Lavina Sahijwani, Sukavanam Nagarajan

Approximate controllability for Riemann-Liouville fractional differential equations

The article objectifies the approximate controllability of fractional nonlinear differential equations having Riemann-Liouville derivatives. The nonlinear term involved in the equation, also depending on the control parameter u(·), is considered to be locally Lipschitz. First, the existence of solutions is deduced using Lipschitz condition, semigroup theory and fixed point approach. Then sufficient condition for approximate controllability of the system is established using Cauchy convergence through iterative and approximate techniques. The theory of semigroup together with probability density function has been uti- lized to reach the desired conclusions. Lastly, an application is provided to support the proposed methodology.

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[1] Oldham, K.B., & Spanier, J. (1974). The frac- tional calculus. Academic Press, New York.
[2] Hernandez, E., O’Regan, D., & Balachan- dran, E. (2010). On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Analy- sis, 73, 3462–3471.
[3] Hilfer, R. (2000). Applications of fractional calculus in physics. World Scientific Publish- ing Co., Singapore.
[4] Kilbas, A.A., Srivastava, H.M., & Trujillo, J.J. (2006). Theory and applications of frac- tional differential equations. North-Holland Mathematics Studies, 204. Elsevier Science, Amsterdam.
[5] Koeller, R.C. (1984). Applications of frac- tional calculus to the theory of viscoelastic- ity. Journal of Applied Mechanics, 51 (2), 299- 307.
[6] Kumar, S., & Sukavanam, N. (2012). Approx- imate controllability of fractional order semi- linear systems with bounded delay. Journal of Differential Equations, 252, 6163–6174.
[7] Liu, Z.H., Zeng, S.D., & Bai, Y.R. (2016) Maximum principles for multi term space time variable order fractional diffusion equa- tions and their applications. Fractional Cal- culus & Applied Analysis, 19(1), 188-211.
[8] Podlubny, I. (1999). Fractional differential equations. Academic Press, San Diego, CA.
[9] Sakthivel, R., Ren, Y., & Mahmudov, N.I. (2011). On the approximate controllability of semilinear fractional differential systems. Computers & Mathematics with Applications, 62, 1451–1459.
[10] Samko, S. G., Kilbas, A. A., & Marichev, O. I. (1993). Fractional integral and derivatives, theory and applications. Gordon and Breach, New York.
[11] Heymans, N., & Podlubny, I. (2006). Physi- cal interpretation of initial conditions for frac- tional differential equations with Riemann- Liouville fractional derivatives. Rheologica Acta, 45, 765-771.
[12] Curtain, R.F., & Zwart, H. (2012). An intro- duction to infinite-dimensional linear systems theory. Springer Science and Business Media, New York.
[13] Barnett, S. (1975). Introduction to mathe- matical control theory. Clarendon Press, Ox- ford.
[14] Kalman, R. E. (1963). Controllability of lin- ear dynamical systems. Contributions to Dif- ferential Equations, 1, 190–213.
[15] Devies, I., & Jackreece, P. (2005). Control- lability and null controllability of linear sys- tems. Journal of Applied Sciences and Envi- ronmental Management, 9, 31-36.
[16] Mahmudov, N.I. (2018). Partial-approximate controllability of nonlocal fractional evolution equations via approximating method. Applied Mathematics and Computation, 334, 227-238.
[17] Klamka,J. (2009). Constrained controllabil- ity of semilinear systems with delays. Nonlin- ear Dynamics, 56, 169–177.
[18] Wen, Y., & Zhou, X.F. (2018). Approxi- mate controllability and complete controlla- bility of semilinear fractional functional dif- ferential systems with control. Advances in Difference Equations, 375, 1-18.
[19] Byszewski, L., & Lakshmikantham, V. (2007). Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Applica- ble Analysis, 40(1), 11–19.
[20] Dauer, J. P., & Mahmudov, N. I. (2002). Ap- proximate controllability of semilinear func- tion equations in Hilbert spaces. Journal of Mathematical Analysis and Applications, 273, 310–327.
[21] Haq, A., & Sukavanam, N. (2020). Existence and approximate controllability of Riemann- Liouville fractional integrodifferential systems with damping. Chaos Solitons & Fractals, 139, 110043-110053.
