Chandan Shukla, Ashwini Kumar Upadhyay, Arun Singh, Ritika Pandey, Anurag Shukla

A new approach on approximate controllability of Sobolev-type Hilfer fractional differential equations

The approximate controllability of Sobolev-type Hilfer fractional control differential systems is the main emphasis of this paper. We use fractional calculus, Gronwall's inequality, semigroup theory, and the Cauchy sequence to examine the main results for the proposed system. The application of well-known fixed point theorem methodologies is avoided in this paper. Finally, a fractional heat equation is discussed as an example.

PDF

___

[1] Baleanu, D., Diethelm, K., Scalas , E., & Tru- jillo, J. J. (2012). Fractional Calculus Models and Numerical Methods, Series on Complex- ity, Nonlinearity and Chaos, World Scientific Publishing, Boston, Mass, USA.
[2] Lakshmikantham, V., Leela, S., & Devi, J. V. (2009). Theory of Fractional Dynamic Sys- tems, Cambridge Scientific Publishers.
[3] Podlubny, I. (1999). Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to method of their solution and some of their applications, San Diego, CA: Academic Press.
[4] Zhou, Y. (2015). Fractional Evolution Equa- tions and Inclusions: Analysis and Control, Elsevier, New York.
[5] Mohan Raja, M., Vijayakumar, V., & Ud- hayakumar R. (2020). Results on the exis- tence and controllability of fractional integro- differential system of order 1 < r < 2 via measure of noncompactness, Chaos, Solitons & Fractals, 139, 1-11.
[6] Shukla, A., Sukavanam, N., & Pandey, D.N. (2015). Complete controllability of semi- linear stochastic system with delay. Ren- diconti del Circolo Matematico di Palermo (1952-), 64(2),209-220.
[7] Shukla, A., Sukavanam, N., & Pandey, D.N. (2015). Approximate Controllability of Semi- linear Fractional Control Systems of Order α ∈ (1, 2]. In 2015 Proceedings of the Con- ference on Control and its Applications (pp. 175-180), Society for Industrial and Applied Mathematics.
[8] Shukla, A., Sukavanam, N., & Pandey, D. N. (2014). Controllability of semilinear sto- chastic system with multiple delays in control. IFAC Proceedings Volumes, 47(1), 306-312.
[9] Shukla, A., Sukavanam, N., & Pandey, D.N. (2018). Approximate controllability of semilinear fractional stochastic control sys- tem. Asian-European Journal of Mathemat- ics, 11(06), p.1850088.
[10] 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 ∈ (1, 2). Chaos, Solitons & Fractals, 154, p.111615.
[11] Mohan Raja, M., Vijayakumar, V., Shukla, A., Sooppy Nisar, K., Sakthivel, N., & Kali- raj, K. (2022). Optimal control and approx- imate controllability for fractional integrod- ifferential evolution equations with infinite delay of order r ∈ (1, 2). Optimal Control Applocations and Methods, 43(4), 996-1019. DOI:https://doi.org/10.1002/oca.2867.
[12] Kavitha, K., Nisar, K.S., Shukla, A., Vijayakumar, V., & Rezapour S. (2021). A discussion concerning the existence re- sults for the Sobolev-type Hilfer frac- tional delay integro-differential systems. Ad- vances in Differerence Equations, 467. DOI: https://doi.org/10.1186/s13662-021-03624-1.
[13] Mohan Raja, M., Vijayakumar, V., Shukla, A., Nisar, K.S. & Rezapour, S. (2021). New discussion on nonlocal controllability for frac- tional evolution system of order 1 < r < 2. Advances in Difference Equations, 481. DOI: https://doi.org/10.1186/s13662-021-03630-3.
[14] Agarwal, S., & Bahuguna, D. (2006). Exis- tence of solutions to Sobolev-type partial neu- tral differential equations, Journal of Applied Mathematics and Stochastic Analysis, 1-10, Article ID 16308.
[15] Brill, H. (1977). A semilinear Sobolev evolu- tion equation in a Banach space. Journal of Differential Equations, 24(3), 412-425.
[16] Chang, Y. K., & Li, W. T. (2006). Control- lability of Sobolev type semilinear functional differential and integrodifferential inclusions with an unbounded delay, Georgian Mathe- matical Journal, 13(1), 11-24.
[17] Lightbourne, J.H., & Rankin, S. (1983). A partial functional differential equation of Sobolev type. Journal of Mathematical Anal- ysis and Applications, 93(2), 328-337.
