# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2021068
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## Controllability of Sobolev type fuzzy differential equation with non-instantaneous impulsive condition

 School of Basic Sciences, Indian Institute of Technology Mandi, Kamand (H.P.) - 175005, India

* Corresponding author: Muslim Malik

Received  August 2020 Revised  March 2021 Early access June 2021

In this manuscript, we investigate the existence, uniqueness and controllability results of a Sobolev type fuzzy differential equation with non-instantaneous impulsive conditions. Non-linear functional analysis, Banach fixed point theorem and fuzzy theory are the main techniques used to establish these results. In support, an example is given to validate the obtained analytical findings.

Citation: Muslim Malik, Anjali Rose, Anil Kumar. Controllability of Sobolev type fuzzy differential equation with non-instantaneous impulsive condition. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021068
##### References:
 [1] S. Agarwal and D. Bahuguna, Existence of solutions to Sobolev-type partial neutral differential equations, International Journal of Stochastic Analysis, 163 (2006), 1-10.  doi: 10.1155/JAMSA/2006/16308.  Google Scholar [2] R. P. Agarwal, D. Baleanu, J. J. Nieto, D. F. Torres and Y. Zhou, A survey on fuzzy fractional differential and optimal control nonlocal evolution equations, Journal of Computational and Applied Mathematics, 339 (2018), 3-29.  doi: 10.1016/j.cam.2017.09.039.  Google Scholar [3] N. U. Ahmed, K. L. Teo and S. H. Hou, Nonlinear impulsive systems on infinite dimensional spaces, Nonlinear Analysis, 54 (2003), 907-925.  doi: 10.1016/S0362-546X(03)00117-2.  Google Scholar [4] S. Arora, M. T. Mohan and J. Dabas, Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces, Mathematical Control and Related Fields, (2020). doi: 10.3934/mcrf. 2020049.  Google Scholar [5] G. Arthi and K. Balachandran, Controllability of second order impulsive evolution systems with infinite delay, Nonlinear Analysis: Hybrid Systems, 11 (2014), 139-153.  doi: 10.1016/j.nahs.2013.08.001.  Google Scholar [6] K. Balachandran, S. Kiruthika and J. J. Trujillo, On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces, Computers and Mathematics with Applications, 62 (2011), 1157-1165.  doi: 10.1016/j.camwa.2011.03.031.  Google Scholar [7] K. Balachandran and J. Y. Park, Sobolev type integrodifferential equation with nonlocal condition in Banach spaces, Taiwanese Journal of Mathematics, 7 (2003), 155-163.  doi: 10.11650/twjm/1500407525.  Google Scholar [8] K. Balachandran and J. P. Dauer, Controllability of functional differential systems of Sobolev type in Banach spaces, Kybernetika, 34 (1998), 349-357.   Google Scholar [9] P. Balasubramaniam and S. Muralisankar, Existence and uniqueness of a fuzzy solution for nonlinear neutral functional differential equiations, Computers and Mathematics with Applications, 42 (2001), 961-967.  doi: 10.1016/S0898-1221(01)00212-7.  Google Scholar [10] P. Balasubramaniam and S. Muralisankar, Existence and uniqueness of fuzzy solution for semilinear fuzzy integrodifferential equations with nonlocal conditions, Computers and Mathematics with Applications, 47 (2004), 1115-1122.  doi: 10.1016/S0898-1221(04)90091-0.  Google Scholar [11] B. Bede and S. G. Gal, Generalizations of the differentiability of fuzzy number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and System, 151 (2005), 581-599.  doi: 10.1016/j.fss.2004.08.001.  Google Scholar [12] A. Bencsik, B. Bede, J. Tar and J. Fodor, Fuzzy differential equations in modeling hydraulic differential servo cylinders, in Third Romanian-Hungarian Joint Symposium on Applied Computational Intelligence (SACI), Timisoara, Romania, (2006). Google Scholar [13] A. Boudaoui and A. Slama, Existence and controllability results for Sobolev-type fractional impulsive stochastic differential equations with infinite delay, Journal of Mathematics and Applications, 40 (2017), 37-58.  doi: 10.7862/rf.2017.3.  Google Scholar [14] J. Casasnovas and F. Rossell, Averaging fuzzy biopolymers, Fuzzy Sets and Systems, 152 (2005), 139-158.  doi: 10.1016/j.fss.2004.10.019.  Google Scholar [15] P. Diamond and P. E. Kloeden, Metric spaces of fuzzy sets: Theory and applications, World Scientific, (1994). doi: 10.1142/2326.  Google Scholar [16] D. Dubois and H. Prade, Towards fuzzy differential calculus part 1: Integration of fuzzy mappings, Fuzzy Sets and System, 8 (1982), 1-7.  