American Institute of Mathematical Sciences

September  2021, 10(3): 619-631. doi: 10.3934/eect.2020083

Results on controllability of non-densely characterized neutral fractional delay differential system

 1 Department of Mathematics, Sri Eshwar College of Engineering(Autonomous), Coimbatore - 641 202, Tamil Nadu, India 2 Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chepauk, Chennai - 600 005, Tamil Nadu, India 3 Department of Mathematics, GMR Institute of Technology, Rajam - 532 127, Andhra Pradesh, India 4 Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin Abdulaziz University, Wadi Aldawasir 11991, Saudi Arabia 5 Post Graduate and Research Department of Mathematics, Kongunadu Arts and Science College(Autonomous), Coimbatore - 641 029, Tamil Nadu, India

* Corresponding author: Chokkalingam Ravichandran

Received  March 2020 Revised  May 2020 Published  September 2021 Early access  July 2020

This work establishes the controllability of nondense fractional neutral delay differential equation under Hille-Yosida condition in Banach space. The outcomes are derived with the aid of fractional calculus theory, semigroup operator theory and Schauder fixed point theorem. Theoretical results are verified through illustration.

Citation: Kasthurisamy Jothimani, Kalimuthu Kaliraj, Sumati Kumari Panda, Kotakkaran Sooppy Nisar, Chokkalingam Ravichandran. Results on controllability of non-densely characterized neutral fractional delay differential system. Evolution Equations and Control Theory, 2021, 10 (3) : 619-631. doi: 10.3934/eect.2020083
References:
 [1] L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162 (1991), 494-505.  doi: 10.1016/0022-247X(91)90164-U. [2] L. Byszewski and H. Akca, On a mild solution of a semilinear functional-differential evolution nonlocal problem, J. Appl. Math. Stoch. Anal., 10 (1997), 265-271.  doi: 10.1155/S1048953397000336. [3] Y. K. Chang, Controllability of impulsive functional differential systems with infinite delay in Banach spaces, Chaos Solitons Fractals, 33 (2007), 1601-1609.  doi: 10.1016/j.chaos.2006.03.006. [4] Y. K. Chang, A. Anguraj and M. Mallika Arjunan, Existence results for non-densely defined neutral impulsive differential inclusions with nonlocal conditions, J. Appl. Math. Comput., 28 (2008), 79-91.  doi: 10.1007/s12190-008-0078-8. [5] X. Fu and X. Liu, Controllability of non-densely defined neutral functional differential systems in abstract space, Chinese Ann. Math. B, 28 (2007), 243-252.  doi: 10.1007/s11401-005-0028-9. [6] B. Ghanbari, S. Kumar and R. Kumar, A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative, Chaos Solitons Fractals, 133 (2020), 109619. doi: 10.1016/j.chaos.2020.109619. [7] W. Gao, P. Veeresha, D. G. Prakasha, H. M. Baskonus and G. Yel, New numerical results for the time-fractional phi-four equation using a novel analytical approach, Symmetry, 12 (2020), 478. doi: 10.3390/sym12030478. [8] E. P. Gatsori, Controllability results for nondensely defined evolution differential inclusions with nonlocal conditions, J. Math. Anal. Appl., 297 (2004), 194-211.  doi: 10.1016/j.jmaa.2004.04.055. [9] H. Gu, Y. Zhou, B. Ahmad and A. Alsaedi, Integral solutions of fractional evolution equations with nondense domain, Electronic J. Differ. Eq., 2017 (2017), 1-15. [10] F. Jarad, T. Abdeljawad and D. Baleanu, Fractional variational optimal control problems with delayed arguments, Nonlinear Dyn., 62 (2010), 609-614.  doi: 10.1007/s11071-010-9748-9. [11] F. Jarad, T. Abdeljawad and D. Baleanu, Fractional variational principles with delay within caputo derivatives, Rep. Math. Phys., 65 (2010), 17-28.  doi: 10.1016/S0034-4877(10)00010-8. [12] V. Kavitha and M. M. Arjunan, Controllability of non-densely defined impulsive neutral functional differential systems with infinite delay in Banach spaces, Nonlinear Anal. Hybri., 4 (2010), 441-450.  doi: 10.1016/j.nahs.2009.11.002. [13] H. Kellerman and M. Hieber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180.  