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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 |
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.
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. |
[2] |
O. Arino, M. 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. |
[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. |
[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. |
[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. |
[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. |
[7] |
V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, New York, 1993.
![]() |
[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. |
[9] |
W. M. Bian,
Approximate controllability of semilinear systems, Acta Math. Hungar., 81 (1998), 41-57.
doi: 10.1023/A:1006510809870. |
[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. |
[11] |
J. M. Borwein and J. Vanderwerff,
Fréchet-Legendre functions and reflexive Banach spaces, J. Convex Anal., 17 (2010), 915-924.
|
[12] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. |
[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. |
[14] |
Y.-K. Chang, A. 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. |
[15] |
P. Chen, X. 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. |
[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. |
[18] |
I. Ekeland and T. Turnbull, Infinite Dimensional Optimization and Convexity, Chicago press, London, 1983.
![]() |
[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. |
[20] |
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. |
[21] |
S. Gao, L. Chen, J. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[27] |
J. Klamka,
Constrained controllability of semilinear systems with delays, Nonlinear Dyn., 56 (2009), 169-177.
doi: 10.1007/s11071-008-9389-4. |
[28] |
J. Klamka,
Schauder's fixed point theorem in nonlinear controllability problems, Control Cybernet., 29 (2000), 153-165.
|
[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. |
[30] |
J. Klamka, A. 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. |
[31] |
J. Klamka, A. 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. |
[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. |
[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. |
[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. |
[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. |
[36] |
A. Lunardi,
On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224.
doi: 10.1137/0521066. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[43] |
K. Naito,
Approximate controllability for a semilinear control system, J. Optim. Theory Appl., 60 (1989), 57-65.
doi: 10.1007/BF00938799. |
[44] |
J. W. Nunziato,
On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204.
doi: 10.1090/qam/295683. |
[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. |
[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. |
[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. |
[48] |
R. Sakthivel, N. 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. |
[49] |
A. M. Samoilenko, N. A. Perestyuk and Y. Chapovsky, Impulsive Differential Equations, World Scientific, Singapore, 1995.
doi: 10.1142/9789812798664. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[56] |
J. Wang, M. 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. |
[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. |
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. |
[2] |
O. Arino, M. 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. |
[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. |
[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. |
[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. |
[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. |
[7] |
V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, New York, 1993.
![]() |
[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. |
[9] |
W. M. Bian,
Approximate controllability of semilinear systems, Acta Math. Hungar., 81 (1998), 41-57.
doi: 10.1023/A:1006510809870. |
[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. |
[11] |
J. M. Borwein and J. Vanderwerff,
Fréchet-Legendre functions and reflexive Banach spaces, J. Convex Anal., 17 (2010), 915-924.
|
[12] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. |
[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. |
[14] |
Y.-K. Chang, A. 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. |
[15] |
P. Chen, X. 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. |
[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. |
[18] |
I. Ekeland and T. Turnbull, Infinite Dimensional Optimization and Convexity, Chicago press, London, 1983.
![]() |
[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. |
[20] |
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. |
[21] |
S. Gao, L. Chen, J. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[27] |
J. Klamka,
Constrained controllability of semilinear systems with delays, Nonlinear Dyn., 56 (2009), 169-177.
doi: 10.1007/s11071-008-9389-4. |
[28] |
J. Klamka,
Schauder's fixed point theorem in nonlinear controllability problems, Control Cybernet., 29 (2000), 153-165.
|
[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. |
[30] |
J. Klamka, A. 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. |
[31] |
J. Klamka, A. 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. |
[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. |
[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. |
[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. |
[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. |
[36] |
A. Lunardi,
On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224.
doi: 10.1137/0521066. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[43] |
K. Naito,
Approximate controllability for a semilinear control system, J. Optim. Theory Appl., 60 (1989), 57-65.
doi: 10.1007/BF00938799. |
[44] |
J. W. Nunziato,
On heat conduction in materials with memory, Quart. Appl. Math., 29 (1971), 187-204.
doi: 10.1090/qam/295683. |
[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. |
[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. |
[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. |
[48] |
R. Sakthivel, N. 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. |
[49] |
A. M. Samoilenko, N. A. Perestyuk and Y. Chapovsky, Impulsive Differential Equations, World Scientific, Singapore, 1995.
doi: 10.1142/9789812798664. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[56] |
J. Wang, M. 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. |
[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. |
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