doi: 10.1016/0165-0114(82)90025-2.  Google Scholar [17] D. Dubois and H. Prade, Towards fuzzy differential calculus part 2: Integration on fuzzy intervals, Fuzzy Sets and System, 8 (1982), 105-116.  doi: 10.1016/0165-0114(82)90001-X.  Google Scholar [18] M. Guo, X. Xue and R. Li, Impulsive functional differential inclusions and fuzzy population models, Fuzzy Sets and Systems, 138 (2003), 601-615.  doi: 10.1016/S0165-0114(02)00522-5.  Google Scholar [19] J. H. Jeong, J. S. Kim, H. E. Youm and J. H. Park, Exact controllability for fuzzy differential equations using extremal solutions, Journal of Computational Analysis and Applications, 23 (2017), 1056-1069.   Google Scholar [20] A. Kandel and W. J. Byatt, Fuzzy differential equations, Proceedings of the International Conference on Cybernetics and Society, Tokyo, Japan, 1 (1978), 1213-1216.   Google Scholar [21] M. Kumar and S. Kumar, Controllability of impulsive second order semilinear fuzzy integrodifferential control systems with nonlocal initial conditions, Applied Soft Computing, 39 (2016), 251-265.  doi: 10.1016/j.asoc.2015.10.006.  Google Scholar [22] 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.  Google Scholar [23] Y. Kwun, J. Kim, M. Park and J. Park, Nonlocal controllability for the semilinear fuzzy integrodifferential equations in n-dimensional fuzzy vector space, Advances in Difference Equations, 2009 (2009), 1-6.  doi: 10.1155/2009/734090.  Google Scholar [24] Y. Kwun, J. Kim, M. Park and J. Park, Controllability for the impulsive semilinear nonlocal fuzzy integrodifferential equations in n-dimensional fuzzy vector spacey, Advances in Difference Equations, 2010 (2010), 1-22.  doi: 10.1186/1687-1847-2010-983483.  Google Scholar [25] B. Liu, A survey of credibility theory, Fuzzy Optimization and Decision Making, 5 (2006), 387-408.  doi: 10.1007/s10700-006-0016-x.  Google Scholar [26] M. Malik, R. Dhayal, S. Abbas and A. Kumar, Controllability of non-autonomous nonlinear differential system with non-instantaneous impulses, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, Serie A, Matemáticas, 113 (2019), 103-118.  doi: 10.1007/s13398-017-0454-z.  Google Scholar [27] A. Meraj and D. N. Pandey, Approximate controllability of nonlocal non-autonomous Sobolev type evolution equations, An International Journal of Optimization and Control: Theories and Applications (IJOCTA), 9 (2019), 86-94.  doi: 10.11121/ijocta.01.2019.00644.  Google Scholar [28] A. Meraj and D. N. Pandey, Approximate controllability of non-autonomous Sobolev type integro-differential equations having nonlocal and non-instantaneous impulsive conditions, Indian Journal of Pure and Applied Mathematics, 51 (2020), 501-518.  doi: 10.1007/s13226-020-0413-9.  Google Scholar [29] M. Mizumoto and K. Tanaka, Some Properties of Fuzzy Numbers, North-Holland, 1979.  Google Scholar [30] M. Muslim, A. Kumar and R. Sakthivel, Exact and trajectory controllability of second order evolution systems with impulses and deviated arguments, Mathematical Methods in the Applied Sciences, 41 (2018), 4259-4272.  doi: 10.1002/mma.4888.  Google Scholar [31] M. Muslim and R. P. Agarwal, Exact controllability of an integro-differential equation with deviated argument, Functional Differential Equations, 21 (2014), 31-45.   Google Scholar [32] J. H. Park, J. S. Park, Y. C. Ahn and Y. C. Kwun, Controllability for the impulsive semilinear fuzzy integrodifferential equations, Springer, 40 (2007), 704-713.  doi: 10.1007/978-3-540-71441-5_76.  Google Scholar [33] J. H. Park, J. S. Park and Y. C. Kwun, Controllability for the semilinear fuzzy integrodifferential equations with nonlocal conditions, in International Conference on Fuzzy Systems and Knowledge Discovery, Springer, Berlin, Heidelberg, (2006), 221-230. doi: 10.1007/11881599_25.  Google Scholar [34] 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.  Google Scholar [35] R. E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space, SIAM Journal on Mathematical Analysis, 3 (1972), 527-543.  doi: 10.1137/0503051.  Google Scholar [36] J. Wang, M. Feckan and A. Debbouche, Time optimal control of a system governed by non-instantaneous impulsive differential equations, Journal of Optimization Theory and Applications, 182 (2019), 573-587.  doi: 10.1007/s10957-018-1313-6.  Google Scholar [37] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.  Google Scholar [38] H. J. Zimmermann, Fuzzy set theory and its applications, Springer Science and Business Media, (2011). doi: 10.1007/978-94-010-0646-0.  Google Scholar

