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Peer-reviewedCC BY 4.0Fractal and Fractional

Finite-Time Stability of Linear Stochastic Proportional Fractional-Order Systems with Time Delay

  • Raouf Fakhfakh1
  • Rabab Alzahrani2
  • Fatimah Alshahrani3
  • Hend Aljahani4
  • Foued Mtiri5
  • Abdellatif Ben Makhlouf6
6 affiliations
  1. 1Department of Mathematics, College of Science, Jouf University, P.O. Box 2014, Sakaka 72388, Saudi Arabia
  2. 2Department of Mathematics, College of Science and Humanities, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
  3. 3Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
  4. 4Department of Mathematics, College of Science, Northern Border University, Arar 91431, Saudi Arabia
  5. 5Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia
  6. 6Mathematics Education Section, Faculty of Education and Arts, Sohar University, P.O. Box 44, Sohar 311, Oman
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Abstract

This paper studies the finite-time stability of linear stochastic fractional-order systems with time delay, in which the dynamics are generated by the proportional Caputo fractional derivative of order ς∈(12,1) with proportional parameter ϖ∈(0,1]. Starting from the integral representation of solutions and using the Itô isometry, we derive a mean-square estimate in which the exponential factor of the proportional kernel sharpens the constants. Combining this estimate with the classical Gronwall inequality and with a generalized one, we establish two finite-time stability criteria: an iterative criterion built on the delay intervals and a closed-form criterion of the Mittag–Leffler type. For ϖ=1, both criteria reduce exactly to known results for Caputo-type stochastic delay systems, so the present results extend the theory to the whole proportional family. The two criteria are compared analytically, a weighted refinement that reduces their conservatism is given, and the influence of the memory of the fractional operator and of the proportional parameter is discussed. Numerical simulations, based on a Monte Carlo scheme for the integral representation of the solution with quantified sampling error, illustrate the two criteria, compare their certified stability horizons, and confirm the theoretical findings.

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Cite

BibTeXRIS
BibTeX
@article{fakhfakh2026finitetime,
  title = {{Finite-Time Stability of Linear Stochastic Proportional Fractional-Order Systems with Time Delay}},
  author = {Raouf Fakhfakh and Rabab Alzahrani and Fatimah Alshahrani and Hend Aljahani and Foued Mtiri and Abdellatif Ben Makhlouf},
  journal = {Fractal and Fractional},
  year = {2026},
  month = oct,
  doi = {10.3390/fractalfract10100697},
  url = {https://www.mdpi.com/2504-3110/10/10/697},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Finite-Time Stability of Linear Stochastic Proportional Fractional-Order Systems with Time Delay
AU  - Raouf Fakhfakh
AU  - Rabab Alzahrani
AU  - Fatimah Alshahrani
AU  - Hend Aljahani
AU  - Foued Mtiri
AU  - Abdellatif Ben Makhlouf
PY  - 2026
DA  - 2026/10/04
JO  - Fractal and Fractional
DO  - 10.3390/fractalfract10100697
UR  - https://www.mdpi.com/2504-3110/10/10/697
AB  - This paper studies the finite-time stability of linear stochastic fractional-order systems with time delay, in which the dynamics are generated by the proportional Caputo fractional derivative of order ς∈(12,1) with proportional parameter ϖ∈(0,1]. Starting from the integral representation of solutions and using the Itô isometry, we derive a mean-square estimate in which the exponential factor of the proportional kernel sharpens the constants. Combining this estimate with the classical Gronwall inequality and with a generalized one, we establish two finite-time stability criteria: an iterative criterion built on the delay intervals and a closed-form criterion of the Mittag–Leffler type. For ϖ=1, both criteria reduce exactly to known results for Caputo-type stochastic delay systems, so the present results extend the theory to the whole proportional family. The two criteria are compared analytically, a weighted refinement that reduces their conservatism is given, and the influence of the memory of the fractional operator and of the proportional parameter is discussed. Numerical simulations, based on a Monte Carlo scheme for the integral representation of the solution with quantified sampling error, illustrate the two criteria, compare their certified stability horizons, and confirm the theoretical findings.
SN  - 2504-3110
ER  -