Trajectory controllability of nonlinear fractional Langevin systems
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Govindaraj Venkatesan
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Suresh Kumar Pitchaikkannu
Abstract
In this paper, we discuss the trajectory controllability of linear and nonlinear fractional Langevin dynamical systems represented by the Caputo fractional derivative by using the Mittag–Leffler function and Gronwall–Bellman inequality. For the nonlinear system, we assume Lipschitz-type conditions on the nonlinearity. Examples are given to illustrate the theoretical results.
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Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
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Research funding: None declared.
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Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] T. Kaczorek, Selected Problems of Fractional Systems Theory, Berlin, Springer-Verlag, 2011.10.1007/978-3-642-20502-6Search in Google Scholar
[2] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam, Elsevier, 2006.Search in Google Scholar
[3] I. Podlubny, Fractional Differential Equations, New York, NY, Academic Press, 1999.Search in Google Scholar
[4] Y. Zhou, Basic Theory of Fractional Differential Equations, Singapore, World Scientific, 2014.10.1142/9069Search in Google Scholar
[5] B. Ahmad, J. J. Nieto, A. Alsaedi, and M. El-Shahed, “A Study of nonlinear Langevin equation involving two fractional orders in different intervals,” Nonlinear Anal. R. World Appl., vol. 13, pp. 599–606, 2012. https://doi.org/10.1016/j.nonrwa.2011.07.052.Search in Google Scholar
[6] F. Mainardi and P. Pironi, “The Fractional Langevin equation: Brownian motion revisited,” Extracta Mathematicae, vol. 1, pp. 140–154, 1996.Search in Google Scholar
[7] K. Fa, “Fractional Langevin equation and Riemann-Liouville fractional derivative,” Eur. Phys. J. E, vol. 24, pp. 139–143, 2007. https://doi.org/10.1140/epje/i2007-10224-2.Search in Google Scholar PubMed
[8] O. Baghani, “On fractional Langevin equation involving two fractional orders,” Commun. Nonlinear Sci. Numer. Simulat., vol. 42, pp. 675–681, 2017. https://doi.org/10.1016/j.cnsns.2016.05.023.Search in Google Scholar
[9] S. C. Lim, M. Li, and L. P. Teo, “Langevin equation with two fractional orders,” Phys. Lett., vol. 372, pp. 6309–6320, 2008. https://doi.org/10.1016/j.physleta.2008.08.045.Search in Google Scholar
[10] P. Guo, C. Zeng, C. Li, and Y. Chen, “Numerics for the fractional Langevin equation driven by the fractional Brownian motion,” Fract. Calc. Appl. Anal., vol. 16, pp. 123–141, 2013. https://doi.org/10.2478/s13540-013-0009-8.Search in Google Scholar
[11] T. Yu, K. Deng, and M. Luo, “Existence and uniqueness of solutions of initial value problems for nonlinear Langevin equation involving two fractional orders,” Commun. Nonlinear Sci. Numer. Simulat., vol. 19, pp. 1661–1668, 2014. https://doi.org/10.1016/j.cnsns.2013.09.035.Search in Google Scholar
[12] K. Balachandran and V. Govindaraj, “Numerical controllability of fractional dynamical systems,” Optimization, vol. 63, pp. 1267–1279, 2014. https://doi.org/10.1080/02331934.2014.906416.Search in Google Scholar
[13] K. Balachandran, V. Govindaraj, M. Rivero, and J. J. Trujillo, “Controllability of fractional damped dynamical systems,” Appl. Math. Comput., vol. 257, pp. 66–73, 2015. https://doi.org/10.1016/j.amc.2014.12.059.Search in Google Scholar
[14] K. Balachandran, V. Govindaraj, L. Rodriguez-Germa, and J. J. Trujillo, “Controllability results for nonlinear fractional-order dynamical systems,” J. Optim. Theor. Appl., vol. 156, pp. 33–44, 2013. https://doi.org/10.1007/s10957-012-0212-5.Search in Google Scholar
[15] K. Balachandran, J. Y. Park, and J. J. Trujillo, “Controllability of nonlinear fractional dynamical systems,” Nonlinear Anal., vol. 75, pp. 1919–1926, 2012. https://doi.org/10.1016/j.na.2011.09.042.Search in Google Scholar
[16] M. Bettayeb and S. Djennoune, “New results on the controllability and observability of fractional dynamical systems,” J. Vib. Control, vol. 14, pp. 1531–1541, 2008. https://doi.org/10.1177/1077546307087432.Search in Google Scholar
[17] Y. Chen, H. Ahn, and D. Xue, “Robust controllability of interval fractional order linear time invariant systems,” Signal Process., vol. 86, pp. 2794–2802, 2006. https://doi.org/10.1016/j.sigpro.2006.02.021.Search in Google Scholar
[18] S. Guermah, S. Djennoune, and M. Bettayeb, “Controllability and observability of linear discrete-time fractional-order systems,” Int. J. Appl. Math. Comput., vol. 18, pp. 213–222, 2008. https://doi.org/10.2478/v10006-008-0019-6.Search in Google Scholar