doi: 10.1016/0022-1236(89)90116-X. [14] A. Kumar and D. N. Pandey, Controllability results for nondensely defined impulsive fractional differential equations in abstract space, Differ. Equ. Dyn. Syst., (2019). doi: 10.1007/s12591-019-00471-1. [15] S. Kumar, R. Kumar, J. Singh, K. S. Nisar and D. Kumar, An efficient numerical scheme for fractional model of HIV-1 infection of $CD4^+$ T-cells with the effect of antiviral drug therapy, Alex. Eng. J., (2020). doi: 10.1016/j.aej.2019.12.046. [16] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, in North-Holland Mathematics Studies, 204, Elsevier Science, Amsterdam, 2006. [17] K. D. Kucche and S. T. Sutar, On Existence and stability results for nonlinear fractional delay differential equations, Bulletin of Parana's Mathematical Society, 36 (2018), 55-75.  doi: 10.5269/bspm.v36i4.33603. [18] N. I. Mahmudov, R. Murugesu, C. Ravichandran and V. Vijayakumar, Approximate controllability results for fractional semilinear integro-differential inclusions in Hilbert spaces, Results Math., 71 (2017), 45-61.  doi: 10.1007/s00025-016-0621-0. [19] T. A. Maraaba, F. Jarad and D. Baleanu, On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives, Sci. China Ser. A, 51 (2008), 1775-1786.  doi: 10.1007/s11425-008-0068-1. [20] T. A. Maraaba, D. Baleanu and F. Jarad, Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, J. Math. Phys., 49 (2008), 083507. doi: 10.1063/1.2970709. [21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, New York, Springer-verlag, 1983. doi: 10.1007/978-1-4612-5561-1. [22] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, an Diego, 1999. [23] C. Ravichandran and J. J. Trujillo, Controllability of impulsive fractional functional integro-differential equations in Banach spaces, J. Funct. Spaces, 2013 (2013), 1-8.  doi: 10.1155/2013/812501. [24] C. Ravichandran, K. Jothimani, H. M. Baskonus and N. Valliammal, New results on nondensely characterized integrodifferential equations with fractional order, Eur. Phys. J. Plus, 133 (2018), 1-10. [25] C. Ravichandran, N. Valliammal and J. J. Nieto, New results on exact controllability of a class of neutral integrodifferential systems with state dependent delay in Banach spaces, J. Franklin Inst., 356 (2019), 1535-1565.  doi: 10.1016/j.jfranklin.2018.12.001. [26] S. J. Sadati, D. Baleanu, A. Ranjbar, R. Ghaderi and T. Abdeljawad, Mittag-Leffler stability theorem for fractional nonlinear systems with delay, Abstr. Appl. Anal., 2010 (2010), 1-7.  doi: 10.1155/2010/108651. [27] R. Sakthivel, R. Ganesh and S. M. Anthoni, Approximate controllability of fractional nonlinear differential inclusions, Appl. Math. Comput., 225 (2013), 708-717.  doi: 10.1016/j.amc.2013.09.068. [28] J. V. D. C. Sousa, E. C. de Oliveira and K. D. Kucche, On the fractional functional differential equation with abstract volterra operator, B. Braz. Math.l Soc., 50 (2019), 803-822.  doi: 10.1007/s00574-019-00139-y. [29] P. Veeresha, D. G. Prakasha, H. M. Baskonus and G. Yel, An efficient analytical approach for fractional Lakshmanan-Porsezian-Daniel model, Math. Methods Appl. Sci., 43 (2020), 4136-4155.  doi: 10.1002/mma.6179. [30] V. Vijayakumar, C. Ravichandran, R. Murugesu and J. J. Trujillo, Controllability results for a class of fractional semilinear integro-differential inclusions via resolvent operators, Appl. Math. Comput., 247 (2014), 152-161.  doi: 10.1016/j.amc.2014.08.080. [31] V. Vijayakumar, Approximate controllability results for nondensely defined fractional neutral differential inclusions with Hille Yosida operators, Internat. J. Control, 92 (2019), 2210-2222.  doi: 10.1080/00207179.2018.1433331. [32] J. Wang and Y. Zhou, Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Anal. Real World Appl., 12 (2011), 3642-3653.  doi: 10.1016/j.nonrwa.2011.06.021. [33] B. Yan, Boundary value problems on the half-line with impulses and infinite delay, J. Math. Anal. Appl., 259 (2001), 94-114.  doi: 10.1006/jmaa.2000.7392. [34] K. Yosida, Functional Analysis, 6$^th$ edition, Springer, Berlin, 1980. [35] Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077.  doi: 10.1016/j.camwa.2009.06.026. [36] Y. Zhou, V. Vijayakumar, C. Ravichandran and R. Murugesu, Controllability results for fractional order neutral functional differential inclusions with infinite delay, Fixed Point Theory, 18 (2017), 773-798.  doi: 10.24193/fpt-ro.2017.2.62. [37] Y. Zhou, V. Vijayakumar and R. Murugesu, Controllability for fractional evolution inclusions without compactness, Evol. Equ. Control The., 4 (2015), 507-524.  doi: 10.3934/eect.2015.4.507. [38] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. doi: 10.1142/9069. [39] Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier, New York, 2015.