show all references

##### References:
 [1] S. Agarwal and D. Bahuguna, Existence of solutions to Sobolev-type partial neutral differential equations, International Journal of Stochastic Analysis, 163 (2006), 1-10.  doi: 10.1155/JAMSA/2006/16308.  Google Scholar [2] R. P. Agarwal, D. Baleanu, J. J. Nieto, D. F. Torres and Y. Zhou, A survey on fuzzy fractional differential and optimal control nonlocal evolution equations, Journal of Computational and Applied Mathematics, 339 (2018), 3-29.  doi: 10.1016/j.cam.2017.09.039.  Google Scholar [3] N. U. Ahmed, K. L. Teo and S. H. Hou, Nonlinear impulsive systems on infinite dimensional spaces, Nonlinear Analysis, 54 (2003), 907-925.  doi: 10.1016/S0362-546X(03)00117-2.  Google Scholar [4] S. Arora, M. T. Mohan and J. Dabas, Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces, Mathematical Control and Related Fields, (2020). doi: 10.3934/mcrf. 2020049.  Google Scholar [5] G. Arthi and K. Balachandran, Controllability of second order impulsive evolution systems with infinite delay, Nonlinear Analysis: Hybrid Systems, 11 (2014), 139-153.  doi: 10.1016/j.nahs.2013.08.001.  Google Scholar [6] K. Balachandran, S. Kiruthika and J. J. Trujillo, On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces, Computers and Mathematics with Applications, 62 (2011), 1157-1165.  doi: 10.1016/j.camwa.2011.03.031.  Google Scholar [7] K. Balachandran and J. Y. Park, Sobolev type integrodifferential equation with nonlocal condition in Banach spaces, Taiwanese Journal of Mathematics, 7 (2003), 155-163.  doi: 10.11650/twjm/1500407525.  Google Scholar [8] K. Balachandran and J. P. Dauer, Controllability of functional differential systems of Sobolev type in Banach spaces, Kybernetika, 34 (1998), 349-357.   Google Scholar [9] P. Balasubramaniam and S. Muralisankar, Existence and uniqueness of a fuzzy solution for nonlinear neutral functional differential equiations, Computers and Mathematics with Applications, 42 (2001), 961-967.  doi: 10.1016/S0898-1221(01)00212-7.  Google Scholar [10] P. Balasubramaniam and S. Muralisankar, Existence and uniqueness of fuzzy solution for semilinear fuzzy integrodifferential equations with nonlocal conditions, Computers and Mathematics with Applications, 47 (2004), 1115-1122.  doi: 10.1016/S0898-1221(04)90091-0.  Google Scholar [11] B. Bede and S. G. Gal, Generalizations of the differentiability of fuzzy number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and System, 151 (2005), 581-599.  doi: 10.1016/j.fss.2004.08.001.  Google Scholar [12] A. Bencsik, B. Bede, J. Tar and J. Fodor, Fuzzy differential equations in modeling hydraulic differential servo cylinders, in Third Romanian-Hungarian Joint Symposium on Applied Computational Intelligence (SACI), Timisoara, Romania, (2006). Google Scholar [13] A. Boudaoui and A. Slama, Existence and controllability results for Sobolev-type fractional impulsive stochastic differential equations with infinite delay, Journal of Mathematics and Applications, 40 (2017), 37-58.  doi: 10.7862/rf.2017.3.  Google Scholar [14] J. Casasnovas and F. Rossell, Averaging fuzzy biopolymers, Fuzzy Sets and Systems, 152 (2005), 139-158.  doi: 10.1016/j.fss.2004.10.019.  Google Scholar [15] P. Diamond and P. E. Kloeden, Metric spaces of fuzzy sets: Theory and applications, World Scientific, (1994). doi: 10.1142/2326.  Google Scholar [16] D. Dubois and H. Prade, Towards fuzzy differential calculus part 1: Integration of fuzzy mappings, Fuzzy Sets and System, 8 (1982), 1-7.  doi: 10.1016/0165-0114(82)90025-2.  Google Scholar [17] D. Dubois and H. Prade, Towards fuzzy differential calculus part 2: Integration on fuzzy intervals, Fuzzy Sets and System, 8 (1982), 105-116.  doi: 10.1016/0165-0114(82)90001-X.  Google Scholar [18] M. Guo, X. Xue and R. Li, Impulsive functional differential inclusions and fuzzy population models, Fuzzy Sets and Systems, 138 (2003), 601-615.  doi: 10.1016/S0165-0114(02)00522-5.  