[19] D. Matignon and B. d’Andréa-Novel, “Some results on controllability and observability of finite dimensional fractional differential systems,” in Proceedings of the IAMCS, IEEE Conference on Systems, Man and Cybernetics Lille, France, July 9–12, 1996, pp. 952–956.Search in Google Scholar
[20] A. B. Shamardan and M. R. A. Moubarak, “Controllability and observability for fractional control systems,” J. Fract. Calc., vol. 15, pp. 25–34, 1999.Search in Google Scholar
[21] X. Ding and J. J. Nieto, “Controllability and optimality of linear time-invariant neutral control systems with different fractional orders,” Acta Math. Sci., vol. 35B, no. 5, pp. 1003–1013, 2015. https://doi.org/10.1016/s0252-9602(15)30034-5.Search in Google Scholar
[22] Y. Zhou, V. Vijayakumar, and R. Murugesu, “Controllability of fractional evolution inclusions without compactness,” Evol. Equ. Control Theor., vol. 4, pp. 407–524, 2015. https://doi.org/10.3934/eect.2015.4.507.Search in Google Scholar
[23] R. K. George, “Trajectory controllability of 1-dimensional nonlinear systems,” in Proceedings of the Research Seminar in honour of Professor M. N. Vasavada Anand, India, S.P. University, 1996, pp. 43–48.Search in Google Scholar
[24] D. N. Chalishajar, R. K. George, A. K. Nandakumaran, and F. S. Acharya, “Trajectory controllability of nonlinear integro-differential system,” J. Franklin Inst., vol. 347, pp. 1065–1075, 2010. https://doi.org/10.1016/j.jfranklin.2010.03.014.Search in Google Scholar
[25] V. Govindaraj, M. Muslim, and R. K. George, “Trajectory controllability of fractional dynamical systems,” J. Control Decis., vol. 4, pp. 114–130, 2017. https://doi.org/10.1080/23307706.2016.1249422.Search in Google Scholar
[26] M. Muslim and R. K. George, “Trajectory controllability of the nonlinear systems governed by fractional differential equations,” Diff. Equ. Dyn. Syst., vol. 27, pp. 529–537, 2019. https://doi.org/10.1007/s12591-016-0292-z.Search in Google Scholar
[27] V. Govindaraj and R. K. George, “Trajectory controllability of fractional integro differential systems in Hilbert space,” Asian J. Control, vol. 20, pp. 1–11, 2018. https://doi.org/10.1002/asjc.1685.Search in Google Scholar
[28] M. Muslim and K. A. Kumar, “Trajectory controllability of fractional differential systems of order α ∈ (1, 2] with deviated argument,” J. Anal., vol. 28, pp. 295–304, 2020. https://doi.org/10.1007/s41478-018-0081-x.Search in Google Scholar
[29] D. Rajesh, M. Muslim, and A. Syed, “Approximate and trajectory controllability of fractional neutral differential equation,” Adv. Oper. Theory, vol. 4, pp. 802–820, 2019. https://doi.org/10.15352/aot.1812-1444.Search in Google Scholar
[30] C. Dineshkumar, R. Udhayakumar, V. Vijayakumar, K. S. Nisar, and A. Shukla, “A note on the approximate controllability of Sobolev type fractional stochastic integro-differential delay inclusions with order 1 < r < 2,” Math. Comput. Simulat., vol. 190, pp. 1003–1026, 2021. https://doi.org/10.1016/j.matcom.2021.06.026.Search in Google Scholar
[31] K. Kavitha, V. Vijayakumar, R. Udhayakumar, and C. Ravichandran, “Results on controllability of Hilfer fractional differential equations with infinite delay via measures of noncompactness,” Asian J. Control, vol. 24, pp. 1406–1415, 2021.10.1002/asjc.2549Search in Google Scholar
[32] K. S. Nisar and V. Vijayakumar, “Results concerning to approximate controllability of non-densely defined Sobolev-type Hilfer fractional neutral delay differential system,” Math. Methods Appl. Sci., vol. 44, pp. 13615–13632, 2021. https://doi.org/10.1002/mma.7647.Search in Google Scholar
[33] V. Vijayakumar, C. Ravichandran, K. S. Nisar, and K. D. Kucche, “New discussion on approximate controllability results for fractional Sobolev type Volterra-Fredholm integro-differential systems of order 1 < r < 2,” Numer. Methods Part. Differ. Equ., 2021. https://doi.org/10.1002/num.22772, In this issue.Search in Google Scholar
[34] V. Vijayakumar, S. K. Panda, K. S. Nisar, and H. M. Baskonus, “Results on approximate controllability results for second-order Sobolev-type impulsive neutral differential evolution inclusions with infinite delay,” Numer. Methods Part. Differ. Equ., vol. 37, pp. 1200–1221, 2021. https://doi.org/10.1002/num.22573.Search in Google Scholar
[35] W. K. Williams, V. Vijayakumar, R. Udhayakumar, S. K. Panda, and K. S. Nisar, “Existence and controllability of nonlocal mixed Volterra–Fredholm type fractional delay integro-differential equations of order 1 < r < 2,” Numer. Methods Part. Differ. Equ., 2020. https://doi.org/10.1002/num.22697, In this issue.Search in Google Scholar
[36] H. K. Khalil, Nonlinear Systems, New Jersey, Prentice-Hall, 1996.Search in Google Scholar
[37] P. Linz, “A survey of methods for the solution of Volterra integral equations of the first kind in the applications and numerical solution of integral equations,” Nonlinear Anal. - TMA, pp. 189–194, 1980.10.1007/978-94-009-9130-9_9Search in Google Scholar