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References:
 [1] L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162 (1991), 494-505.  doi: 10.1016/0022-247X(91)90164-U. [2] L. Byszewski and H. Akca, On a mild solution of a semilinear functional-differential evolution nonlocal problem, J. Appl. Math. Stoch. Anal., 10 (1997), 265-271.  doi: 10.1155/S1048953397000336. [3] Y. K. Chang, Controllability of impulsive functional differential systems with infinite delay in Banach spaces, Chaos Solitons Fractals, 33 (2007), 1601-1609.  doi: 10.1016/j.chaos.2006.03.006. [4] Y. K. Chang, A. Anguraj and M. Mallika Arjunan, Existence results for non-densely defined neutral impulsive differential inclusions with nonlocal conditions, J. Appl. Math. Comput., 28 (2008), 79-91.  doi: 10.1007/s12190-008-0078-8. [5] X. Fu and X. Liu, Controllability of non-densely defined neutral functional differential systems in abstract space, Chinese Ann. Math. B, 28 (2007), 243-252.  doi: 10.1007/s11401-005-0028-9. [6] B. Ghanbari, S. Kumar and R. Kumar, A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative, Chaos Solitons Fractals, 133 (2020), 109619. doi: 10.1016/j.chaos.2020.109619. [7] W. Gao, P. Veeresha, D. G. Prakasha, H. M. Baskonus and G. Yel, New numerical results for the time-fractional phi-four equation using a novel analytical approach, Symmetry, 12 (2020), 478. doi: 10.3390/sym12030478. [8] E. P. Gatsori, Controllability results for nondensely defined evolution differential inclusions with nonlocal conditions, J. Math. Anal. Appl., 297 (2004), 194-211.  doi: 10.1016/j.jmaa.2004.04.055. [9] H. Gu, Y. Zhou, B. Ahmad and A. Alsaedi, Integral solutions of fractional evolution equations with nondense domain, Electronic J. Differ. Eq., 2017 (2017), 1-15. [10] F. Jarad, T. Abdeljawad and D. Baleanu, Fractional variational optimal control problems with delayed arguments, Nonlinear Dyn., 62 (2010), 609-614.  doi: 10.1007/s11071-010-9748-9. [11] F. Jarad, T. Abdeljawad and D. Baleanu, Fractional variational principles with delay within caputo derivatives, Rep. Math. Phys., 65 (2010), 17-28.  doi: 10.1016/S0034-4877(10)00010-8. [12] V. Kavitha and M. M. Arjunan, Controllability of non-densely defined impulsive neutral functional differential systems with infinite delay in Banach spaces, Nonlinear Anal. Hybri., 4 (2010), 441-450.  doi: 10.1016/j.nahs.2009.11.002. [13] H. Kellerman and M. Hieber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180.  doi: 10.1016/0022-1236(89)90116-X. [14] A. Kumar and D. N. Pandey, Controllability results for nondensely defined impulsive fractional differential equations in abstract space, Differ. Equ. Dyn. Syst., (2019). doi: 10.1007/s12591-019-00471-1. [15] S. Kumar, R. Kumar, J. Singh, K. S. Nisar and D. Kumar, An efficient numerical scheme for fractional model of HIV-1 infection of $CD4^+$ T-cells with the effect of antiviral drug therapy, Alex. Eng. J., (2020). doi: 10.1016/j.aej.2019.12.046. [16] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, in North-Holland Mathematics Studies, 204, Elsevier Science, Amsterdam, 2006. [17] K. D. Kucche and S. T. Sutar, On Existence and stability results for nonlinear fractional delay differential equations, Bulletin of Parana's Mathematical Society, 36 (2018), 55-75.  doi: 10.5269/bspm.v36i4.33603. [18] N. I. Mahmudov, R. Murugesu, C. Ravichandran and V. Vijayakumar, Approximate controllability results for fractional semilinear integro-differential inclusions in Hilbert spaces, Results Math., 71 (2017), 45-61.  doi: 10.1007/s00025-016-0621-0. [19] T. A. Maraaba, F. Jarad and D. Baleanu, On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives, Sci. China Ser. A, 51 (2008), 1775-1786.  