# American Institute of Mathematical Sciences

doi: 10.3934/eect.2021058
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## Approximate controllability of neutral delay integro-differential inclusion of order $\alpha\in (1, 2)$ with non-instantaneous impulses

 1 Department of Mathematics & Computer Science, Sri Sathya Sai Institute of Higher Learning, Prasanthi Nilayam (A.P.)-515134, India 2 Department of Mathematics & Scientific Computing, National Institute of Technology, Hamirpur (H.P.)-177005, India

* Corresponding author: Avadhesh Kumar (soni.iitkgp@gmail.com)

Received  June 2021 Revised  September 2021 Early access November 2021

This paper aims to establish the approximate controllability results for fractional neutral integro-differential inclusions with non-instantaneous impulse and infinite delay. Sufficient conditions for approximate controllability have been established for the proposed control problem. The tools for study include the fixed point theorem for discontinuous multi-valued operators with the $\alpha-$resolvent operator. Finally, the proposed results are illustrated with the help of an example.

Citation: Avadhesh Kumar, Ankit Kumar, Ramesh Kumar Vats, Parveen Kumar. Approximate controllability of neutral delay integro-differential inclusion of order $\alpha\in (1, 2)$ with non-instantaneous impulses. Evolution Equations and Control Theory, doi: 10.3934/eect.2021058
##### References:
 [1] N. Abada, M. Benchohra and H. Hammouche, Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, Journal of Differential Equations., 246 (2009), 3834-3863.  doi: 10.1016/j.jde.2009.03.004. [2] P. Balasubramaniam and P. Tamilalagan, Approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay by using Mainardis function, Applied Mathematics and Computation, 256 (2015), 232-246.  doi: 10.1016/j.amc.2015.01.035. [3] P. Balasubramaniam, V. Vembarasan and T. Senthilkumar, Approximate controllability of impulsive fractional integro-differential systems with nonlocal conditions in Hilbert space, Numerical and Functional Analysis, Optimization, 35 (2014), 177-197.  doi: 10.1080/01630563.2013.811420. [4] M. Benchohra, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, vol. 2, New York, 2006. doi: 10.1155/9789775945501. [5] B. C. Dhage, Fixed-point theorems for discontinuous multi-valued operators on ordered spaces with applications, Comput. Math. Appl., 51 (2006), 589-604.  doi: 10.1016/j.camwa.2005.07.017. [6] X. Fu and K. Mei, Approximate controllability of semilinear partial functional differential systems, Journal of Dynamics Control Systems, 15 (2009), 425-443.  doi: 10.1007/s10883-009-9068-x. [7] M. Guo, X. Xue and R. Li, Controllability of impulsive evolution inclusions with nonlocal conditions, Journal of Optimization Theory and Applications, 120 (2004), 355-374.  doi: 10.1023/B:JOTA.0000015688.53162.eb. [8] J. K. Hale and J. Kato, Phase spaces for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. [9] E. Hern and D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Am. Math. Soc., 141 (2013), 1641-1649.  doi: 10.1090/S0002-9939-2012-11613-2. [10] E. Hernández, A. Prokopczyk and L. Ladeira, A note on partial functional differential equations with state-dependent delay, Nonlinear Analysis: Real World Applications, 7 (2006), 510-519.  doi: 10.1016/j.nonrwa.2005.03.014. [11] A. Kumar, K. Jeet and R. K. Vats, Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space, Evolution Equation and Control Theory, 2021. doi: 10.3934/eect.2021016. [12] A. Kumar, M. Malik and R. Sakthivel, Controllability of the second order nonlinear differential equations with non-instantaneous impulses, J. Dynam. Control Systems, 24 (2018), 325-342.  