# American Institute of Mathematical Sciences

June  2021, 10(2): 411-429. doi: 10.3934/eect.2020073

## Approximation theorems for controllability problem governed by fractional differential equation

 1 School of Basic Sciences, Indian Institute of Technology Mandi, Kamand (H.P.) - 175 005, India 2 Department of Mathematics, BITS Pilani KK Birla Goa Campus, Zuarinagar, Goa-403 726, India 3 Department of Applied Mathematics, Bharathiar University, Coimbatore, Tamilnadu-641 046, India

* Corresponding author: Rathinasamy Sakthivel

Received  July 2019 Revised  March 2020 Published  June 2021 Early access  June 2020

In this manuscript, we discuss the optimal control problem for a nonlinear system governed by the fractional differential equation in a separable Hilbert space $X$. We utilize the fixed point technique and $\eta$-resolvent family to present the existence of control for the fractional system. The optimal pair is obtained as the limit of the optimal pair sequence of the unconstrained problem. Further, we derive some approximation results, which guarantee the convergence of the numerical method to optimal pair sequence. Finally, the main results are validated with the aid of an example.

Citation: Rajesh Dhayal, Muslim Malik, Syed Abbas, Anil Kumar, Rathinasamy Sakthivel. Approximation theorems for controllability problem governed by fractional differential equation. Evolution Equations and Control Theory, 2021, 10 (2) : 411-429. doi: 10.3934/eect.2020073
##### References:
 [1] R. P. Agarwal, D. Baleanu, J. J. Nieto, D. F. M. Torres and Y. Zhou, A survey on fuzzy fractional differential and optimal control nonlocal evolution equations, J. Comput. Appl. Math., 339 (2018), 3-29.  doi: 10.1016/j.cam.2017.09.039. [2] D. Aimene, D. Baleanu and D. Seba, Controllability of semilinear impulsive Atangana-Baleanu fractional differential equations with delay, Chaos Solitons Fractals, 128 (2019), 51-57.  doi: 10.1016/j.chaos.2019.07.027. [3] D. Araya and C. Lizama, Almost automorphic mild solutions to fractional differential equations, Nonlinear Anal., 69 (2008), 3692-3705.  doi: 10.1016/j.na.2007.10.004. [4] Y. Cao, Numerical approximations of exact controllability problems by optimal control problems for parabolic differential equations, Appl. Math. Comput., 119 (2001), 127-145.  doi: 10.1016/S0096-3003(99)00251-9. [5] J. Dabas, A. Chauhan and M. Kumar, Existence of the mild solutions for impulsive fractional equations with infinite delay, Int. J. Differ. Equ., 2011 (2011), Art. ID 793023, 20 pp. doi: 10.1155/2011/793023. [6] A. Debbouche and V. Antonov, Approximate controllability of semilinear Hilfer fractional differential inclusions with impulsive control inclusion conditions in Banach spaces, Chaos Solitons Fractals, 102 (2017), 140-148.  doi: 10.1016/j.chaos.2017.03.023. [7] A. Debbouche, J. J. Nieto and D. F. M. Torres, Optimal solutions to relaxation in multiple control problems of Sobolev type with nonlocal nonlinear fractional differential equations, J. Optim. Theory Appl., 174 (2017), 7-31.  doi: 10.1007/s10957-015-0743-7. [8] J. I. Diaz and A. M. Ramos, Numerical experiments regarding the distributed control of semilinear parabolic problems, Comput. Math. Appl., 48 (2004), 1575-1586.  doi: 10.1016/j.camwa.2004.04.033. [9] M. Hasse, The Functional Calculus for Sectorial Operators, operator theory : advance and applications, 169. Birkhauser-Verlag, Basel, 2006. doi: 10.1007/3-7643-7698-8. [10] H. Huang and X. Fu, Approximate controllability of semi-linear neutral integro-differential equations with nonlocal conditions, J. Dyn. Control Syst., 26 (2020), 127-147.  