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On some damped 2 body problems
Approximate controllability of second order impulsive systems with state-dependent delay 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 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.
References:
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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. |
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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 |
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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 |
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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. |
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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. |
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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. |
[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. |
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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. |
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V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems, Mathematics in
Science and Engineering, 190, Academic Press, Inc., Boston, MA, 1993. |
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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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
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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.
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H.O. Fattorini,
Ordinary differential equations in linear topological spaces. II, J. Differential Equations, 6 (1969), 50-70.
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[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[39] |
E. Marschall,
Remarks on normal operators on Banach spaces, Rend. Circ. Mat. Palermo (2), 35 (1986), 317-329.
doi: 10.1007/BF02843901. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems, Mathematics in
Science and Engineering, 190, Academic Press, Inc., Boston, MA, 1993. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
I. Ekeland and T. Turnbull, Infinite-Dimensional Optimization and Convexity, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1983. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[39] |
E. Marschall,
Remarks on normal operators on Banach spaces, Rend. Circ. Mat. Palermo (2), 35 (1986), 317-329.
doi: 10.1007/BF02843901. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
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