[18] Hilfer, R. (2002). Experimental evidence for fractional time evolution in glass forming ma- terials. Chemical physics, 284(1-2), 399-408.
[19] Abbas, S., Benchohra, M., Lazreg, J.E., & Zhou, Y. (2017). A survey on Hadamard and Hilfer fractional differential equations: analy- sis and stability. Chaos, Solitons & Fractals, 102, 47-71.
[20] Debbouche, A., & Antonov, V. (2017). Ap- proximate controllability of semilinear Hilfer fractional differential inclusions with impul- sive control inclusion conditions in Banach spaces. Chaos, Solitons & Fractals, 102, 140-148.
[21] Dineshkumar, C., Sooppy Nisar, K., Ud- hayakumar, R., & Vijayakumar, V. (2022). A discussion on approximate controllability of Sobolev-type Hilfer neutral fractional sto- chastic differential inclusions. Asian Journal of Control, 24(5), 2378-2394.
[22] Furati, K.M., & Kassim, M.D. (2012). Ex- istence and uniqueness for a problem involv- ing Hilfer fractional derivative. Computers & Mathematics with Applications, 64(6), 1616- 1626.
[23] Kavitha, K., Vijayakumar, V., & Udhayaku- mar, R. (2020). Results on controllability of Hilfer fractional neutral differential equations with infinite delay via measures of noncom- pactness. Chaos, Solitons & Fractals, 139, p.110035.
[24] Kavitha, K., Vijayakumar, V., Udhayaku- mar, R., Sakthivel, N., & Sooppy Nisar, K. (2021). A note on approximate controllabil- ity of the Hilfer fractional neutral differential inclusions with infinite delay. Mathematical Methods in the Applied Sciences, 44(6), 4428- 4447.
[25] Gu, H., & Trujillo, J.J. (2015). Existence of mild solution for evolution equation with Hil- fer fractional derivative. Applied Mathemat- ics and Computation, 257, 344-354.
[26] Nisar, K.S., & Vijayakumar, V. (2021). Re- sults concerning to approximate controllabil- ity of non-densely defined Sobolev-type Hil- fer fractional neutral delay differential sys- tem. Mathematical Methods in the Applied Sciences, 44(17), 13615-13632.
[27] Yang, M., & Wang, Q.R. (2017). Approx- imate controllability of Hilfer fractional dif- ferential inclusions with nonlocal conditions. Mathematical Methods in the Applied Sci- ences, 40(4), 1126-1138.
[28] Belmor, S., Ravichandran, C., & Jarad, F. (2020). Nonlinear generalized fractional dif- ferential equations with generalized fractional integral conditions. Journal of Taibah Univer- sity for Science, 14(1), 114-123.
[29] Jothimani, K., Kaliraj, K., Panda, S.K., Nisar, K.S., & Ravichandran, C. (2021). Re- sults on controllability of non-densely char- acterized neutral fractional delay differential system. Evolution Equations & Control The- ory, 10(3), p.619.
[30] Vijayaraj, V., Ravichandran, C., Botmart, T., Nisar, K.S., & Jothimani, K. (2023). Exis- tence and data dependence results for neutral fractional order integro-differential equations. AIMS Mathematics, 8(1), 1055-1071.
[31] Kaliraj, K., Priya, P.L., & Ravichandran, C. (2022). An Explication of Finite-Time Stabil- ity for Fractional Delay Model with Neutral Impulsive Conditions. Qualitative Theory of Dynamical Systems, 21(4), p.161.
[32] Jothimani, K., Ravichandran, C., Kumar, V., Djemai, M., & Nisar, K.S. (2022). In- terpretation of Trajectory Control and Opti- mization for the Nondense Fractional System. International Journal of Applied and Compu- tational Mathematics, 8(6), p.273.
[33] Nisar, K.S., Vijayaraj, V., Valliammal, N., Logeswari, K., Ravichandran, C., Abdel- Aty, A.H., & Yahia, I.S. (2022). A note on controllability of noninstantaneous impulsive atangana-baleanu-caputo neutral fractional integrodifferential systems. Fractals, 30(08), p.2240203.
[34] Nisar, K.S., Logeswari, K., Vijayaraj, V., Baskonus, H.M., & Ravichandran, C. (2022). Fractional order modeling the gemini virus in capsicum annuum with optimal control. Frac- tal and Fractional, 6(2), p.61.
[35] Miller, K. S., & Ross, B. (1993). An Introduc- tion to the Fractional Calculus and Fractional Differential Equations, Wiley, New York.
[36] Zhou, Y., & Jiao, F. (2010). Existence of mild solutions for fractional neutral evolution equations. Computers & Mathematics with Applications, 59(3), 1063-1077.