Optimal controllability of stochastic multi-term Hilfer fractional systems with delayed controls

Javed Akhtar; Abdur Raheem; Faizan Ahmad Khan; Adel Alatawi; Fahad M. Alamrani; Asma Afreen; Areefa Khatoon; Javed Akhtar; Abdur Raheem; Faizan Ahmad Khan; Adel Alatawi; Fahad M. Alamrani; Asma Afreen; Areefa Khatoon

doi:10.3934/math.2026285

AIMS Mathematics

2026, Volume 11, Issue 3: 6935-6954. doi: 10.3934/math.2026285

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Optimal controllability of stochastic multi-term Hilfer fractional systems with delayed controls

1.
Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India
2.
Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk-71491, Saudi Arabia
3.
Department of Mathematics, C. V. Raman Global University, Bhubaneswar-752054, India
4.
Symbiosis Institute of Technology, Pune Campus, Symbiosis International, Deemed University, Pune-412115, India

Received: 23 January 2026 Revised: 05 March 2026 Accepted: 09 March 2026 Published: 17 March 2026
MSC : 26A33, 34K50, 47H10, 60H15, 93E20

This paper investigates the optimal controllability of stochastic multi-term Hilfer fractional differential systems with finite delays in the control function. The considered model combines multi-order fractional dynamics, stochastic perturbations, and delayed control actions, which provide a unified framework to describe complex dynamical processes with memory and uncertainty. By employing the theory of fractional calculus and the concept of a $ (\delta, \beta_{\kappa}) $-resolvent family, we establish the existence and uniqueness of mild solutions for the proposed system. Balder's theorem and variational techniques demonstrate the existence of an optimal state-control pair minimising a quadratic cost functional. A diffusion-type fractional stochastic model with delayed control is presented to illustrate the applicability of the theoretical findings. The obtained results not only extend several existing works on fractional stochastic control systems, but also provide new insights into the analysis of Hilfer-type operators with multiple fractional orders and control delays, which are relevant in viscoelastic materials, heat transfer processes, and neural network models with memory effects.
- Hilfer fractional derivative,
- optimal control,
- stochastic system,
- delay equation,
- fixed-point theorem
Citation: Javed Akhtar, Abdur Raheem, Faizan Ahmad Khan, Adel Alatawi, Fahad M. Alamrani, Asma Afreen, Areefa Khatoon. Optimal controllability of stochastic multi-term Hilfer fractional systems with delayed controls[J]. AIMS Mathematics, 2026, 11(3): 6935-6954. doi: 10.3934/math.2026285

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Abstract

This paper investigates the optimal controllability of stochastic multi-term Hilfer fractional differential systems with finite delays in the control function. The considered model combines multi-order fractional dynamics, stochastic perturbations, and delayed control actions, which provide a unified framework to describe complex dynamical processes with memory and uncertainty. By employing the theory of fractional calculus and the concept of a $ (\delta, \beta_{\kappa}) $-resolvent family, we establish the existence and uniqueness of mild solutions for the proposed system. Balder's theorem and variational techniques demonstrate the existence of an optimal state-control pair minimising a quadratic cost functional. A diffusion-type fractional stochastic model with delayed control is presented to illustrate the applicability of the theoretical findings. The obtained results not only extend several existing works on fractional stochastic control systems, but also provide new insights into the analysis of Hilfer-type operators with multiple fractional orders and control delays, which are relevant in viscoelastic materials, heat transfer processes, and neural network models with memory effects.

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