Results Concerning Exact Controllability of Nonlinear Volterra-Fredholm Integro-differential Equations of Fractional Order Via a Semigroup Approach
DOI:
https://doi.org/10.64229/n56qyp91Keywords:
NNVFFIE, Exact controllability, Fractional calculus, Fixed point theorem, Semigroup theoryAbstract
This manuscript explored the exact controllability of nonlinear nonlocal Volterra–Fredholm fractional integro-differential equations (NNVFFIE) in Banach spaces. Inspired the effectiveness of fractional-order models to capture memory and hereditary effects in biological, engineering, and physical systems, a fractional control system with the Caputo derivative is considered due to its relevance for initial value problems. Using tools from fractional calculus and the theory of strongly continuous semigroups, the control system is reformulated in an abstract integral form. Sufficient conditions for exact controllability are established using a fixed point theorem under appropriate compactness, boundedness, and growth assumptions on the nonlinear terms. The proposed framework provides a systematic approach to steering the system from a given initial state to a desired final state within a finite time interval. The obtained results extend existing exact controllability criteria for fractional integro-differential systems. An illustrative example is included to demonstrate the applicability of the abstract findings.
References
[1]Kulmus K, Essex C, Hoffmann KH, Prehl J. Fractional diffusion: A structured approach to application with examples. Mathematical and Computational Applications, 2025, 30(2), 40. DOI: 10.3390/mca30020040
[2]Žecová M, Terpák J. Heat conduction modeling by using fractional-order derivatives. Applied Mathematics and Computation, 2015, 257, 365-373. DOI: 10.1016/j.amc.2014.12.136
[3]Oldham KB. Fractional differential equations in electrochemistry. Advances in Engineering software, 2010, 41(1), 9-12. DOI: 10.1016/j.advengsoft.2008.12.012
[4]Ortigueira MD. Principles of fractional signal processing. Digital Signal Processing, 2024, 149, 104490. DOI: 10.1016/j.dsp.2024.104490
[5]Thomson D, Padula F. Introduction to fractional-order control: A practical laboratory approach. IFAC-PapersOnLine, 2022, 55(17), 126-131. DOI: 10.1016/j.ifacol.2022.09.268
[6]Podlubny, I. Fractional differential equations, Academic Press, New York, 1999. Available at: https://igor.podlubny.website.tuke.sk/fde.html (accessed on 18 December 2025).
[7]Zhou Y. Basic theory of fractional differential equations.World Scientific, Singapore, 2014. DOI: 10.1142/9069
[8]Malar K, Ilavarasi R. Existence, uniqueness and controllability results for fractional neutral integro-differential equations with non-instantaneous impulses and delay. Kragujevac Journal of Mathematics, 2026, 50(5), 683-724. DOI: 10.46793/KgJMat2605.683M
[9]Kumar K, Kumar P. Existence of solutions of fractional integrodifferential system with nonlocal conditions. IIUM Engineering Journal, 2015, 16(1), 33-41. DOI: 10.31436/iiumej.v16i1.535
[10]Hamoud AA, Mohammed NM. Existence and uniqueness results for fractional Volterra-Fredholmintegro differential equations with integral boundary conditions. Dynamics of Continuous, Discrete and Impulsive Systems, 2023, 30(1), 75-86. Available at: https://online.watsci.org/abstract_pdf/2023v30/v30n1a-pdf/4.pdf (accessed on 18 December 2025).
[11]Hettadj DE, Djourdem H. Existence and uniqueness for Caputo fractional differential equations with integral boundary condition. Gulf Journal of Mathematics, 2025, 21(1), 382-397. DOI: 10.56947/gjom.v21i1.3417
[12]Brociek R, Hetmaniok E, Słota D. Application of fractional derivatives in modeling the heat flow in the thermal protection system. Communications in Nonlinear Science and Numerical Simulation, 2025, 150, 109040. DOI: 10.1016/j.cnsns.2025.109040
[13]El Hassani A, Hattaf K, Achtaich N. Spatiotemporal dynamics of a fractional model for growth of coral in a tank with anomalous diffusion. Partial Differential Equations in Applied Mathematics, 2024, 9, 100656. DOI: 10.1016/j.padiff.2024.100656
[14]Mashayekhi S, Sedaghat S. Fractional model of stem cell population dynamics. Chaos, Solitons & Fractals, 2021, 146, 110919. DOI: 10.1016/j.chaos.2021.110919
[15]Bhangale N, Kachhia KB, Gómez‐Aguilar JF. Fractional viscoelastic models with Caputo generalized fractional derivative. Mathematical Methods in the Applied Sciences, 2023, 46(7), 7835-7846. DOI: 10.1002/mma.7229
[16]Duman O. Controllability analysis of fractional-order delay differential equations via contraction principle. Journal of Mathematical Sciences and Modelling, 2024, 7(3), 121-127. DOI: 10.33187/jmsm.1504151
[17]Ramos PS, Sousa JV, de Oliveira EC. Controllability of fractional impulsive integro-differential control system. HAL Open Science, 2021. Available at: https://hal.science/hal-03258774/file/Controllability_of_fractional_impulsive_integro_differential_control_system.pdf (accessed on 18 December 2025).