doi: 10.1007/s10883-019-09438-5. [11] M. C. Joshi and R. K. Bose, Some Topics in Nonlinear Functional Analysis, John Wiley and Sons, Inc., New York, 1985. [12] M. C. Joshi and A. Kumar, Approximation of exact controllability problem involving parabolic differential equations, IMA J. Math. Control Inform., 22 (2005), 350-360.  doi: 10.1093/imamci/dni032. [13] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. [14] J. Klamka, Controllability of dynamical systems. A survey, Bulletin of the Polish Academy of Sciences: Tech. Sci., 61 (2013), 335-342. [15] G. Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim., 20 (1982), 414-427.  doi: 10.1137/0320032. [16] A. Kumar, M. C. Joshi and A. K. Pani, On approximation theorems for controllability of nonlinear parabolic problems, IMA J. Math. Control Inform., 24 (2007), 115-136.  doi: 10.1093/imamci/dnl012. [17] M. Malik and R. Agarwal, Exact controllability of an integro-differential equation with deviated argument, Funct. Differ. Equ., 21 (2004), 31-45. [18] M. Malik, R. Dhayal and S. Abbas, Exact controllability of a retarded fractional differential equation with non-instantaneous impulses, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 26 (2019), 53-69. [19] M. Malik, R. Dhayal, S. Abbas and A. Kumar, Controllability of non-autonomous nonlinear differential system with non-instantaneous impulses, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, 113 (2019), 103-118.  doi: 10.1007/s13398-017-0454-z. [20] M. Malik and R. K. George, Trajectory controllability of the nonlinear systems governed by fractional differential equations, Differ. Equ. Dyn. Syst., 27 (2019), 529-537.  doi: 10.1007/s12591-016-0292-z. [21] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, 1993. [22] D. A. Murio, Implicit finite difference approximation for time fractional diffusion equations, Comput. Math. Appl., 56 (2008), 1138-1145.  doi: 10.1016/j.camwa.2008.02.015. [23] K. B. Oldham and J. Spanier, The Fractional Calculus, Mathematics in Science and Engineering, 111. Academic Press, New York-London, 1974. [24] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. [25] 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, J. Franklin Inst., 356 (2019), 1535-1565.  doi: 10.1016/j.jfranklin.2018.12.001. [26] R. Sakthivel, N. I. Mahmudov and J. J. Nieto, Controllability for a class of fractional-order neutral evolution control systems, Appl. Math. Comput., 218 (2012), 10334-10340.  doi: 10.1016/j.amc.2012.03.093. [27] S. Suganya, M. M. Arjunan and J. J. Trujillo, Existence results for an impulsive fractional integro-differential equation with state-dependent delay, Appl. Math. Comput., 266 (2015), 54-69.  doi: 10.1016/j.amc.2015.05.031. [28] N. H. Sweilam, M. M. Khader and A. M. S. Mahdy, Crank-Nicolson finite difference method for solving time-fractional diffusion equation, J. Fraction. Calculus Appl., 2 (2012), 1-9. [29] J. Wang, A. G. Ibrahim, D. O'Regan and Y. Zhou, Controllability for noninstantaneous impulsive semilinear functional differential inclusions without compactness, Indag. Math., 29 (2018), 1362-1392.  doi: 10.1016/j.indag.2018.07.002. [30] J. Wang and Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal. Real World Appl., 12 (2011), 262-272.  doi: 10.1016/j.nonrwa.2010.06.013. [31] J. Wang and Y. Zhou, Complete controllability of fractional evolution systems, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 4346-4355.  doi: 10.1016/j.cnsns.2012.02.029. [32] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061. [33] D. Zhang and Y. Liang, Existence and controllability of fractional evolution equation with sectorial operator and impulse, Adv. Differ. Equ., 2018, Paper No. 219, 12 pp. doi: 10.1186/s13662-018-1664-1.

