Approximate controllability of higher-order Sobolev-type stochastic AB-fractional differential inclusions with impulses and Clarke sub-differentials

A. M. Sayed Ahmed; Taha Radwan; Yakup Yildirim; Hamdy M. Ahmed; A. M. Sayed Ahmed; Taha Radwan; Yakup Yildirim; Hamdy M. Ahmed

doi:10.3934/math.2026347

AIMS Mathematics

2026, Volume 11, Issue 3: 8428-8466. doi: 10.3934/math.2026347

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Approximate controllability of higher-order Sobolev-type stochastic AB-fractional differential inclusions with impulses and Clarke sub-differentials

1.
Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Alexandria, Egypt
2.
Department of Management Information Systems, College of Business and Economics, Qassim University, Buraydah 51452, Saudi Arabia
3.
Department of Computer Engineering, Biruni University, Istanbul–34010, Turkey
4.
Mathematics Research Center, Near East University, 99138 Nicosia, Cyprus
5.
Department of Physics and Engineering Mathematics, Higher Institute of Engineering, El-Shorouk Academy, El-Shorouk City, Cairo, Egypt

Received: 08 January 2026 Revised: 13 March 2026 Accepted: 24 March 2026 Published: 30 March 2026
MSC : 26A33, 34A60, 34K45, 60H15, 93B05, 93E03

In this paper, we investigated the existence of mild solutions and the approximate controllability of a novel class of Sobolev-type stochastic impulsive differential inclusions driven by a higher-order Atangana–Baleanu fractional derivative in the Caputo sense. The system is formulated in an infinite-dimensional framework and incorporates Brownian motion processes, impulsive effects, and non-smooth multi-valued nonlinearities described via Clarke's generalized sub-differential. By employing methods from fractional evolution theory, stochastic analysis, and multi-valued fixed-point theory, we established sufficient conditions for solvability and approximate controllability. The results extended classical controllability frameworks to systems exhibiting memory, randomness, and impulsive dynamics. An illustrative example is provided to demonstrate the applicability of the theoretical findings.
- fractional derivatives,
- Sobolev-type,
- stochastic differential inclusions,
- impulsive systems,
- approximate controllability,
- Clarke sub-differential,
- nonlocal conditions,
- brownian motion process,
- mild solutions
Citation: A. M. Sayed Ahmed, Taha Radwan, Yakup Yildirim, Hamdy M. Ahmed. Approximate controllability of higher-order Sobolev-type stochastic AB-fractional differential inclusions with impulses and Clarke sub-differentials[J]. AIMS Mathematics, 2026, 11(3): 8428-8466. doi: 10.3934/math.2026347

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Abstract

In this paper, we investigated the existence of mild solutions and the approximate controllability of a novel class of Sobolev-type stochastic impulsive differential inclusions driven by a higher-order Atangana–Baleanu fractional derivative in the Caputo sense. The system is formulated in an infinite-dimensional framework and incorporates Brownian motion processes, impulsive effects, and non-smooth multi-valued nonlinearities described via Clarke's generalized sub-differential. By employing methods from fractional evolution theory, stochastic analysis, and multi-valued fixed-point theory, we established sufficient conditions for solvability and approximate controllability. The results extended classical controllability frameworks to systems exhibiting memory, randomness, and impulsive dynamics. An illustrative example is provided to demonstrate the applicability of the theoretical findings.

References

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