[18]Kumar K, Kumar R. Boundary controllability of delay differential systems of fractional order with nonlocal condition. Journal of Applied Nonlinear Dynamics, 2017, 6(4), 465-472. DOI:10.5890/JAND.2017.12.002
[19]Hamoud AA, Jameel SA, Mohammed NM, Emadifar H, Parvaneh F, Khademi M. On controllability for fractional Volterra-Fredholm system. Nonlinear Functional Analysis and Applications, 2023, 407-420. DOI: 10.22771/nfaa.2023.28.02.06
[20]Gautam P, Shukla A, Johnson M, Vijayakumar V. Approximate controllability of third order dispersion systems. Bulletin des Sciences Mathématiques, 2024, 191, 103394. DOI: 10.1016/j.bulsci.2024.103394
[21]Hussain S, Sarwar M, Nisar KS, Shah K. Controllability of fractional differential evolution equation of order γ∈(1, 2) with nonlocal conditions. AIMS Math, 2023, 8(6), 14188-14206. DOI: 10.3934/math.2023726
[22]Aimene D, Seba D, Laoubi K. Controllability of impulsive fractional functional evolution equations with infinite state‐dependent delay in Banach spaces. Mathematical Methods in the Applied Sciences, 2021, 44(10), 7979-7994. DOI: 10.1002/mma.5644
[23]Aissani K, Benchohra M. Controllability of fractional integrodifferential equations with state-dependent delay. Journal of Integral Equations and Applications, 2016, 28(2), 149-167. DOI:10.1216/JIE-2016-28-2-149
[24]Matar MM. On controllability of linear and nonlinear fractional integrodifferential systems. Fractional Differential Calculus 2019, 9(1), 19-32. DOI: 10.7153/fdc-2019-09-02
[25]Cheng Y, Gao S, Wu Y. Exact controllability of fractional order evolution equations in Banach spaces. Advances in Difference Equations, 2018, 2018(1), 332. DOI: 10.1186/s13662-018-1794-5
[26]Górniewicz L, Ntouyas SK, O'regan D. Controllability of semilinear differential equations and inclusions via semigroup theory in Banach spaces. Reports on Mathematical Physics, 2005, 56(3), 437-470. DOI: 10.1016/S0034-4877(05)80096-5
[27]Naimi A, Brahim T, Zennir K. Existence and stability results for the solution of neutral fractional integro-differential equation with nonlocal conditions. Tamkang Journal of Mathematics, 2022, 53(3), 239-257. DOI: 10.5556/j.tkjm.53.2022.3550
[28]Chalishajar DN. Controllability of mixed Volterra–Fredholm-type integro-differential systems in Banach space. Journal of the Franklin Institute, 2007, 344(1), 12-21. DOI: 10.1016/j.jfranklin.2006.04.002
[29]Pazy A. Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, 1983. DOI: 10.1007/978-1-4612-5561-1
[30]Kumar S. Approximate controllability of nonlocal impulsive fractional order semilinear time varying delay systems. Nonlinear Dynamics and Systems Theory, 2016, 16(4), 420-30. Available at: https://e-ndst.kiev.ua/v16n4/V16N4.pdf#page=90 (accessed on 18 December 2025).
[31]Kaliraj K, Muthuvel K. A study on the approximate controllability results of fractional stochastic integro-differential inclusion systems via sectorial operators. International Journal of Optimization & Control: Theories & Applications, 2023, 13(2), 193-204. DOI: 10.11121/ijocta.2023.1348
[32]Alsulami HH, Ntouyas SK, Al-Mezel SA, Ahmad B, Alsaedi A. A study of third-order single-valued and multi-valued problems with integral boundary conditions. Boundary Value Problems, 2015, 2015(1), 25. DOI: 10.1186/s13661-014-0271-7
[33]Liang Y, Fan Z, Li G. Existence, uniqueness and regularity of solutions for fractional integro-differential equations with state-dependent delay. Journal of Applied Analysis and Computation, 2024, 14, 623-641. DOI: 10.11948/20220263
[34]Quinn MD, Carmichael N. An approach to non-linear control problems using fixed-point methods, degree theory and pseudo-inverses. Numerical Functional Analysis and Optimization, 1985, 7(2-3), 197-219. DOI: 10.1080/01630568508816189
[35]Kumar K, Kumar R. A discussion on controllability of semilinear impulsive functional differential equations of second order. TWMS Journal of Applied and Engineering Mathematics. 2025, 15(3), 590-601. Available at: https://jaem.isikun.edu.tr/web/index.php/current/129-vol15no3/1350 (accessed on 18 December 2025).
[36]Diop A, Fall M, Diop MA, Ezzinbi K. Existence and controllability results for integrodifferential equations with state-dependent delay and random effects. Filomat, 2022, 36(4), 1363-1379. DOI: 10.2298/FIL2204363D
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Kamalendra Kumar, Manish Nath Tripathi (Author)
This work is licensed under a Creative Commons Attribution 4.0 International License.