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##### References:
 [1] R. P. Agarwal, D. Baleanu, J. J. Nieto, D. F. M. Torres and Y. Zhou, A survey on fuzzy fractional differential and optimal control nonlocal evolution equations, J. Comput. Appl. Math., 339 (2018), 3-29.  doi: 10.1016/j.cam.2017.09.039. [2] D. Aimene, D. Baleanu and D. Seba, Controllability of semilinear impulsive Atangana-Baleanu fractional differential equations with delay, Chaos Solitons Fractals, 128 (2019), 51-57.  doi: 10.1016/j.chaos.2019.07.027. [3] D. Araya and C. Lizama, Almost automorphic mild solutions to fractional differential equations, Nonlinear Anal., 69 (2008), 3692-3705.  doi: 10.1016/j.na.2007.10.004. [4] Y. Cao, Numerical approximations of exact controllability problems by optimal control problems for parabolic differential equations, Appl. Math. Comput., 119 (2001), 127-145.  doi: 10.1016/S0096-3003(99)00251-9. [5] J. Dabas, A. Chauhan and M. Kumar, Existence of the mild solutions for impulsive fractional equations with infinite delay, Int. J. Differ. Equ., 2011 (2011), Art. ID 793023, 20 pp. doi: 10.1155/2011/793023. [6] A. Debbouche and V. Antonov, Approximate controllability of semilinear Hilfer fractional differential inclusions with impulsive control inclusion conditions in Banach spaces, Chaos Solitons Fractals, 102 (2017), 140-148.  doi: 10.1016/j.chaos.2017.03.023. [7] A. Debbouche, J. J. Nieto and D. F. M. Torres, Optimal solutions to relaxation in multiple control problems of Sobolev type with nonlocal nonlinear fractional differential equations, J. Optim. Theory Appl., 174 (2017), 7-31.  doi: 10.1007/s10957-015-0743-7. [8] J. I. Diaz and A. M. Ramos, Numerical experiments regarding the distributed control of semilinear parabolic problems, Comput. Math. Appl., 48 (2004), 1575-1586.  doi: 10.1016/j.camwa.2004.04.033. [9] M. Hasse, The Functional Calculus for Sectorial Operators, operator theory : advance and applications, 169. Birkhauser-Verlag, Basel, 2006. doi: 10.1007/3-7643-7698-8. [10] H. Huang and X. Fu, Approximate controllability of semi-linear neutral integro-differential equations with nonlocal conditions, J. Dyn. Control Syst., 26 (2020), 127-147.  doi: 10.1007/s10883-019-09438-5. [11] M. C. Joshi and R. K. Bose, Some Topics in Nonlinear Functional Analysis, John Wiley and Sons, Inc., New York, 1985. [12] M. C. Joshi and A. Kumar, Approximation of exact controllability problem involving parabolic differential equations, IMA J. Math. Control Inform., 22 (2005), 350-360.  doi: 10.1093/imamci/dni032. [13] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. [14] J. Klamka, Controllability of dynamical systems. A survey, Bulletin of the Polish Academy of Sciences: Tech. Sci., 61 (2013), 335-342. [15] G. Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim., 20 (1982), 414-427.  doi: 10.1137/0320032. [16] A. Kumar, M. C. Joshi and A. K. Pani, On approximation theorems for controllability of nonlinear parabolic problems, IMA J. Math. Control Inform., 24 (2007), 115-136.  doi: 10.1093/imamci/dnl012. [17] M. Malik and R. Agarwal, Exact controllability of an integro-differential equation with deviated argument, Funct. Differ. Equ., 21 (2004), 31-45. [18] M. Malik, R. Dhayal and S. Abbas, Exact controllability of a retarded fractional differential equation with non-instantaneous impulses, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 26 (2019), 53-69. [19] M. Malik, R. Dhayal, S. Abbas and A. Kumar, Controllability of non-autonomous nonlinear differential system with non-instantaneous impulses, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, 113 (2019), 103-118.  doi: 10.1007/s13398-017-0454-z. [20] M. Malik and R. K. George, Trajectory controllability of the nonlinear systems governed by fractional differential equations, Differ. Equ. Dyn. Syst., 27 (2019), 529-537.  doi: 10.1007/s12591-016-0292-z. [21] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, 1993. [22] D. A. Murio, Implicit finite difference approximation for time fractional diffusion equations, Comput. Math. Appl., 56 (2008), 1138-1145.  doi: 10.1016/j.camwa.2008.02.015. [23] K. B. Oldham and J. Spanier, The Fractional Calculus, Mathematics in Science and Engineering, 111. Academic Press, New York-London, 1974. [24] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. [25] 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, J. Franklin Inst., 356 (2019), 1535-1565.  doi: 10.1016/j.jfranklin.2018.12.001. [26] R. Sakthivel, N. I. Mahmudov and J. J. Nieto, Controllability for a class of fractional-order neutral evolution control systems, Appl. Math. Comput., 218 (2012), 10334-10340.  doi: 10.1016/j.amc.2012.03.093. [27] S. Suganya, M. M. Arjunan and J. J. Trujillo, Existence results for an impulsive fractional integro-differential equation with state-dependent delay, Appl. Math. Comput., 266 (2015), 54-69.  doi: 10.1016/j.amc.2015.05.031. [28] N. H. Sweilam, M. M. Khader and A. M. S. Mahdy, Crank-Nicolson finite difference method for solving time-fractional diffusion equation, J. Fraction. Calculus Appl., 2 (2012), 1-9. [29] J. Wang, A. G. Ibrahim, D. O'Regan and Y. Zhou, Controllability for noninstantaneous impulsive semilinear functional differential inclusions without compactness, Indag. Math., 29 (2018), 1362-1392.  doi: 10.1016/j.indag.2018.07.002. [30] J. Wang and Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal. Real World Appl., 12 (2011), 262-272.  doi: 10.1016/j.nonrwa.2010.06.013. [31] J. Wang and Y. Zhou, Complete controllability of fractional evolution systems, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 4346-4355.  doi: 10.1016/j.cnsns.2012.02.029. [32] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061. [33] D. Zhang and Y. Liang, Existence and controllability of fractional evolution equation with sectorial operator and impulse, Adv. Differ. Equ., 2018, Paper No. 219, 12 pp. doi: 10.1186/s13662-018-1664-1.
Comparison between $z_{b}$ and $\overline{z}(b)$
Approximated optimal control $\overline{u}$
Numerical solution $\overline{z}$ corresponding to optimal control $\overline{u}$ with $\eta = 0.75$
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