doi: 10.1007/s10883-017-9376-5. [13] A. Kumar, R. K. Vats and A. Kumar, Approximate controllability of second-order non-autonomous system with finite delay, Journal of Dynamical and Control System, 26 (2020), 611-627.  doi: 10.1007/s10883-019-09475-0. [14] A. Kumar, R. K. Vats, A. Kumar and D. N. Chalishajar, Numerical approach to the controllability of fractional order impulsive differential equations, Demonstratio Mathematica., 53 (2020), 193-207.  doi: 10.1515/dema-2020-0015. [15] P. Kumar, D. N. Pandey and D. Bahuguna, On a new class of abstract impulsive functional differential equations of fractional order, Journal of Nonlinear Science and Applications, 7 (2014), 102-114.  doi: 10.22436/jnsa.007.02.04. [16] S. Kumar and R. Sakthivel, Constrained controllability of second order retarded nonlinear systems with nonlocal condition, IMA Journal of Mathematical Control and Information, 37 (2020), 441-454.  doi: 10.1093/imamci/dnz007. [17] M. Li and M. Liu, Approximate controllability of semilinear neutral stochastic integro-differential inclusions with infinite delay, Discrete Dynamics in Nature and Society, 2015 (2015), Art. ID 420826, 16 pp. doi: 10.1155/2015/420826. [18] M. D. P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems, Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-7614-8. [19] M. Muslim and A. Kumar, Trajectory controllability of fractional differential systems of order $\alpha \in (1, 2]$ with deviated argument, The Journal of Analysis, 28 (2020), 295-304.  doi: 10.1007/s41478-018-0081-x. [20] M. Muslim, A. Kumar and M. Fečkan, Existence, uniqueness and stability of solutions to second order nonlinear differential equations with non-instantaneous impulses, Journal of King Saud University-Science, 30 (2018), 204-213.  doi: 10.1016/j.jksus.2016.11.005. [21] D. N. Pandey, S. Das and N. Sukavanam, Existence of solution for a second-order neutral differential equation with state dependent delay and non-instantaneous impulses, International Journal of Nonlinear Science, 18 (2014), 145-155. [22] C. Ravichandran, N. Valliammal and J. J. Nieto, New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces, Journal of the Franklin Institute, 356 (2019), 1535-1565.  doi: 10.1016/j.jfranklin.2018.12.001. [23] R. Sakthivel, R. Ganesh, Y. Ren and S. M. Anthoni, Approximate controllability of nonlinear fractional dynamical systems, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 3498-3508.  doi: 10.1016/j.cnsns.2013.05.015. [24] R. Sakthivel, N. I. Mahmudov and J. H. Kim, On controllability of second-order nonlinear impulsive differential systems, Nonlinear Analysis, Theory Methods & Applications, 71 (2009), 45-52.  doi: 10.1016/j.na.2008.10.029. [25] J. P. C. dos Santos, M. M. Arjunan and C. Cuevas, Existence results for fractional neutral integro-differential equations with state-dependent delay, Computers and Mathematics with Applications, 62 (2011), 1275-1283.  doi: 10.1016/j.camwa.2011.03.048. [26] G. Shen, R. Sakthivel, Y. Ren and M. Li, Controllability and stability of fractional stochastic functional systems driven by Rosenblatt process, Collectanea Mathematica, 71 (2020), 63-82.  doi: 10.1007/s13348-019-00248-3. [27] Z. Yan, Approximate controllability of fractional neutral integro-differential inclusions with state-dependent delay in Hilbert spaces, IMA Journal of Mathematical Control and Information, 30 (2013), 443-462.  doi: 10.1093/imamci/dns033. [28] Z. Yan, On a nonlocal problem for fractional integrodifferential inclusions in Banach spaces, Annales Polonici Mathematici, 101 (2011), 87-103.  doi: 10.4064/ap101-1-9. [29] Z. Yan, Controllability of fractional-order partial neutral functional integrodifferential inclusions with infinite delay, Journal of the Franklin Institute, 348 (2011), 2156-2173.  doi: 10.1016/j.jfranklin.2011.06.009.

