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December  2021, 11(4): 857-883. 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  December 2021 Early access  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 and Related Fields, 2021, 11 (4) : 857-883. doi: 10.3934/mcrf.2020049
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. 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.

[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. 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.

[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.

[16]

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

[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. 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.

[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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

[16]

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

[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. 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.

[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.

[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. 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.

[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.

[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. 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.

[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. 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.

[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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