# American Institute of Mathematical Sciences

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  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 & Control Theory, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] H. Kellerman and M. Hieber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180.  doi: 10.1016/0022-1236(89)90116-X.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] N. I. Mahmudov, R. Murugesu, C. Ravichandran and V. 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An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, an Diego, 1999.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [34] K. Yosida, Functional Analysis, 6$^th$ edition, Springer, Berlin, 1980.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [38] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. doi: 10.1142/9069.  Google Scholar [39] Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier, New York, 2015.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] H. Kellerman and M. Hieber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180.  doi: 10.1016/0022-1236(89)90116-X.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [34] K. Yosida, Functional Analysis, 6$^th$ edition, Springer, Berlin, 1980.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [38] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. doi: 10.1142/9069.  Google Scholar [39] Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier, New York, 2015.  Google Scholar
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