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##### References:
 [1] N. Abada, M. Benchohra and H. Hammouche, Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, Journal of Differential Equations., 246 (2009), 3834-3863.  doi: 10.1016/j.jde.2009.03.004. [2] P. Balasubramaniam and P. Tamilalagan, Approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay by using Mainardis function, Applied Mathematics and Computation, 256 (2015), 232-246.  doi: 10.1016/j.amc.2015.01.035. [3] P. Balasubramaniam, V. Vembarasan and T. Senthilkumar, Approximate controllability of impulsive fractional integro-differential systems with nonlocal conditions in Hilbert space, Numerical and Functional Analysis, Optimization, 35 (2014), 177-197.  doi: 10.1080/01630563.2013.811420. [4] M. Benchohra, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, vol. 2, New York, 2006. doi: 10.1155/9789775945501. [5] B. C. Dhage, Fixed-point theorems for discontinuous multi-valued operators on ordered spaces with applications, Comput. Math. Appl., 51 (2006), 589-604.  doi: 10.1016/j.camwa.2005.07.017. [6] X. Fu and K. Mei, Approximate controllability of semilinear partial functional differential systems, Journal of Dynamics Control Systems, 15 (2009), 425-443.  doi: 10.1007/s10883-009-9068-x. [7] M. Guo, X. Xue and R. Li, Controllability of impulsive evolution inclusions with nonlocal conditions, Journal of Optimization Theory and Applications, 120 (2004), 355-374.  doi: 10.1023/B:JOTA.0000015688.53162.eb. [8] J. K. Hale and J. Kato, Phase spaces for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. [9] E. Hern and D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Am. Math. Soc., 141 (2013), 1641-1649.  doi: 10.1090/S0002-9939-2012-11613-2. [10] E. Hernández, A. Prokopczyk and L. Ladeira, A note on partial functional differential equations with state-dependent delay, Nonlinear Analysis: Real World Applications, 7 (2006), 510-519.  doi: 10.1016/j.nonrwa.2005.03.014. [11] A. Kumar, K. Jeet and R. K. Vats, Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space, Evolution Equation and Control Theory, 2021. doi: 10.3934/eect.2021016. [12] A. Kumar, M. Malik and R. Sakthivel, Controllability of the second order nonlinear differential equations with non-instantaneous impulses, J. Dynam. Control Systems, 24 (2018), 325-342.  doi: 10.1007/s10883-017-9376-5. [13] A. Kumar, R. K. Vats and A. Kumar, Approximate controllability of second-order non-autonomous system with finite delay, Journal of Dynamical and Control System, 26 (2020), 611-627.  doi: 10.1007/s10883-019-09475-0. [14] A. Kumar, R. K. Vats, A. Kumar and D. N. Chalishajar, Numerical approach to the controllability of fractional order impulsive differential equations, Demonstratio Mathematica., 53 (2020), 193-207.  doi: 10.1515/dema-2020-0015. [15] P. Kumar, D. N. Pandey and D. Bahuguna, On a new class of abstract impulsive functional differential equations of fractional order, Journal of Nonlinear Science and Applications, 7 (2014), 102-114.  doi: 10.22436/jnsa.007.02.04. [16] S. Kumar and R. Sakthivel, Constrained controllability of second order retarded nonlinear systems with nonlocal condition, IMA Journal of Mathematical Control and Information, 37 (2020), 441-454.  doi: 10.1093/imamci/dnz007. [17] M. Li and M. Liu, Approximate controllability of semilinear neutral stochastic integro-differential inclusions with infinite delay, Discrete Dynamics in Nature and Society, 2015 (2015), Art. ID 420826, 16 pp. doi: 10.1155/2015/420826. [18] M. D. P. Monteiro Marques, Differential Inclusions in Nonsmooth Mechanical Problems, Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-7614-8. [19] M. Muslim and A. Kumar, Trajectory controllability of fractional differential systems of order $\alpha \in (1, 2]$ with deviated argument, The Journal of Analysis, 28 (2020), 295-304.  doi: 10.1007/s41478-018-0081-x. [20] M. Muslim, A. Kumar and M. Fečkan, Existence, uniqueness and stability of solutions to second order nonlinear differential equations with non-instantaneous impulses, Journal of King Saud University-Science, 30 (2018), 204-213.  doi: 10.1016/j.jksus.2016.11.005. [21] D. N. Pandey, S. Das and N. Sukavanam, Existence of solution for a second-order neutral differential equation with state dependent delay and non-instantaneous impulses, International Journal of Nonlinear Science, 18 (2014), 145-155. [22] C. Ravichandran, N. Valliammal and J. J. Nieto, New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces, Journal of the Franklin Institute, 356 (2019), 1535-1565.  doi: 10.1016/j.jfranklin.2018.12.001. [23] R. Sakthivel, R. Ganesh, Y. Ren and S. M. Anthoni, Approximate controllability of nonlinear fractional dynamical systems, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 3498-3508.  doi: 10.1016/j.cnsns.2013.05.015. [24] R. Sakthivel, N. I. Mahmudov and J. H. Kim, On controllability of second-order nonlinear impulsive differential systems, Nonlinear Analysis, Theory Methods & Applications, 71 (2009), 45-52.  doi: 10.1016/j.na.2008.10.029. [25] J. P. C. dos Santos, M. M. Arjunan and C. Cuevas, Existence results for fractional neutral integro-differential equations with state-dependent delay, Computers and Mathematics with Applications, 62 (2011), 1275-1283.  doi: 10.1016/j.camwa.2011.03.048. [26] G. Shen, R. Sakthivel, Y. Ren and M. Li, Controllability and stability of fractional stochastic functional systems driven by Rosenblatt process, Collectanea Mathematica, 71 (2020), 63-82.  doi: 10.1007/s13348-019-00248-3. [27] Z. Yan, Approximate controllability of fractional neutral integro-differential inclusions with state-dependent delay in Hilbert spaces, IMA Journal of Mathematical Control and Information, 30 (2013), 443-462.  doi: 10.1093/imamci/dns033. [28] Z. Yan, On a nonlocal problem for fractional integrodifferential inclusions in Banach spaces, Annales Polonici Mathematici, 101 (2011), 87-103.  doi: 10.4064/ap101-1-9. [29] Z. Yan, Controllability of fractional-order partial neutral functional integrodifferential inclusions with infinite delay, Journal of the Franklin Institute, 348 (2011), 2156-2173.  doi: 10.1016/j.jfranklin.2011.06.009.
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