TY  - JOUR
T1  - Exact Controllability for Nonlinear Implicit Caputo Fractional Models with Hybrid Structure
AU  - Duman, Okan
PY  - 2025
DA  - November
Y2  - 2025
DO  - 10.36753/mathenot.1783161
JF  - Mathematical Sciences and Applications E-Notes
JO  - Math. Sci. Appl. E-Notes
PB  - Murat TOSUN
WT  - DergiPark
SN  - 2147-6268
SP  - 190
EP  - 200
VL  - 13
IS  - 4
LA  - en
AB  - Existing research on fractional hybrid differential equations (abbrev. FHDEs) has largely focused on results concerning the existence, uniqueness or stability of solutions, often under restrictive assumptions and relying on contraction mappings with constants less than one or auxiliary conditions. However, the controllability of nonlinear hybrid implicit fractional systems with control inputs has only been partially addressed. In this paper, we advance the theory by analysing a generalised class of Caputo FHDEs that include explicit control functions. Our contributions are threefold. Firstly, we establish an equivalence theorem that clarifies the precise notion of a solution, forming the basis for all subsequent arguments. Secondly, we prove the existence of solutions under fixed control using Schauder's fixed-point theorem, with auxiliary constructions supported by the Banach contraction principle. Third, we demonstrate exact controllability under assumptions that are strictly weaker than those required in earlier works. These results broaden the class of admissible systems and highlight the flexibility of fixed-point techniques beyond conventional approaches. A physical example illustrates the applicability of the theoretical findings.
KW  - Controllability
KW  - Existence of solution
KW  - Fractional hybrid differential equation
CR  - [1] Boulaaras, S., Jan, R., Pham, V.: Recent advancement of fractional calculus and its applications in physical systems. The European Physical Journal Special Topics. 232 (14), 2347–2350 (2023).
CR  - [2] Tarasov, V. E.: Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer Science and Business Media, 2011.
CR  - [3] Sun, H., Zhang, Y., Baleanu, D., Chen, W., Chen Y.: A new collection of real world applications of fractional calculus in science and engineering. Communications in Nonlinear Science and Numerical Simulation. 64, 213–231 (2018).
CR  - [4] Vijayakumar, V., Nisar, K. S., Chalishajar, D., Shukla, A., Malik, M., Alsaadi, A., Aldosary, S.: A note on approximate controllability of fractional semilinear integrodifferential control systems via resolvent operators. Fractal and Fractional. 6 (2), 73 (2022).
CR  - [5] Raja, M., Vijayakumar, V., Shukla, A., Nisar, K., Baskonus, M.:On the approximate controllability results for fractional integrodifferential systems of order 1 < r < 2 with sectorial operators. Journal of Computational and Applied Mathematics. 415, 114492 (2022).
CR  - [6] Duman, O.: Controllability analysis of fractional-order delay differential equations via contraction principle. Journal of Mathematical Sciences and Modelling. 7 (3), 120–126 (2024).
CR  - [7] Nisar, K. S., Muthuselvan, K.: A new effective technique of nonlocal controllability criteria for state delay with impulsive fractional integro-differential equation. Results in Applied Mathematics. 21, 100437 (2024).
CR  - [8] Shukla, A., Vijayakumar, V., Nisar, K. S.: 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 (2022).
CR  - [9] Jothilakshmi, G., Sundaravadivoo, B., Nisar, K. S., Alsaeed, S.: Impulsive fractional integro-delay differential equation-controllability through delayed Mittag-Leffler function perturbation. International Journal of Dynamics and Control. 12 (11), 4178–4187 (2024).
CR  - [10] Dhage, B. C., Lakshmikantham, V.: Basic results on hybrid differential equations. Nonlinear Analysis: Hybrid Systems. 4 (3), 414–424 (2010).
CR  - [11] Zhao, Y., Sun, S., Han, Z., Li, Q.: Theory of fractional hybrid differential equations. Computers & Mathematics with Applications. 62 (3), 1312–1324 (2011).
CR  - [12] Benchohra, M., Lazreg, J. E.: On stability for nonlinear implicit fractional differential equations. Le Matematiche. 70 (2), 49–61 (2015).
CR  - [13] Ardjouni, A., Djoudi, A.: Initial-value problems for nonlinear hybrid implicit Caputo fractional differential equations. Malaya Journal of Matematik. 7 (2), 314–317 (2019).
CR  - [14] Lachouri, A., Ardjouni, A., Djoudi, A.: Existence and Ulam stability results for nonlinear hybrid implicit Caputo fractional differential equations. Mathematica Moravica. 24 (1), 109–122 (2020).
CR  - [15] Burton, T. A.: A fixed-point theorem of Krasnoselskii. Applied Mathematics Letters. 1(1), 85–88 (1998).
CR  - [16] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and Applications of Fractional Differential Equations. Elsevier, Boston, 2006.
UR  - https://doi.org/10.36753/mathenot.1783161
L1  - https://dergipark.org.tr/en/download/article-file/5237947
ER  -