<p>This paper investigates the controllability and stability of semilinear stochastic fractional functional differential equations with impulsive effects and piecewise Caputo derivatives. The main challenge is establishing controllability, which arises from the combined influence of impulsive discontinuities and the piecewise fractional structure, which naturally splits the solution and the associated control into different subintervals. This interval-wise structure complicates the direct application of standard controllability techniques. To address this analytical difficulty, the control function is decomposed according to the underlying subintervals, and auxiliary constants <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\beta _1, \beta _2 \in {\mathbb {R}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> satisfying <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\beta _1 + \beta _2 = 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>β</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>β</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> are introduced into the control representation. This formulation enables a unified controllability analysis without treating each subinterval separately and leads to sufficient conditions ensuring controllability of the system. Furthermore, sufficient conditions are derived for the periodicity of solutions. The system is also shown to possess exponential stability and Mittag-Leffler Hyers–Ulam–Rassias stability. Finally, a numerical example is presented to illustrate the applicability and effectiveness of the theoretical results.</p>

Controllability and Stability Results for Impulsive Stochastic Piecewise Fractional Differential Equations

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发表日期：

2026年4月24日

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B. K. Bharath,
J. J. Nieto,
P. Prakash

摘要

This paper investigates the controllability and stability of semilinear stochastic fractional functional differential equations with impulsive effects and piecewise Caputo derivatives. The main challenge is establishing controllability, which arises from the combined influence of impulsive discontinuities and the piecewise fractional structure, which naturally splits the solution and the associated control into different subintervals. This interval-wise structure complicates the direct application of standard controllability techniques. To address this analytical difficulty, the control function is decomposed according to the underlying subintervals, and auxiliary constants $\beta _1, \beta _2 \in {\mathbb {R}}$ $β_{1}, β_{2} \in R$ satisfying $\beta _1 + \beta _2 = 1$ $β_{1} + β_{2} = 1$ are introduced into the control representation. This formulation enables a unified controllability analysis without treating each subinterval separately and leads to sufficient conditions ensuring controllability of the system. Furthermore, sufficient conditions are derived for the periodicity of solutions. The system is also shown to possess exponential stability and Mittag-Leffler Hyers–Ulam–Rassias stability. Finally, a numerical example is presented to illustrate the applicability and effectiveness of the theoretical results.