Controllability of Nonlocal Impulsive Semilinear Differential Inclusions with Fractional Sectorial Operators and Infinite Delay

(Feryal Abdullah Al Adsani and Ahmed Gamal Ibrahim)

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Abstract

This paper demonstrates the controllability of two fractional nonlocal impulsive semilinear differential inclusions with infinite delay, where the linear part is a fractional sectorial operator and the nonlinear term is a multivalued function. The operator families generated by the linear part are not assumed to be compact. The objective is achieved using the properties of fractional sectorial operators and the Hausdorff measure of noncompactness. The results generalise several recent findings, and the method can be used to extend further contributions to cases where the linear term is a fractional sectorial operator and the nonlinear term is a multivalued function, in the presence of instantaneous impulses and infinite delays. The novelty of this work lies in initiating the study of the controllability of a system involving a fractional Caputo derivative under infinite impulses and delays. An example is presented to verify the theoretical developments. Given the wide-ranging applications of fractional calculus in medicine, energy and other scientific fields, this work contributes to those domains.
KEYWORDS
Caputo derivative, mazur's lemma, mild solutions, multivalued functions, noncompact measure, phase space

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References

Abbas, M.I. (2020). On the controllability of Hilfer-Katugampola fractional differential equations. Acta et Commentationes Universitatis Tartuensis de Mathematica, 24(2), 195–204. DOI: 10.12697/ACUTM.2020.24.13 
Al Zubi, M.A., Afef, K. and Az-Zo’bi, E.A. (2024). Assorted spatial optical dynamics of a generalized fractional quadruple nematic liquid crystal system in non-local media. Symmetry, 16(6), 778. DOI: 10.3390/sym16060778
Aladsani, F. and Ibrahim, A.G. (2024). Existence and stability of solutions for p-proportional ω-weighted κ-hilfer fractional differential inclusions in the presence of non-instantaneous impulses in banach spaces. Fractal and Fractional, 8(8), 475. DOI: 10.3390/fractalfract8080475
Almarri, B. and Elshenhab, A.M. (2022). Controllability of fractional stochastic delay systems driven by the Rosenblatt process. Fractal and Fractional, 6(11), 664. DOI: 10.3390/fractalfract6110664
Alsarori, N. and Ghadle, K. (2022). Existence and controllability of fractional evolution inclusions with impulse and sectorial operator. Results in Nonlinear Analysis, 5(3), 235–49. DOI: 10.53006/rna.1018780
Alsheekhhussain, Z. and Ibrahim, A.G. (2021). Controllability of Semilinear Multi-Valued Differential Inclusions with Non-Instantaneous Impulses of Order α∈(1, 2) without Compactness. Symmetry, 13(4), 566. DOI: 10.3390/sym13040566
Az-Zo’bi, E.A., Afef, K., Ur Rahman, R., Akinyemi, L., Bekir, A., Ahmad, H. and Mahariq, I. (2024). Novel topological, non-topological, and more solitons of the generalized cubic p-system describing isothermal flux. Optical and Quantum Electronics, 56(1), 84. DOI: 10.1007/s11082-023-05642-7
Bader, R., Kamenskiĭ, M. and Obukhovskiĭ, V. (2001). On some classes of operator inclusions with lower semicontinuous nonlinearities. Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center. 17(1): 143–56
Baleanu, D. and Lopes, A.M. (2019). Handbook of fractional calculus with applications. Applications in Engineering, Life and Social Sciences, Part A, Southampton: Comput Mech Publicat, 7. De Gruyter. DOI: 10.1515/9783110571929
Bose, C.V. and Udhayakumar, R. (2023). Analysis on the controllability of Hilfer fractional neutral differential equations with almost sectorial operators and infinite delay via measure of noncompactness. Qualitative Theory of Dynamical Systems, 22(1), 22. DOI: 10.1007/s12346-022-00719-2
Butt, A.I.K., Imran, M., Batool, S. and Nuwairan, M.A. (2023). Theoretical analysis of a COVID-19 CF-fractional model to optimally control the spread of pandemic. Symmetry, 15(2), 380. DOI: 10.3390/sym15020380
Cardinali, T. and Rubbioni, P. (2012). Impulsive mild solutions for semilinear differential inclusions with nonlocal conditions in Banach spaces. Nonlinear Analysis: Theory, Methods and Applications, 75(2), 871–9. DOI: 10.1016/j.na.2011.09.023
Chalishajar, D., Ravikumar, K., Ramkumar, K. and Anguraj, A. (2024). Null controllability of Hilfer fractional stochastic differential equations with nonlocal conditions. Numerical Algebra, Control and Optimization, 14(2), 322–38. DOI: 10.3934/naco.2022029
Dineshkumar, C. and Udhayakumar, R. (2022). Results on approximate controllability of fractional stochastic Sobolev‐type Volterra–Fredholm integro‐differential equation of order 1< r< 2. Mathematical Methods in the Applied Sciences, 45(11), 6691–704. DOI: 10.1002/mma.8200