Google Scholar [19] J. H. Jeong, J. S. Kim, H. E. Youm and J. H. Park, Exact controllability for fuzzy differential equations using extremal solutions, Journal of Computational Analysis and Applications, 23 (2017), 1056-1069.   Google Scholar [20] A. Kandel and W. J. Byatt, Fuzzy differential equations, Proceedings of the International Conference on Cybernetics and Society, Tokyo, Japan, 1 (1978), 1213-1216.   Google Scholar [21] M. Kumar and S. Kumar, Controllability of impulsive second order semilinear fuzzy integrodifferential control systems with nonlocal initial conditions, Applied Soft Computing, 39 (2016), 251-265.  doi: 10.1016/j.asoc.2015.10.006.  Google Scholar [22] 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.  Google Scholar [23] Y. Kwun, J. Kim, M. Park and J. Park, Nonlocal controllability for the semilinear fuzzy integrodifferential equations in n-dimensional fuzzy vector space, Advances in Difference Equations, 2009 (2009), 1-6.  doi: 10.1155/2009/734090.  Google Scholar [24] Y. Kwun, J. Kim, M. Park and J. Park, Controllability for the impulsive semilinear nonlocal fuzzy integrodifferential equations in n-dimensional fuzzy vector spacey, Advances in Difference Equations, 2010 (2010), 1-22.  doi: 10.1186/1687-1847-2010-983483.  Google Scholar [25] B. Liu, A survey of credibility theory, Fuzzy Optimization and Decision Making, 5 (2006), 387-408.  doi: 10.1007/s10700-006-0016-x.  Google Scholar [26] M. Malik, R. Dhayal, S. Abbas and A. Kumar, Controllability of non-autonomous nonlinear differential system with non-instantaneous impulses, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, Serie A, Matemáticas, 113 (2019), 103-118.  doi: 10.1007/s13398-017-0454-z.  Google Scholar [27] A. Meraj and D. N. Pandey, Approximate controllability of nonlocal non-autonomous Sobolev type evolution equations, An International Journal of Optimization and Control: Theories and Applications (IJOCTA), 9 (2019), 86-94.  doi: 10.11121/ijocta.01.2019.00644.  Google Scholar [28] A. Meraj and D. N. Pandey, Approximate controllability of non-autonomous Sobolev type integro-differential equations having nonlocal and non-instantaneous impulsive conditions, Indian Journal of Pure and Applied Mathematics, 51 (2020), 501-518.  doi: 10.1007/s13226-020-0413-9.  Google Scholar [29] M. Mizumoto and K. Tanaka, Some Properties of Fuzzy Numbers, North-Holland, 1979.  Google Scholar [30] M. Muslim, A. Kumar and R. Sakthivel, Exact and trajectory controllability of second order evolution systems with impulses and deviated arguments, Mathematical Methods in the Applied Sciences, 41 (2018), 4259-4272.  doi: 10.1002/mma.4888.  Google Scholar [31] M. Muslim and R. P. Agarwal, Exact controllability of an integro-differential equation with deviated argument, Functional Differential Equations, 21 (2014), 31-45.   Google Scholar [32] J. H. Park, J. S. Park, Y. C. Ahn and Y. C. Kwun, Controllability for the impulsive semilinear fuzzy integrodifferential equations, Springer, 40 (2007), 704-713.  doi: 10.1007/978-3-540-71441-5_76.  Google Scholar [33] J. H. Park, J. S. Park and Y. C. Kwun, Controllability for the semilinear fuzzy integrodifferential equations with nonlocal conditions, in International Conference on Fuzzy Systems and Knowledge Discovery, Springer, Berlin, Heidelberg, (2006), 221-230. doi: 10.1007/11881599_25.  Google Scholar [34] 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.  Google Scholar [35] R. E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space, SIAM Journal on Mathematical Analysis, 3 (1972), 527-543.  doi: 10.1137/0503051.  Google Scholar [36] J. Wang, M. Feckan and A. Debbouche, Time optimal control of a system governed by non-instantaneous impulsive differential equations, Journal of Optimization Theory and Applications, 182 (2019), 573-587.  doi: 10.1007/s10957-018-1313-6.  Google Scholar [37] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.  Google Scholar [38] H. J. Zimmermann, Fuzzy set theory and its applications, Springer Science and Business Media, (2011). doi: 10.1007/978-94-010-0646-0.  Google Scholar
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