doi: 10.1007/s11425-008-0068-1. [20] T. A. Maraaba, D. Baleanu and F. Jarad, Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, J. Math. Phys., 49 (2008), 083507. doi: 10.1063/1.2970709. [21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, New York, Springer-verlag, 1983. doi: 10.1007/978-1-4612-5561-1. [22] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, an Diego, 1999. [23] C. Ravichandran and J. J. Trujillo, Controllability of impulsive fractional functional integro-differential equations in Banach spaces, J. Funct. Spaces, 2013 (2013), 1-8.  doi: 10.1155/2013/812501. [24] C. Ravichandran, K. Jothimani, H. M. Baskonus and N. Valliammal, New results on nondensely characterized integrodifferential equations with fractional order, Eur. Phys. J. Plus, 133 (2018), 1-10. [25] C. Ravichandran, N. Valliammal and J. J. Nieto, New results on exact controllability of a class of neutral integrodifferential systems with state dependent delay in Banach spaces, J. Franklin Inst., 356 (2019), 1535-1565.  doi: 10.1016/j.jfranklin.2018.12.001. [26] S. J. Sadati, D. Baleanu, A. Ranjbar, R. Ghaderi and T. Abdeljawad, Mittag-Leffler stability theorem for fractional nonlinear systems with delay, Abstr. Appl. Anal., 2010 (2010), 1-7.  doi: 10.1155/2010/108651. [27] R. Sakthivel, R. Ganesh and S. M. Anthoni, Approximate controllability of fractional nonlinear differential inclusions, Appl. Math. Comput., 225 (2013), 708-717.  doi: 10.1016/j.amc.2013.09.068. [28] J. V. D. C. Sousa, E. C. de Oliveira and K. D. Kucche, On the fractional functional differential equation with abstract volterra operator, B. Braz. Math.l Soc., 50 (2019), 803-822.  doi: 10.1007/s00574-019-00139-y. [29] P. Veeresha, D. G. Prakasha, H. M. Baskonus and G. Yel, An efficient analytical approach for fractional Lakshmanan-Porsezian-Daniel model, Math. Methods Appl. Sci., 43 (2020), 4136-4155.  doi: 10.1002/mma.6179. [30] V. Vijayakumar, C. Ravichandran, R. Murugesu and J. J. Trujillo, Controllability results for a class of fractional semilinear integro-differential inclusions via resolvent operators, Appl. Math. Comput., 247 (2014), 152-161.  doi: 10.1016/j.amc.2014.08.080. [31] V. Vijayakumar, Approximate controllability results for nondensely defined fractional neutral differential inclusions with Hille Yosida operators, Internat. J. Control, 92 (2019), 2210-2222.  doi: 10.1080/00207179.2018.1433331. [32] J. Wang and Y. Zhou, Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Anal. Real World Appl., 12 (2011), 3642-3653.  doi: 10.1016/j.nonrwa.2011.06.021. [33] B. Yan, Boundary value problems on the half-line with impulses and infinite delay, J. Math. Anal. Appl., 259 (2001), 94-114.  doi: 10.1006/jmaa.2000.7392. [34] K. Yosida, Functional Analysis, 6$^th$ edition, Springer, Berlin, 1980. [35] Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077.  doi: 10.1016/j.camwa.2009.06.026. [36] Y. Zhou, V. Vijayakumar, C. Ravichandran and R. Murugesu, Controllability results for fractional order neutral functional differential inclusions with infinite delay, Fixed Point Theory, 18 (2017), 773-798.  doi: 10.24193/fpt-ro.2017.2.62. [37] Y. Zhou, V. Vijayakumar and R. Murugesu, Controllability for fractional evolution inclusions without compactness, Evol. Equ. Control The., 4 (2015), 507-524.  doi: 10.3934/eect.2015.4.507. [38] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. doi: 10.1142/9069. [39] Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier, New York, 2015.
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