Approximate controllability for systems of fractional nonlinear differential equations involving Riemann-Liouville derivatives

Lavina  Sahijwani; Nagarajan Sukavanam

doi:10.11121/ijocta.2023.1178

Authors

Lavina Sahijwani Department of Mathematics, IIT Roorkee, India https://orcid.org/0000-0001-7809-210X
Nagarajan Sukavanam Department of Mathematics, IIT Roorkee, India https://orcid.org/0000-0001-9627-8653

DOI:

https://doi.org/10.11121/ijocta.2023.1178

Keywords:

Nonlinear systems, Riemann-Liouville fractional derivatives, Fixed point theorem, Approximate controllability

Abstract

The article objectifies the approximate controllability of fractional nonlinear differential equations having Riemann-Liouville derivatives. First, the existence of solutions is deduced through fixed point approach and then approximate controllability is proved using Cauchy convergence through iterative and approximate techniques. The theory of semigroup together with probability density function has been utilized to reach the desired conclusions.

Downloads

Download data is not yet available.

Author Biographies

Lavina Sahijwani, Department of Mathematics, IIT Roorkee, India

is presently working as a Ph.D. scholar in the Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India. She has done her M.Sc. Mathematics from Maharshi Dayanand Saraswati University, Ajmer, India.

Nagarajan Sukavanam, Department of Mathematics, IIT Roorkee, India

is a Professor & former Head of the Department at Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India. He has obtained his Ph.D. from Indian Institute of Science, Bangalore, India in the year 1985. He has supervised more than 25 scholars to obtain their doctorate degrees and has adjudicated several doctoral thesis from different Universities. He has published more than 150 research articles in International journals of high repute.

References

Oldham, K.B., & Spanier, J. (1974). The fractional calculus. Academic Press, New York.

Hernandez, E., O’Regan, D., & Balachan- dran, E. (2010). On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Analysis, 73, 3462–3471.

Hilfer, R. (2000). Applications of fractional calculus in physics. World Scientific Publishing Co., Singapore.

Kilbas, A.A., Srivastava, H.M., & Trujillo, J.J. (2006). Theory and applications of fractional differential equations. North-Holland Mathematics Studies, 204. Elsevier Science, Amsterdam.

Koeller, R.C. (1984). Applications of fractional calculus to the theory of viscoelasticity. Journal of Applied Mechanics, 51 (2), 299- 307.

Kumar, S., & Sukavanam, N. (2012). Approximate controllability of fractional order semilinear systems with bounded delay. Journal of Differential Equations, 252, 6163–6174.

Liu, Z.H., Zeng, S.D., & Bai, Y.R. (2016) Maximum principles for multi term space time variable order fractional diffusion equations and their applications. Fractional Calculus & Applied Analysis, 19(1), 188-211.

Podlubny, I. (1999). Fractional differential equations. Academic Press, San Diego, CA.

Sakthivel, R., Ren, Y., & Mahmudov, N.I. (2011). On the approximate controllability of semilinear fractional differential systems. Computers & Mathematics with Applications, 62, 1451–1459.

Samko, S. G., Kilbas, A. A., & Marichev, O. I. (1993). Fractional integral and derivatives, theory and applications. Gordon and Breach, New York.

Heymans, N., & Podlubny, I. (2006). Physi- cal interpretation of initial conditions for fractional differential equations with Riemann- Liouville fractional derivatives. Rheologica Acta, 45, 765-771.

Curtain, R.F., & Zwart, H. (2012). An introduction to infinite-dimensional linear systems theory. Springer Science and Business Media, New York.

Barnett, S. (1975). Introduction to mathematical control theory. Clarendon Press, Oxford.

Kalman, R. E. (1963). Controllability of linear dynamical systems. Contributions to Differential Equations, 1, 190–213.

Devies, I., & Jackreece, P. (2005). Controllability and null controllability of linear systems. Journal of Applied Sciences and Environmental Management, 9, 31-36.

Mahmudov, N.I. (2018). Partial-approximate controllability of nonlocal fractional evolution equations via approximating method. Applied Mathematics and Computation, 334, 227-238.

Klamka,J. (2009). Constrained controllability of semilinear systems with delays. Nonlin- ear Dynamics, 56, 169–177.

Wen, Y., & Zhou, X.F. (2018). Approximate controllability and complete controllability of semilinear fractional functional differential systems with control. Advances in Difference Equations, 375, 1-18.

Byszewski, L., & Lakshmikantham, V. (2007). Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Applicable Analysis, 40(1), 11–19.

