Syst., 26 (2020), 1-16.  doi: 10.1007/s10883-018-9423-x.  Google Scholar

[16]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, New York, 1999. Google Scholar

[17]

V. N. Do, A note on approximate controllability of semilinear systems, Systems Control Lett., 12 (1989), 365-371.  doi: 10.1016/0167-6911(89)90047-9.  Google Scholar

[18] I. Ekeland and T. Turnbull, Infinite Dimensional Optimization and Convexity, Chicago press, London, 1983.   Google Scholar
[19]

W. E. Fitzgibbon, Semilinear functional differential equations in Banach spaces, J. Differential Equations, 29 (1978), 1-14.  doi: 10.1016/0022-0396(78)90037-2.  Google Scholar

[20]

C. GaoK. LiE. 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

[21]

S. GaoL. ChenJ. J. Nieto and A. Torres, Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, 24 (2006), 6037-6045.  doi: 10.1016/j.vaccine.2006.05.018.  Google Scholar

[22]

R. K. George, Approximate controllability of non-autonomous semilinear systems, Nonlinear Anal., 24 (1995), 1377-1393.  doi: 10.1016/0362-546X(94)E0082-R.  Google Scholar

[23]

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

[24]

E. Hernández, R. Sakthivel and S. Tanaka Aki, Existence results for impulsive evolution differential equations with state-dependent delay, Electron. J. Differential Equations, 2008 (2008), 28, 1–11. doi: EJDE-2008/28.  Google Scholar

[25]

J.-M. Jeong and H.-H. Roh, Approximate controllability for semilinear retarded systems, J. Appl. Math. Anal. Appl., 321 (2006), 961-975.  doi: 10.1016/j.jmaa.2005.09.005.  Google Scholar

[26]

M. Kerboua, A. Debbouche and D. Baleanu, Approximate controllability of Sobolev type nonlocal fractional stochastic dynamic systems in Hilbert spaces, Abstr. Appl. Anal., 2013 (2013), 262191, 10pp. doi: 10.1155/2013/262191.  Google Scholar

[27]

J. Klamka, Constrained controllability of semilinear systems with delays, Nonlinear Dyn., 56 (2009), 169-177.  doi: 10.1007/s11071-008-9389-4.  Google Scholar

[28]

J. Klamka, Schauder's fixed point theorem in nonlinear controllability problems, Control Cybernet., 29 (2000), 153-165.   Google Scholar

[29]

J. Klamka, Controllability and Minimum Energy Control, in Series Studies in Systems, Decision and Control, Springer-Verlag, New York, 2019. doi: 10.1007/978-3-319-92540-0.  Google Scholar

[30]

J. KlamkaA. Babiarz and M. Niezabitowski, Banach fixed-point theorem in semilinear controllability problems–a survey, Bull. Polish Acad. Sci. Tech. Sci., 64 (2016), 21-35.  doi: 10.1515/bpasts-2016-0004.  Google Scholar

[31]

J. KlamkaA. Babiarz and M. Niezabitowski, Schauder's fixed point theorem in approximate controllability problems, Int. J. Appl. Math. Comput. Sci., 26 (2016), 263-275.  doi: 10.1515/amcs-2016-0018.  Google Scholar

[32]

K. D. Kucche and M. B. Dhakne, Sobolev typen Volterra-Fredholmfunctional integrodifferential equations in Banach spaces, Bol. Soc. Parana. Mat., 32 (2014), 239-255.  doi: 10.5269/bspm.v32i1.19901.  Google Scholar

[33]

H. Leiva and P. Sundar, Approximate controllability of the Burgers equation with impulses and delay, Far East J. Math. Sci., 102 (2017), 2291-2306.  doi: 10.17654/MS102102291.  Google Scholar

[34]

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

[35]

J. H. Lightbourne and S. M. Rankin, A partial functional differential equation of Sobolev type, J. Appl. Math. Anal. Appl., 93 (1983), 328-337.  doi: 10.1016/0022-247X(83)90178-6.  Google Scholar

[36]

A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224.  doi: 10.1137/0521066.  Google Scholar

[37]

N. I. Mahmudov, Approximate controllability of fractional Sobolev type evolution equations in Banach Spaces, Abstr. Appl. Anal., 2013 (2013), 502839, 1–9. doi: 10.1155/2013/502839.  Google Scholar

[38]

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

[39]

N. I. Mahmudov, Existence and approximate controllability of Sobolev type fractional stochastic evolution equations, Bull. Polish Acad. Sci. Tech. Sci., 62 (2014), 205-215.  doi: 10.2478/bpasts-2014-0020.  Google Scholar

[40]

M. McKibben, A note on the approximate controllability of a class of abstract semilinear evolution equations, Far East J. Math. Sci., 5 (2002), 113-133.   Google Scholar

[41]

M. T. Mohan, On the three dimensional Kelvin-Voigt fluids: Global solvability, exponential stability and exact controllability of Galerkin approximations, Evol. Equ. Control Theory, 9 (2020), 301-339.  doi: 10.3934/eect.2020007.  Google Scholar

[42]

K. Naito, Controllability of semilinear control systems dominated by the linear part, SIAM J. Math. Anal., 25 (1987), 715-722.  doi: 10.1137/0325040.  Google Scholar

[43]

K. Naito, Approximate controllability for a semilinear control system, J. Optim. Theory Appl., 60 (1989), 57-65.  doi: 10.1007/BF00938799.  Google Scholar

[44]

J. W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204.  doi: 10.1090/qam/295683.  Google Scholar

[45]

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

[46]

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

[47]

R. Sakthivel and E. R. Anandhi, Approximate controllability of impulsive differential equations with state-dependent delay, Internat. J. Control, 83 (2010), 387-393.  doi: 10.1080/00207170903171348.  Google Scholar

[48]

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

[49]

A. M. Samoilenko, N. A. Perestyuk and Y. Chapovsky, Impulsive Differential Equations, World Scientific, Singapore, 1995. doi: 10.1142/9789812798664.  Google Scholar

[50]

R. E. Showalter, Existence and representation theorem for a semilinear Sobolev equation in Banach space, SIAM J. Math. Anal., 3 (1972), 527-543.  doi: 10.1137/0503051.  Google Scholar

[51]

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

[52]

R. Triggiani, Addendum:A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optim., 18 (1980), 98-99.  doi: 10.1137/0318007.  Google Scholar

[53]

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

[54]

V. Vijayakumar, Approximate controllability results for impulsive neutral differential inclusions of Sobolev type with infinite delay, Internat. J. Control, 91 (2018), 2366-2386.  doi: 10.1080/00207179.2017.1346300.  Google Scholar

[55]

L. Wang, Approximate controllability of delayed semilinear control systems, J. Appl. Math. Stoch. Anal., 2005 (2005), 67-76.  doi: 10.1155/JAMSA.2005.67.  Google Scholar

[56]

J. WangM. Fečkan and Y. Zhou, Approximate controllability of Sobolev type fractional evolution systems with nonlocal conditions, Evol. Equ. Control Theory, 6 (2017), 471-486.  doi: 10.3934/eect.2017024.  Google Scholar

[57]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, Handbook of Differential Equations: Evolutionary Equations, 3 (2007), 527-621.  doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

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K. Ravikumar, Manil T. Mohan, A. Anguraj. Approximate controllability of a non-autonomous evolution equation in Banach spaces. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 461-485. doi: 10.3934/naco.2020038

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