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
EECT Home
This Issue
Next Article
Well-posedness of infinite-dimensional non-autonomous passive boundary control systems

June 2021, 10(2): 411-429. 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 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.

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, 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. 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]	Daliang Zhao, Yansheng Liu, Xiaodi Li. Controllability for a class of semilinear fractional evolution systems via resolvent operators. Communications on Pure & Applied Analysis, 2019, 18 (1) : 455-478. doi: 10.3934/cpaa.2019023
[2]	Harbir Antil, Mahamadi Warma. Optimal control of the coefficient for the regional fractional $p$-Laplace equation: Approximation and convergence. Mathematical Control & Related Fields, 2019, 9 (1) : 1-38. doi: 10.3934/mcrf.2019001
[3]	Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics & Games, 2021, 8 (4) : 381-402. doi: 10.3934/jdg.2021013
[4]	Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control & Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040
[5]	Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105
[6]	Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213
[7]	Qing Tang. On an optimal control problem of time-fractional advection-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 761-779. doi: 10.3934/dcdsb.2019266
[8]	Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129
[9]	Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709
[10]	Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199
[11]	Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017
[12]	Steven Richardson, Song Wang. The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains. Journal of Industrial & Management Optimization, 2010, 6 (1) : 161-175. doi: 10.3934/jimo.2010.6.161
[13]	Robert Baier, Matthias Gerdts, Ilaria Xausa. Approximation of reachable sets using optimal control algorithms. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 519-548. doi: 10.3934/naco.2013.3.519
[14]	Robert J. Kipka, Yuri S. Ledyaev. Optimal control of differential inclusions on manifolds. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4455-4475. doi: 10.3934/dcds.2015.35.4455
[15]	Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305
[16]	Mohamed Ouzahra. Controllability of the semilinear wave equation governed by a multiplicative control. Evolution Equations & Control Theory, 2019, 8 (4) : 669-686. doi: 10.3934/eect.2019039
[17]	Angkana Rüland, Mikko Salo. Quantitative approximation properties for the fractional heat equation. Mathematical Control & Related Fields, 2020, 10 (1) : 1-26. doi: 10.3934/mcrf.2019027
[18]	Kevin Zumbrun. L^∞ resolvent bounds for steady Boltzmann's Equation. Kinetic & Related Models, 2017, 10 (4) : 1255-1257. doi: 10.3934/krm.2017048
[19]	Jingbo Dou, Huaiyu Zhou. Liouville theorems for fractional Hénon equation and system on $\mathbb{R}^n$. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1915-1927. doi: 10.3934/cpaa.2015.14.1915
[20]	H. W. J. Lee, Y. C. E. Lee, Kar Hung Wong. Differential equation approximation and enhancing control method for finding the PID gain of a quarter-car suspension model with state-dependent ODE. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2305-2330. doi: 10.3934/jimo.2019055

2020 Impact Factor: 1.081

Tools

Metrics

Tools

Article outline

Figures and Tables

2020 Impact Factor: 1.081

Tools

Figures and Tables

2020 Impact Factor: 1.081

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