[38] K. Deimling, Multivalued Differential Equations, New York, NY, Walter de Gruyter, 1992.10.1515/9783110874228Search in Google Scholar
[39] K. Deimling, “Nonlinear Volterra integral equation of the first kind,” Nonlinear Anal. Theor. Methods Appl., vol. 25, pp. 951–957, 1995. https://doi.org/10.1016/0362-546x(95)00090-i.Search in Google Scholar
[40] M. C. Joshi and R. K. Bose, Some Topics in Nonlinear Functional Analysis, New Delhi, Wiley Eastern Limited, 1985.Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Original Research Articles
- One-dimensional optimal system and similarity transformations for the 3 + 1 Kudryashov–Sinelshchikov equation
- Hopf bifurcations in a network of FitzHugh–Nagumo biological neurons
- Gravity modulation effect on weakly nonlinear thermal convection in a fluid layer bounded by rigid boundaries
- A new numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ equation
- Chebyshev spectral method for solving a class of local and nonlocal elliptic boundary value problems
- Adaptive extragradient methods for solving variational inequalities in real Hilbert spaces
- Elastic trend filtering
- A new approach to representations of homothetic motions in Lorentz space
- Numerical simulation of variable-order fractional differential equation of nonlinear Lane–Emden type appearing in astrophysics
- Stability analysis and numerical simulations of the fractional COVID-19 pandemic model
- Numerical solutions of the Bagley–Torvik equation by using generalized functions with fractional powers of Laguerre polynomials
- N-fold Darboux transformation and exact solutions for the nonlocal Fokas–Lenells equation on the vanishing and plane wave backgrounds
- Closed form soliton solutions to the space-time fractional foam drainage equation and coupled mKdV evolution equations
- Master-slave synchronization in the Duffing-van der Pol and Φ6 Duffing oscillators
- Singularity analysis and analytic solutions for the Benney–Gjevik equations
- Trajectory controllability of nonlinear fractional Langevin systems
- Similarity transformations for modified shallow water equations with density dependence on the average temperature
- Non-equidistant partition predictor–corrector method for fractional differential equations with exponential memory
- Dynamical analysis of fractional-order of IVGTT glucose–insulin interaction
- Dynamic response of Mathieu–Duffing oscillator with Caputo derivative
- Non-Newtonian fluid flow having fluid–particle interaction through a porous zone in a channel with permeable walls
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Articles in the same Issue
- Frontmatter
- Original Research Articles
- One-dimensional optimal system and similarity transformations for the 3 + 1 Kudryashov–Sinelshchikov equation
- Hopf bifurcations in a network of FitzHugh–Nagumo biological neurons
- Gravity modulation effect on weakly nonlinear thermal convection in a fluid layer bounded by rigid boundaries
- A new numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ equation
- Chebyshev spectral method for solving a class of local and nonlocal elliptic boundary value problems
- Adaptive extragradient methods for solving variational inequalities in real Hilbert spaces
- Elastic trend filtering
- A new approach to representations of homothetic motions in Lorentz space
- Numerical simulation of variable-order fractional differential equation of nonlinear Lane–Emden type appearing in astrophysics
- Stability analysis and numerical simulations of the fractional COVID-19 pandemic model
- Numerical solutions of the Bagley–Torvik equation by using generalized functions with fractional powers of Laguerre polynomials
- N-fold Darboux transformation and exact solutions for the nonlocal Fokas–Lenells equation on the vanishing and plane wave backgrounds
- Closed form soliton solutions to the space-time fractional foam drainage equation and coupled mKdV evolution equations
- Master-slave synchronization in the Duffing-van der Pol and Φ6 Duffing oscillators
- Singularity analysis and analytic solutions for the Benney–Gjevik equations
- Trajectory controllability of nonlinear fractional Langevin systems
- Similarity transformations for modified shallow water equations with density dependence on the average temperature
- Non-equidistant partition predictor–corrector method for fractional differential equations with exponential memory
- Dynamical analysis of fractional-order of IVGTT glucose–insulin interaction
- Dynamic response of Mathieu–Duffing oscillator with Caputo derivative
- Non-Newtonian fluid flow having fluid–particle interaction through a porous zone in a channel with permeable walls
- A class of computationally efficient Newton-like methods with frozen inverse operator for nonlinear systems