Dineshkumar, C., Udhayakumar, R., Vijayakumar, V., Nisar, K.S. and Shukla, A. (2022). A note concerning to approximate controllability of Atangana-Baleanu fractional neutral stochastic systems with infinite delay. Chaos, Solitons and Fractals, 157(n/a), 111916. DOI: 10.1016/j.chaos.2022.111916
Hale, J.K. and Kato, J. (1978). Phase spaces for retarded equations with infinite delay. Funkcialaj Ekvacioj, 21(n/a), 11–41.
Hassan, T.S., Gamal Ahmed, R., El-Sayed, A.M., El-Nabulsi, R.A., Moaaz, O. and Mesmouli, M.B. (2022). Solvability of a State–Dependence Functional Integro-Differential Inclusion with Delay Nonlocal Condition. Mathematics, 10(14), 2420. DOI: 10.3390/math10142420
Kamenskii, M.I., Obukhovskii, V.V. and Zecca, P. (2011). Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces (Vol. 7). Germany: Walter de Gruyter.
Karthikeyan, K., Tamizharasan, D., Nieto, J.J. and Nisar, K.S. (2021). Controllability of second-order differential equations with state-dependent delay. IMA Journal of Mathematical Control and Information, 38(4), 1072–83. DOI: 10.1093/imamci/dnab027
Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations (Vol. 204). Netherland: Elsevier.
Kumar, A., Jeet, K. and Vats, R.K. (2022). Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations and Control Theory, 11(2), 605–19. DOI: 10.3934/eect.2021016
Kumar, A., Vats, R.K., Kumar, A. and Chalishajar, D.N. (2020). Numerical approach to the controllability of fractional order impulsive differential equations. Demonstratio Mathematica, 53(1), 193–207. DOI: 10.1515/dema-2020-0015
Kumar, S. (2023). On approximate controllability of non-autonomous measure driven systems with non-instantaneous impulse. Applied Mathematics and Computation, 441(n/a), 127695. DOI: 10.1016/j.amc.2022.127695
Mohan Raja, M., Vijayakumar, V., Udhayakumar, R. and Nisar, K.S. (2024). Results on existence and controllability results for fractional evolution inclusions of order 1< r< 2 with Clarke's subdifferential type. Numerical Methods for Partial Differential Equations, 40(1), e22691. DOI: 10.1002/num.22691
O'Regan, D. and Precup, R. (2000). Fixed point theorems for set-valued maps and existence principles for integral inclusions. Journal of Mathematical Analysis and Applications, 245(2), 594–612. DOI: 10.1006/jmaa.2000.6789
Raja, M.M., Vijayakumar, V. and Veluvolu, K.C. (2025). Higher-order caputo fractional integrodifferential inclusions of Volterra–Fredholm type with impulses and infinite delay: existence results. Journal of Applied Mathematics and Computing, n/a(n/a)1–26. DOI: 10.1007/s12190-025-02412-4
Raja, M.M., Vijayakumar, V., Shukla, A., Nisar, K.S. and Baskonus, H.M. (2022). On the approximate controllability results for fractional integrodifferential systems of order 1< r< 2 with sectorial operators. Journal of Computational and Applied Mathematics, 415(n/a), 114492. DOI: 10.1016/j.cam.2022.114492
Ren, L., Wang, J. and O’Regan, D. (2019). Asymptotically periodic behavior of solutions of fractional evolution equations of order 1< α< 2. Mathematica Slovaca, 69(3), 599–610. DOI: 10.1515/ms-2017-0250
Salem, A. and Alharbi, K.N. (2023). Controllability for fractional evolution equations with infinite time-delay and non-local conditions in compact and noncompact cases. Axioms, 12(3), 264. DOI: 10.3390/axioms12030264
Slama, A. and Boudaoui, A. (2017). Approximate controllability of fractional nonlinear neutral stochastic differential inclusion with nonlocal conditions and infinite delay. Arabian Journal of Mathematics, 6(n/a), 31–54. DOI: 10.1007/s40065-017-0163-7
Sudsutad, W., Thaiprayoon, C., Kongson, J. and Sae-dan, W. (2024). A mathematical model for fractal-fractional monkeypox disease and its application to real data. AIMS Mathematics, 9(4), 8516–8563. DOI: 10.3934/math.2024414
Varun Bose, C.S., Udhayakumar, R., Elshenhab, A.M., Kumar, M.S. and Ro, J.S. (2022). Discussion on the approximate controllability of Hilfer fractional neutral integro-differential inclusions via almost sectorial operators. Fractal and Fractional, 6(10), 607. DOI: 10.3390/fractalfract6100607
Wang, J., Ibrahim, A.G. and Fečkan, M. (2015). Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces. Applied Mathematics and Computation, 257(n/a), 103–18. DOI: 10.1016/j.amc.2014.04.093
Wang, J., Ibrahim, G. and O’Regan, D.D. (2019). Controllability of Hilfer fractional noninstantaneous impulsive semilinear differential inclusions with nonlocal conditions. Nonlinear Analysis: Modelling and Control, 24(6), 958–84. DOI: 10.15388/NA.2019.6.7
Wang, J., IbrahimA, A.G. and O'Regan, D. (2020). Finite approximate controllability of Hilfer fractional semilinear differential equations. Miskolc Mathematical Notes, 21(1), 489–507. DOI: 10.18514/MMN.2020.2921 
Yang, M. and Wang, Q. (2016). Approximate controllability of Riemann–Liouville fractional differential inclusions. Applied Mathematics and Computation, 274(n/a), 267–81. DOI: 10.1016/j.amc.2015.11.017
Zhang, X., Chen, P., Abdelmonem, A. and Li, Y. (2019). Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroups. Mathematica Slovaca, 69(1), 111–24. DOI: 10.1515/ms-2017-0207