[22] Haq, A., & Sukavanam, N. (2021). Partial ap- proximate controllability of fractional systems with Riemann-Liouville derivatives and non- local conditions. Rendiconti del Circolo Math- ematico di Palermo Series 2, 70, 1099-1114.
[23] Lakshmikantham, V. (2008). Theory of frac- tional functional differential equations. Non- linear Analysis, 69, 3337-3343.
[24] Liu, Z.H., Sun, J.H., & Szanto, I. (2013). Monotone iterative technique for Riemann- Liouville fractional integro-differential equa- tions with advanced arguments. Results in Mathematics, 63, 1277-1287.
[25] Mahmudov, N.I. (2017). Finite-approximate controllability of evolution equations. Ap- plied and Computational Mathematics, 16, 159–167.
[26] Monje, A., Chen, Y.Q., Vinagre, B.M., Xue, D., & Feliu, V. (2010). Fractional-order sys- tems and controls, fundamentals and applica- tions. Springer-Verlag, London.
[27] Naito, K. (1987). Controllability of semilin- ear control systems dominated by the linear part. SIAM Journal on Control and Opti- mization, 25(3), 715–722.
[28] Sukavanam, N., & Kumar, M. (2010). S- controllability of an abstract first order semi- linear control system. Numerical Functional Analysis and Optimization, 31, 1023-1034.
[29] Triggiani, R. (1975). Controllability and ob- servability in Banach spaces with bounded op- erators. SIAM Journal on Control and Opti- mization, 13, 462-291.
[30] Wang, J.R., & Zhou, Y. (2011). A class of fractional evolution equations and optimal controls. Nonlinear Analysis: Real World Ap- plications, 12, 262-272.
[31] Zhou, Y., & Jiao, F. (2010). Existence of mild solutions for fractional neutral evolution equations. Computers & Mathematics with Applications, 59, 1063-1077.
[32] Scherer, R., Kalla, S.L., Boyadjiev, L., & Al-Saqabi, B. (2008). Numerical treatment of fractional heat equations. Applied Numerical Mathematics, 58, 1212-1223.
[33] Bora, S.N., & Roy, B. (2021). Approximate controllability of a class of semilinear Hil- fer fractional differential equations. Results in Mathematics, 76, 1-20.
[34] Dineshkumar, C., Udhayakumar, R., Vi- jayakumar, V., Nisar, K.S., & Shukla, A. (2022). A note concerning to approx- imate controllability of Atangana-Baleanu fractional neutral stochastic systems with in- finite delay. Chaos Solitons & Fractals, 157, 111916.
[35] Raja, M.M., Vijayakumar, V., Shukla, A., Nisar K.S., Sakthivel, N., & Kaliraj, K. (2022). Optimal control and approximate con- trollability for fractional integrodifferential evolution equations with infinite delay of or- der r ∈ (1,2). Optimal Control Applications and Methods, 1-24.
[36] Shukla, A., Vijayakumar, V., & Nisar, K.S. (2022). A new exploration on the existence and approximate controllability for fractional semilinear impulsive control systems of or- der r∈ (1,2). Chaos Solitons & Fractals, 154, 111615.
[37] Ma, Y.K., Kavitha, K., Albalawi, W., Shukla A., Nisar K.S., & Vijayakumar, V. (2022). An analysis on the approximate controllability of Hilfer fractional neutral differential systems in Hilbert spaces. Alexandria Engineering Jour- nal, 61(9), 7291-7302.
[38] Dineshkumar, C., Udhayakumar, R., Vi- jayakumar, V., Shukla, A., & Nisar, K.S. (2021). A note on approximate controllabil- ity for nonlocal fractional evolution stochas- tic integrodifferential inclusions of order r ∈ (1,2) with delay. Chaos Solitons & Fractals, 153, 111565.
[39] Dineshkumar, C., Nisar, K.S., Udhayaku- mar, R., & Vijayakumar, V. (2021). New dis- cussion about the approximate controllability of fractional stochastic differential inclusions with order 1 < r < 2. Asian Journal of Con- trol, 1 − 25.
[40] Vijayakumar, V., Nisar, K.S., Chalishajar, D., Shukla, A., Malik, M., Alsaadi, A., & Al- dosary S.F. (2022). A note on approximate controllability of fractional semilinear inte- grodifferential control systems via resolvent operators. Fractal and Fractional, 6(2), 1-14.
[41] Liu, Z., & Li, X. (2015). Approximate con- trollability of fractional evolution systems with Riemann–Liouville fractional deriva- tives. SIAM Journal on Control and Opti- mization, 53(1), 1920-1933.
[42] Ye, H.P., Gao, J.M., & Ding, Y.S. (2007). A generalised Gronwall inequality and its ap- plications to a fractional differential equation. Journal of Mathematical Analysis and Appli- cations, 328, 1075-1081.