Dauer, J. P., & Mahmudov, N. I. (2002). Approximate controllability of semilinear function equations in Hilbert spaces. Journal of Mathematical Analysis and Applications, 273, 310–327.

Haq, A., & Sukavanam, N. (2020). Existence and approximate controllability of Riemann- Liouville fractional integrodifferential systems with damping. Chaos Solitons & Fractals, 139, 110043-110053.

Haq, A., & Sukavanam, N. (2021). Partial approximate controllability of fractional systems with Riemann-Liouville derivatives and nonlocal conditions. Rendiconti del Circolo Mathematico di Palermo Series 2, 70, 1099-1114.

Lakshmikantham, V. (2008). Theory of fractional functional differential equations. Nonlinear Analysis, 69, 3337-3343.

Liu, Z.H., Sun, J.H., & Szanto, I. (2013). Monotone iterative technique for Riemann- Liouville fractional integro-differential equations with advanced arguments. Results in Mathematics, 63, 1277-1287.

Mahmudov, N.I. (2017). Finite-approximate controllability of evolution equations. Applied and Computational Mathematics, 16, 159–167.

Monje, A., Chen, Y.Q., Vinagre, B.M., Xue, D., & Feliu, V. (2010). Fractional-order systems and controls, fundamentals and applications. Springer-Verlag, London.

Naito, K. (1987). Controllability of semilinear control systems dominated by the linear part. SIAM Journal on Control and Optimization, 25(3), 715–722.

Sukavanam, N., & Kumar, M. (2010). S-controllability of an abstract first order semilinear control system. Numerical Functional Analysis and Optimization, 31, 1023-1034.

Triggiani, R. (1975). Controllability and observability in Banach spaces with bounded operators. SIAM Journal on Control and Optimization, 13, 462-291.

Wang, J.R., & Zhou, Y. (2011). A class of fractional evolution equations and optimal controls. Nonlinear Analysis: Real World Applications, 12, 262-272.

Zhou, Y., & Jiao, F. (2010). Existence of mild solutions for fractional neutral evolution equations. Computers & Mathematics with Applications, 59, 1063-1077.

Scherer, R., Kalla, S.L., Boyadjiev, L., & Al-Saqabi, B. (2008). Numerical treatment of fractional heat equations. Applied Numerical Mathematics, 58, 1212-1223.

Bora, S.N., & Roy, B. (2021). Approximate controllability of a class of semilinear Hilfer fractional differential equations. Results in Mathematics, 76, 1-20.

Dineshkumar, C., Udhayakumar, R., Vijayakumar, V., Nisar, K.S., & Shukla, A. (2022). A note concerning to approximate controllability of Atangana-Baleanu fractional neutral stochastic systems with infinite delay. Chaos Solitons & Fractals, 157, 111916.

Raja, M.M., Vijayakumar, V., Shukla, A., Nisar K.S., Sakthivel, N., & Kaliraj, K. (2022). Optimal control and approximate controllability for fractional integrodifferential evolution equations with infinite delay of order r in (1,2). Optimal Control Applications and Methods, 1-24.

Shukla, A., Vijayakumar, V., & Nisar, K.S. (2022). A new exploration on the existence and approximate controllability for fractional semilinear impulsive control systems of order r in (1,2). Chaos Solitons & Fractals, 154, 111615.

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 Journal, 61(9), 7291-7302.

Dineshkumar, C., Udhayakumar, R., Vijayakumar, V., Shukla, A., & Nisar, K.S. (2021). A note on approximate controllability for nonlocal fractional evolution stochastic integrodifferential inclusions of order r in (1,2) with delay. Chaos Solitons & Fractals, 153, 111565.

Dineshkumar, C., Nisar, K.S., Udhayakumar, R., & Vijayakumar, V. (2021). New discussion about the approximate controllability of fractional stochastic differential inclusions with order 1 < r < 2. Asian Journal of Control, 1 ? 25.

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 integrodifferential control systems via resolvent operators. Fractal and Fractional, 6(2), 1-14.

Liu, Z., & Li, X. (2015). Approximate controllability of fractional evolution systems with Riemann–Liouville fractional derivatives. SIAM Journal on Control and Optimization, 53(1), 1920-1933.

Ye, H.P., Gao, J.M., & Ding, Y.S. (2007). A generalised Gronwall inequality and its applications to a fractional differential equation. Journal of Mathematical Analysis and Applications, 328, 1075-1081.