Approximation theorems for controllability problem governed by fractional differential equation

Rajesh Dhayal; Muslim Malik; Syed Abbas; Anil Kumar; Rathinasamy Sakthivel

doi:10.3934/eect.2020073

Previous Article
Blow-up criteria for linearly damped nonlinear Schrödinger equations
EECT Home
This Issue
Next Article
Decay rates for the moore-gibson-thompson equation with memory

doi: 10.3934/eect.2020073

Approximation theorems for controllability problem governed by fractional differential equation

Rajesh Dhayal ^1, , Muslim Malik ^1, , Syed Abbas ^1, , Anil Kumar ^2, and Rathinasamy Sakthivel ^3,,

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 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.

Keywords: Fractional differential equation, controllability, optimal control, resolvent family, approximation theorems.

Mathematics Subject Classification: Primary: 34A08; Secondary: 93B05, 49J20, 93C25.

Citation: Rajesh Dhayal, Muslim Malik, Syed Abbas, Anil Kumar, Rathinasamy Sakthivel. Approximation theorems for controllability problem governed by fractional differential equation. Evolution Equations & Control Theory, 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	M. C. Joshi and R. K. Bose, Some Topics in Nonlinear Functional Analysis, John Wiley and Sons, Inc., New York, 1985. Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	J. Klamka, Controllability of dynamical systems. A survey, Bulletin of the Polish Academy of Sciences: Tech. Sci., 61 (2013), 335-342. Google Scholar
[15]	G. Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim., 20 (1982), 414-427. doi: 10.1137/0320032. Google Scholar
[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. Google Scholar
[17]	M. Malik and R. Agarwal, Exact controllability of an integro-differential equation with deviated argument, Funct. Differ. Equ., 21 (2004), 31-45. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	K. B. Oldham and J. Spanier, The Fractional Calculus, Mathematics in Science and Engineering, 111. Academic Press, New York-London, 1974. Google Scholar
[24]	I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

show all references

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	M. C. Joshi and R. K. Bose, Some Topics in Nonlinear Functional Analysis, John Wiley and Sons, Inc., New York, 1985. Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	J. Klamka, Controllability of dynamical systems. A survey, Bulletin of the Polish Academy of Sciences: Tech. Sci., 61 (2013), 335-342. Google Scholar
[15]	G. Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim., 20 (1982), 414-427. doi: 10.1137/0320032. Google Scholar
[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. Google Scholar
[17]	M. Malik and R. Agarwal, Exact controllability of an integro-differential equation with deviated argument, Funct. Differ. Equ., 21 (2004), 31-45. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	K. B. Oldham and J. Spanier, The Fractional Calculus, Mathematics in Science and Engineering, 111. Academic Press, New York-London, 1974. Google Scholar
[24]	I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

Figure 1. Comparison between $ z_{b} $ and $ \overline{z}(b) $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. Approximated optimal control $ \overline{u} $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3. Numerical solution $ \overline{z} $ corresponding to optimal control $ \overline{u} $ with $ \eta = 0.75 $

Figure Options

Download full-size image

Download as PowerPoint slide

[1]	Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019
[2]	Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020046
[3]	Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020317
[4]	Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020347
[5]	Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020432
[6]	Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011
[7]	Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020 doi: 10.3934/jgm.2020032
[8]	Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077
[9]	Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020319
[10]	S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020435
[11]	Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020445
[12]	Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296
[13]	Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020463
[14]	Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020049
[15]	Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240
[16]	José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020271
[17]	Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264
[18]	Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050
[19]	Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020103
[20]	Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020047

2019 Impact Factor: 0.953

Tools

Metrics

Tools

Article outline

Figures and Tables

2019 Impact Factor: 0.953

Tools

Figures and Tables

2019 Impact Factor: 0.953

American Institute of Mathematical Sciences

Approximation theorems for controllability problem governed by fractional differential equation

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Approximation theorems for controllability problem governed by fractional differential equation

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors