EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6820
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Stochastic Stability and Controllability Analysis of
Impulsive Multi-Term Fractional Differential System

M. Lavanya1, B. Sundara Vadivoo2, Ibrahim Alraddadi3,∗, Dragan Pamucar4,∗,
Osama Oqilat5, Hijaz Ahmad6,3,7,8

1 Department of Mathematics, Alagappa University, Karaikudi-630 004, India
2 Department of Mathematics, Central University of Tamil Nadu, Thiruvarur-610 005, India
3 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah,
Saudi Arabia
4 Széchenyi István University, Győr, Hungary
5 Department of Basic Sciences, Faculty of Arts and Science, Hourani Center for Applied
Scientific Research, Al-Ahliyya Amman University, Amman, Jordan
6 Operational Research Center in Healthcare, Near East University, Nicosia/TRNC,
99138 Mersin 10, Turkey
7 VIZJA University, Okopowa 59, 01-043 Warsaw, Poland
8 Department of Mathematics, College of Science, Korea University, 145 Anam-ro,
Seongbuk-gu, Seoul 02841, South Korea

Abstract. This study investigates the existence, stability, and controllability of multi-term stochas-
tic fractional impulsive differential equations. By employing the contraction mapping principle,
sufficient conditions ensuring stochastic stability are rigorously established. Two classes of sys-
tems—linear and nonlinear are analyzed in detail. Furthermore, the controllability of these sys-
tems is demonstrated using the Gramian operator approach. To validate the theoretical findings,
numerical simulations are performed in MATLAB, illustrating the effectiveness and applicability
of the proposed results.
2020 Mathematics Subject Classifications: 26A33, 44A10, 60H20, 93E03, 93E15
Key Words and Phrases: Stochastic impulsive systems, multi-term fractional derivative, Laplace
transform, stochastic stability, controllability, Rosenblatt process

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6820

Email addresses: lavanyamuruganantham1997s@gmail.com (M. Lavanya),
bsundaravadivoo@gmail.com (B. S. Vadivoo), ialraddadi@iu.edu.sa (I. Alraddadi),
pamucar.dragan@sze.hu (D. Pamucar), o.oqilat@ammanu.edu.jo (O. Oqilat),
hijazahmad@korea.ac.kr (H. Ahmad)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



M. Lavanya et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6820 2 of 20

1. Introduction

Fractional stochastic differential equations (FSDEs) combine the concepts of fractional
calculus and stochastic processes. They extend the classical theory of stochastic differen-
tial equations (SDEs) by introducing fractional derivatives, which capture non-local and
memory-dependent behaviors in the underlying stochastic processes. FSDEs are employed
to model systems in which both randomness and fractional dynamics play significant roles.
Recent studies have developed efficient analytical and numerical approaches for solving
such equations, including shooting methods, bilinear time-series frameworks, and frac-
tional wave-based models [1–3]. These advances demonstrate the versatility of FSDEs in
modeling complex systems that exhibit both stochastic and long-memory characteristics.
Their applications span multiple disciplines and contribute to a deeper understanding of
processes with intricate dynamical behaviors. As technology and research continue to
evolve, FSDEs are expected to find broader and more innovative applications across vari-
ous scientific and engineering domains (see also [4–7]). The Caputo fractional derivative is
commonly used to extend the concept of differentiation to non-integer orders. It provides
a practical means of describing fractional-order dynamics, being defined as a combination
of the classical derivative and the Riemann–Liouville fractional integral.

A multi-term Caputo fractional derivative extends this concept by considering a sum
of multiple terms, each involving a Caputo fractional derivative with different orders and
weights. This can lead to more flexible and accurate modeling of complex systems that
exhibit a combination of different fractional-order behaviors. In materials science and engi-
neering, viscoelastic materials exhibit time-dependent responses that can be characterized
by fractional calculus. Multi-term Caputo fractional derivatives can model viscoelastic ma-
terials with different relaxation times and memory effects. Multi-term fractional deriva-
tives can model networks with different types of interactions that have fractional-order
characteristics. This is particularly relevant in studies of complex systems and network
dynamics (see [8–15]). An impulsive fractional stochastic differential equation (IF-SDE)
combines elements of fractional calculus, impulsive differential equations, and stochastic
processes. An impulsive differential equation involves sudden changes in the state of a
system at discrete moments in time. These changes are modeled as impulses or jumps.
IF-SDEs can have various specific forms depending on the precise nature of the frac-
tional derivative, the impulsive term, and the underlying dynamics. These equations find
applications in modeling phenomena with memory effects, sudden jumps, and random
influences, such as financial processes, biological systems, and anomalous diffusion (see
[16]). Stochastic stability refers to the property of a stochastic system to remain bounded
and exhibit well-behaved behavior over time, even in the presence of random fluctuations
or noise. In other words, a stochastically stable system does not undergo excessive or
unpredictable changes due to the inherent randomness in its dynamics. Stochastic sta-
bility is particularly relevant when dealing with real-world systems that are subject to
inherent randomness, such as biological systems, financial markets, and complex physical
systems. Stability analysis provides insights into the long-term behavior of these systems
and helps ensure their reliability and predictability despite the presence of uncertain and



M. Lavanya et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6820 3 of 20

random factors (see [10, 12, 17, 18]). The controllability analysis of fractional stochas-
tic differential equations presents unique challenges due to the combination of fractional
derivatives and stochastic components. Effective strategies require innovative approaches
that account for memory effects, stochasticity, and nonlinearity. This field is at the inter-
section of fractional calculus, stochastic control theory, and applied mathematics, offering
opportunities to devise novel control methods for a wide range of real-world systems (see
[19, 20]). The Rosenblatt process, also known as the Rosenblatt-Levy process, is a type
of stochastic process used in probability theory and statistical analysis. It’s named af-
ter Murray Rosenblatt, who introduced it in the 1950s. The process is closely related to
fractional Brownian motion and has applications in fields like finance, physics, and signal
processing. The Rosenblatt process captures this artistic wandering. It’s like an intricate
dance between memory and randomness. The past informs the artist’s decisions, while the
randomness of the postcards adds a touch of unpredictability. And just like a piece of art,
the Rosenblatt process’s path is unique every time you look at it(see [? ]). We are aware
of no study in the literature that deals with the stochastic stability and controllability
analysis of a stochastic multi-term fractional impulsive differential system.
Our work makes the following significant contributions: A Fractional stochastic delay
differential equation with the stochastic term as a Rosenblatt process is considered for
multi-term for the first time. It can reflect the transfer process more effectively compared
to existing literature. In this work, we extend the stochastic stability result for multi-term
fractional stochastic impulsive differential systems by providing suitable hypothesis. The
fixed point theorem was used to introduce controllability for the multi-term equation in
the existing system. There are mathematical instances presented.
The following theory is structured as follows: Section 2 introduces basic definitions and
lemmas. In Section 3, the multi-term fractional impulsive differential system’s state reac-
tion is illustrated. Section 4 deduces the FSDDE solution and the required and sufficient
requirements of Controllability, with examples. Section 5 discusses stochastic stability
criteria and provides instances. Section 6 provided the conclusion.

2. Preliminaries

Definition 1. [6] Let a(r) be a function of a real variable r ∈ R+. The Laplace transform
(LT) of a(r) is defined as

L[a(r)] =
∫ ∞

0
e−sra(r) dr = A(s).

The Laplace transform of a convolution satisfies

L[a ∗ n] = L[a(r)] · L[n(r)]
= A(s)N(s),

where
(a ∗ n)(r) =

∫ r

0
a(r − s)n(s) ds.



M. Lavanya et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6820 4 of 20

Moreover, the inverse Laplace transform of a product can be written as

L−1[A(s)N(s)] =
(
L−1[A(s)] ∗ L−1[N(s)]

)
(r).

For the Caputo fractional derivative of order α > 0, the Laplace transform is given by

L
[
CDα

0+f(t)
]
= sαF (s)−

n−1∑
k=0

sα−k−1f (k)(0+),

Where n− 1 < α ≤ n, n ∈ N, and F (s) = L[f(t)].

Definition 2. [6] The fractional Caputo derivative(CD) can be defined by

(CDxa)(r) =
1

Γ(n− x)

∫ r

0
(r − s)n−x−1a(n)(s)ds, n− 1 < x ≤ n, n ∈ N

If 0 < x ≤ 1,

(CDxa)(r) =
1

Γ(n− x)

∫ r

0

a1(s)

(r − s)x
ds

The LT of CDis

L[CDxa(r)](s) = sxA(s)−
n−1∑
l=0

al(0+)sx−1−l

If 0 < x ≤ 1,
L[CDxa(r)](s) = sxA(s)− sx−1a(0+)

For 1 < x ≤ 2,
L[CDxa(r)](s) = sxA(s)− sx−1a(0+)− sx−2a′(0+)

Lemma 1. [21] Banach Contraction Principle
	 has a singular fixed point if X is a Banach space and 	 : X → X is a contraction
mapping.

2.1. Rosenblatt process (RP)

In a filtered probability space(Ω,F , {Fq}q≥0, P ) where {Fq}q≥0 is the filtration and it
is right continuous. The Rosenblatt process, which is defined on this filtered probability
space is denoted by(see [? ])

RH(q) = d(H)

∫ q

0

∫ q

0

{∫ q

t1∨t2

∂KH′

∂u
(u, t1)

∂KH′

∂u
(u, t2)du

}
dB(t1)dB(t2) (1)

Where q, r ∈ [0, a], a > 0, q ≤ r, {B(q)} is a Brownian motion, Hurst parameter H > 1
2 ,

d(H) = 1
H+1(

H
2(2H−1))

−1
2 and H′ = H+1

2 .
Here KH(q, r) is the kernel,



M. Lavanya et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6820 5 of 20

KH(q, r) = CHr
1
2
−H

∫ q
r (u− r)H− 3

2uH− 1
2du

for q > r, where CH =

√
H(2H−1)

β(2−2H,H− 1
2
)

and KH(q, r) = 0 if q ≤ r.

• Let A be a Separable Hilbert Space with < ., . >A and |.|A

• Υ is a Hilbert space of all square integrable and Fr- measurable processes.

3. System and Solution Representation

Evaluate the following linear system

CDMa(r) +

x∑
g=1

ug
CDNga(r) = Ka(r) +H(r)dξ(r), r ∈ i = (0, z] r 6= rt, (2)

∆a(rt) = Jt(a(r
−
t )), t = 1, 2, ..., s,

a′(0) = δ

• Where, CDM is M order Caputo derivative and 1 < M < 2, CDNg is Ng order
Caputo
derivative and 0 < Ng < M .

• K < 0 and ug ∈ R,

• a(r) ∈ R,

• H : i→ L0
2 is a Hilbert-Schmidt operator,

• dξ(r) is a Rosenblatt process,

• ∆a(rt) = a(r+t ) − a(r−t ) is the impulse term, a(r+t ) = limh→0 a(rt + h) is the right
limit and a(r−t ) = limh→0 a(rt − h) is the left limit of a(r) on r = rt.

• δ is initial function,

Consider

CDMa(r) +

x∑
g=1

ug
CDNga(r) = Ka(r) +H(r)dξ(r), r ∈ i = (0, z], (3)

a′(0) = δ

By taking LT of (3),

a(r) =

∫ r

0
pm(r − q)δdq +

x∑
g=1

ug r
m−1pm,1+m−Ng(r)δ (4)



M. Lavanya et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6820 6 of 20

+

∫ r

0
(r − q)m−1pm,m(r − q)H(q)dξ(q)

Here,

pm(r) = L−1
[ qm−1

qm +
∑x

g=1 ugq
Ng −K

]
rm−1pm,1+m−Ng(r) = L−1

[ qNg−2

qm +
∑x

g=1 ugq
Ng −K

]
rm−1pm,m(r) = L−1

[ 1

qm +
∑x

g=1 ugq
Ng −K

]
Lemma 2. For 1 < M < 2, the solution of the system (2) can be represented as

a(r) =

∫ r

0
pm(r − q)δdq +

x∑
g=1

ug r
m−1pm,1+m−Ng(r)δ

+

∫ r

0
(r − q)m−1pm,m(r − q)H(q)dξ(q) for r ∈ (o, r1]

a(r) =

∫ r

0
pm(r − q)δdq +

x∑
g=1

ug r
m−1pm,1+m−Ng(r)δ + J1(a(r

−
1 ))

+

∫ r

0
(r − q)m−1pm,m(r − q)H(q)dξ(q) for r ∈ (r1, r2]

.

.

.

a(r) =

∫ r

0
pm(r − q)δdq +

x∑
g=1

ug r
m−1pm,1+m−Ng(r)δ +

s∑
θ=1

Jθ(a(r
−
θ ))

+

∫ r

0
(r − q)m−1pm,m(r − q)H(q)dξ(q)

for r ∈ (rt, rt + 1], t = 1, 2, 3, ..., s

Proof: If r ∈ (0, r1], then

a(r) =

∫ r

0
pm(r − q)δdq +

x∑
g=1

ug r
m−1pm,1+m−Ng(r)δ

+

∫ r

0
(r − q)m−1pm,m(r − q)H(q)dξ(q)

a(r1) =

∫ r1

0
pm(r1 − q)δdq +

x∑
g=1

ug r
m−1
1 pm,1+m−Ng(r1)δ



M. Lavanya et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6820 7 of 20

+

∫ r1

0
(r1 − q)m−1pm,m(r1 − q)H(q)dξ(q)

If r ∈ (r1, r2],

a(r) = a(r+1 ) + J1(a(r
−
1 ))−

∫ r1

0
pm(r1 − q)δdq +

x∑
g=1

ug r
m−1
1 pm,1+m−Ng(r1)δ

+

∫ r1

0
(r1 − q)m−1pm,m(r1 − q)H(q)dξ(q) +

∫ r

0
pm(r − q)δdq

+
x∑

g=1

ug r
m−1pm,1+m−Ng(r)δ

+

∫ r

0
(r − q)m−1pm,m(r − q)H(q)dξ(q)

a(r) =

∫ r

0
pm(r − q)δdq +

x∑
g=1

ug r
m−1pm,1+m−Ng(r)δ + J1(a(r

−
1 ))

+

∫ r

0
(r − q)m−1pm,m(r − q)H(q)dξ(q)

And so on, If (rt, rt+1], t = 1, 2, 3, ..., s, then the same argument implies the following
expression as

a(r) =

∫ r

0
pm(r − q)δdq +

x∑
g=1

ug r
m−1pm,1+m−Ng(r)δ +

s∑
θ=1

Jθ(a(r
−
θ ))

+

∫ r

0
(r − q)m−1pm,m(r − q)H(q)dξ(q)

4. Controllability Analysis

CDMa(r) +

x∑
g=1

ug
CDNga(r) = Ka(r) + Ey(r) +H(r)dξ(r), r ∈ i = (0, z] r 6= rt,(5)

∆a(rt) = Jt(a(r
−
t )), t = 1, 2, ..., s,

a′(0) = δ



M. Lavanya et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6820 8 of 20

Where E : Υ → A is a linear bounded operator and y(.) ∈ Υ.
The solution of the system (5) as:

a(r) =



∫ r
0 pm(r − q)δdq +

∑x
g=1 ug r

m−1pm,1+m−Ng(r)δ

+
∫ r
0 (r − q)m−1pm,m(r − q)Ey(q)dq

+
∫ r
0 (r − q)m−1pm,m(r − q)H(q)dξ(q) for r ∈ (o, r1]

∫ r
0 pm(r − q)δdq +

∑x
g=1 ug r

m−1pm,1+m−Ng(r)δ

+
∑s

θ=1 Jθ(a(r
−
θ )) +

∫ r
0 (r − q)m−1pm,m(r − q)Ey(q)dq

+
∫ r
0 (r − q)m−1pm,m(r − q)H(q)dξ(q)

for r ∈ (rt, rt + 1], t = 1, 2, 3, ..., s

Now, Consider

CDMa(r) +

x∑
g=1

ug
CDNga(r) = Ka(r) + Ey(r) +H(r)dξ(r), r ∈ i = (0, z], (6)

a′(0) = δ

Theorem 1. The system (6) is controllable on [0,z] iff the Grammian operator

BK
M,Ng

[0, z] =

∫ z

0
(z − q)2M−2[pm,m(z − q)]EE∗[pm,m(z − q)]∗dq (7)

is positive definite, here * represents adjoint operator.

Proof: Let system (6) is controllable on [0,z]. Now, our claim is BK
M,Ng

[0, z] 6= 0

Suppose take the operator BK
M,Ng

[0, z] = 0, ∃ a vector d ∈ R 3:

d∗BK
M,Ng

[0, z]d = 0

ie)

d∗
∫ z

0
(z − q)2M−2[pm,m(z − q)]EE∗[pm,m(z − q)]∗dqd = 0

Then,
d∗(z − q)M−1[pm,m(z − q)]E = 0 (8)

on [0,z]. ∃ a control function y(r) steers the response from initial 0 to final a(z) = d at
r = z. Then,

d =

∫ z

0
(z − q)2M−2[pm,m(z − q)]Ey(q)dq

By apply d∗ on both side,

d∗d =

∫ z

0
(z − q)2M−2d∗[pm,m(z − q)]Ey(q)dq



M. Lavanya et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6820 9 of 20

Which implies d∗d = 0 by equation (8) ⇒ d = 0. which is a ⇒⇐ to assumption d 6= 0.
Hence BK

M,Ng
[0, z] 6= 0.

Conversely, BK
M,Ng

[0, z] 6= 0, so the linear operator BK
M,Ng

[0, z] is invertible. If r = 0 and
r = z on the solution of the equation (6), we get

a(0) = 0; a(z) = δ

Evaluate the following Non-linear control system

CDMa(r) +

x∑
g=1

ug
CDNga(r) = Ka(r) + Ey(r) + V (r, a(r)) +H(r, a(r))dξ(r), (9)

r ∈ i = (0, z] r 6= rt,

∆a(rt) = Jt(a(r
−
t )), t = 1, 2, ..., s,

a′(0) = δ

Where H : i × R → L0
2 is a Hilbert Schmidt operator,and V : i × R → R is a real

continuous function in R.
The solution of the system (9) as:

a(r) =



∫ r
0 pm(r − q)δdq +

∑x
g=1 ug r

m−1pm,1+m−Ng(r)δ

+
∫ r
0 (r − q)m−1pm,m(r − q)[V (q, a(q)) + Ey(q)]dq

+
∫ r
0 (r − q)m−1pm,m(r − q)H(q, a(q))dξ(q) for r ∈ (o, r1]

∫ r
0 pm(r − q)δdq +

∑x
g=1 ug r

m−1pm,1+m−Ng(r)δ

+
∑s

θ=1 Jθ(a(r
−
θ )) +

∫ r
0 (r − q)m−1pm,m(r − q)[V (q, a(q)) + Ey(q)]dq

+
∫ r
0 (r − q)m−1pm,m(r − q)H(q, a(q))dξ(q)

for r ∈ (rt, rt + 1], t = 1, 2, 3, ..., s

Let us consider the assumptions
(F11) : For any a, γ ∈ C ′(i,R) and r ∈ [0, i] ∃ a constant β0 > 0 3:

‖V (r, a)− V (r, γ)‖2 ≤ β0‖a− γ‖2 and V (r, 0) = 0

(F12) : For any a, γ ∈ C ′(i,R) and r ∈ [0, i] ∃ a constant β1 > 0 3:

‖H(r, a)−H(r, γ)‖2 ≤ β1‖a− γ‖2 and H(r, 0) = 0

(F13) : [β0 + β1]Q0 < 1, Where Q0 =
∫ r
0 (r − q)m−1pm,m(r − q)dqs

Theorem 2. The system (9) is completely controllable on [0,i] with m ∈ (1, 2) and
Ng ∈ (0,m), g= 1,2,..,x provided that

[β0 + β1]Q0 < 1



M. Lavanya et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6820 10 of 20

Proof: Define Λ : c′(i,R) → c′(i,R), for each r ∈ (0, r1] we obtain

(Λa)(r) =

∫ r

0
pm(r − q)δdq +

x∑
g=1

ug r
m−1pm,1+m−Ng(r)δ (10)

+

∫ r

0
(r − q)m−1pm,m(r − q)[V (q, a(q)) + Ey(q)]dq

+

∫ r

0
(r − q)m−1pm,m(r − q)H(q, a(q))dξ(q)

For each r ∈ (rt, rt+1], t = 1, 2, 3..., s, we obtain

(Λa)(r) =

∫ r

0
pm(r − q)δdq +

x∑
g=1

ug r
m−1pm,1+m−Ng(r)δ +

s∑
θ=1

Jθ(a(r
−
θ )) (11)

+

∫ r

0
(r − q)m−1pm,m(r − q)[V (q, a(q)) + Ey(q)]dq

+

∫ r

0
(r − q)m−1pm,m(r − q)H(q, a(q))dξ(q)

The control of (9), for each r ∈ (0, r1], for a0 ∈ R

y(r) = (r − q)m−1d∗pm,m(r − q)E{a0 −
∫ r

0
pm(r − q)δdq (12)

−
x∑

g=1

ug r
m−1pm,1+m−Ng(r)δ

−
∫ r

0
(r − q)m−1pm,m(r − q)[V (q, a(q))]dq

−
∫ r

0
(r − q)m−1pm,m(r − q)H(q, a(q))dξ(q)/Gs}

The control of (9), for each r ∈ (rt, rt+1], t = 1, 2, 3, ..., s, for a0 ∈ R

y(r) = (r − q)m−1d∗pm,m(r − q)E{a0 −
∫ r

0
pm(r − q)δdq (13)

−
x∑

g=1

ug r
m−1pm,1+m−Ng(r)δ −

s∑
θ=1

Jθ(a(r
−
θ ))

−
∫ r

0
(r − q)m−1pm,m(r − q)[V (q, a(q))]dq

−
∫ r

0
(r − q)m−1pm,m(r − q)H(q, a(q))dξ(q)/Gs}

By taking the norm

sup
q∈i

E‖y(r)‖2 ≤ E‖(r − q)m−1d∗pm,m(r − q)‖2 ∗ E‖a0‖2 + E‖
∫ r

0
pm(r − q)δdq‖2



M. Lavanya et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6820 11 of 20

+E‖
x∑

g=1

ug r
m−1pm,1+m−Ng(r)δ‖2 + E‖

s∑
θ=1

Jθ(a(r
−
θ ))‖

2

+E‖
∫ r

0
(r − q)m−1pm,m(r − q)[V (q, a(q))]dq‖2

+E‖
∫ r

0
(r − q)m−1pm,m(r − q)H(q, a(q))dξ(q)‖2

≤ ‖d∗‖2‖B−1‖2{‖a0‖2 + α+ ψ + [β2 + β3]Q0} := φ

Where α := E‖
∑s

θ=1 Jθ(a(r
−
θ ))‖

2, ψ := E‖
∫ r
0 pm(r−q)δdq‖2+E‖

∑x
g=1 ug r

m−1pm,1+m−Ng(r)δ‖2,
β2 := ‖V (q, a(q))‖2 and β3 := ‖H(q, a(q))‖2

‖Λa(r)− Λγ(r)‖2 ≤ ‖
∫ r

0
(r − q)m−1pm,m(r − q)[V (q, a(q))− V (q, γ(q))]dq

+

∫ r

0
(r − q)m−1pm,m(r − q)[H(q, a(q))−H(q, γ(q))]dξ(q)‖2

≤ [β0 + β1]Q0‖a− γ‖2

From (F13), [β0 + β1]Q0 < 1. This concludes that Λ is a contraction map, from Lemma
(1) it has a unique fixed point on the interval [0,z].

sup
q∈i

E‖Λa(r)‖2 ≤ ‖
∫ r

0
pm(r − q)δdq +

x∑
g=1

ug r
m−1pm,1+m−Ng(r)δ +

s∑
θ=1

Jθ(a(r
−
θ ))

+

∫ r

0
(r − q)m−1pm,m(r − q)[V (q, a(q)) + Ey(q)]dq

+

∫ r

0
(r − q)m−1pm,m(r − q)H(q, a(q))dξ(q)‖2

≤ ‖ψ + α+ [β2φ+ β3]Q0‖2 := µ

Hence, supq∈iE‖Λa(r)‖2 ≤ µ, we conclude that the equation (9) is controllable on i.

4.1. Examples

Example 1. Evaluate the following linear control system

CDMa(r) +
3∑

g=1

ug
CDNga(r) = Ka(r) + Ey(r) +H(r)dξ(r), r ∈ i = (0, 5] r 6= rt,(14)

∆a(rt) =
1.5 + a(t−i )

4
, ti = i, i = 1, 2, 3.

a′(0) = δ



M. Lavanya et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6820 12 of 20

Figure 1: Solution of the system (14) on [0, 5] for M = 1.5, K = −0.0834.

Where M = 1.5,K = −0.0834, H = 10, E = 0.25, R = 2.5, u1 = 1, u2 = 2, u3 = 3, and
a′(0) = 1 The solution of the system (14) as:

a(r) =



∫ r
0 p1.5(r − q)δdq +

∑3
g=1 ug r

1.5−1p1.5,1+1.5−Ng(r)δ

+
∫ r
0 (r − q)1.5−1p1.5,1.5(r − q)Ey(q)dq

+
∫ r
0 (r − q)1.5−1p1.5,1.5(r − q)H(q)dξ(q) for r ∈ (o, 5]

∫ r
0 p1.5(r − q)δdq +

∑3
g=1 ug r

1.5−1p1.5,1+1.5−Ng(r)δ

+
∑s

θ=1 Jθ(a(r
−
θ )) +

∫ r
0 (r − q)1.5−1p1.5,1.5(r − q)Ey(q)dq

+
∫ r
0 (r − q)1.5−1p1.5,1.5(r − q)H(q)dξ(q)

for r ∈ (rt, rt + 1], t = 1, 2, 3.

The controllability Grammian operator,

BK
M,Ng

[0, z] = 3.9338 > 0

Which is positive definite, hence by theorem (1) system (14) is completely controllable on
[0,5]

Example 2. Evaluate the following non-linear control system

CDMa(r) +
2∑

g=1

ug
CDNga(r) = Ka(r) + Ey(r) + V (r, a(r)) +H(r, a(r))dξ(r), (15)

r ∈ i = (0, 8] r 6= rt,



M. Lavanya et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6820 13 of 20

Figure 2: Solution of the system (14) on [0, 10] for M = 1.9, K = −0.0632.

∆a(rt) =
1.8 + a(t−i )

3
t =

i

2
, i = 1, 2, 3, 4

a′(0) = δ

Where M = 1.8,K − 0.1429, H = 5, E = 2, u1 = 0.5, u2 = 1 V (r, a(r)) = a(r)+0.074
r+0.34 and

a′(0) = 0.2 The solution of the system (15) as:

a(r) =



∫ r
0 p1.8(r − q)δdq +

∑2
g=1 ug r

1.8−1p1.8,1+1.8−Ng(r)δ

+
∫ r
0 (r − q)1.8−1p1.8,1.8(r − q)[V (q, a(q)) + Ey(q)]dq

+
∫ r
0 (r − q)1.8−1p1.8,1.8(r − q)H(q, a(q))dξ(q) for r ∈ (o, 8]

∫ r
0 p1.8(r − q)δdq +

∑2
g=1 ug r

1.8−1p1.8,1+1.8−Ng(r)δ

+
∑s

θ=1 Jθ(a(r
−
θ )) +

∫ r
0 (r − q)1.8−1p1.8,1.8(r − q)[V (q, a(q)) + Ey(q)]dq

+
∫ r
0 (r − q)1.8−1p1.8,1.8(r − q)H(q, a(q))dξ(q)

for r ∈ (rt, rt + 1], t = 1, 2, 3, 4.

The controllability Grammian operator,

BK
M,Ng

[0, z] = 2.7391 > 0

Which is positive definite, hence by theorem (2) system (15) is completely controllable on
[0,8]



M. Lavanya et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6820 14 of 20

Figure 3: Solution and Control of the system (14) on [0, 5] for M = 1.2, K = −0.0543.

5. Stochastic Stability

Evaluate the following system

CDMa(r) +
x∑

g=1

ug
CDNga(r) = Ka(r) + V (r, a(r)) +H(r, a(r))dξ(r), (16)

r ∈ i = (0, z] r 6= rt,

∆a(rt) = Jt(a(r
−
t )), t = 1, 2, ..., s,

a′(0) = δ

The solution of (16) is

a(r) =



∫ r
0 pm(r − q)δdq +

∑x
g=1 ug r

m−1pm,1+m−Ng(r)δ

+
∫ r
0 (r − q)m−1pm,m(r − q)[V (q, a(q))]dq

+
∫ r
0 (r − q)m−1pm,m(r − q)H(q, a(q))dξ(q) for r ∈ (o, r1]

∫ r
0 pm(r − q)δdq +

∑x
g=1 ug r

m−1pm,1+m−Ng(r)δ

+
∑s

θ=1 Jθ(a(r
−
θ )) +

∫ r
0 (r − q)m−1pm,m(r − q)[V (q, a(q))]dq

+
∫ r
0 (r − q)m−1pm,m(r − q)H(q, a(q))dξ(q)

for r ∈ (rt, rt + 1], t = 1, 2, 3, ..., s

Definition 3. ([17]) The solution a(r) = 0 of equation (16) is stochastically stable in
the norm. If, given ε > 0 , ∃ δ(ε) > 0 3: for any initial condition whose norm satisfies
‖a1‖2 < δ the norm of the solution process satisfies E‖a(r)‖2 < ε, ∀g ≥ g0



M. Lavanya et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6820 15 of 20

Theorem 3. Under the assumptions (F11)− (F13), the system (16) is stochastic stable
on [0, j].

Proof: Define Ω : c′(i,R) → c′(i,R)

(Ωa)(r) =



∫ r
0 pm(r − q)δdq +

∑x
g=1 ug r

m−1pm,1+m−Ng(r)δ

+
∫ r
0 (r − q)m−1pm,m(r − q)[V (q, a(q))]dq

+
∫ r
0 (r − q)m−1pm,m(r − q)H(q, a(q))dξ(q) for r ∈ (o, r1]

∫ r
0 pm(r − q)δdq +

∑x
g=1 ug r

m−1pm,1+m−Ng(r)δ

+
∑s

θ=1 Jθ(a(r
−
θ )) +

∫ r
0 (r − q)m−1pm,m(r − q)[V (q, a(q))]dq

+
∫ r
0 (r − q)m−1pm,m(r − q)H(q, a(q))dξ(q)

for r ∈ (rt, rt + 1], t = 1, 2, 3, ..., s

We omit the proof. Since we have enough to prove Ω is a contraction mapping, and this
proof is similar to the proof in the theorem (2).

5.1. Examples

Example 3. Stochastic stability in impulsive fractional stochastic differential equations
can be illustrated with a real-life example involving a financial model. Consider a scenario
where an investor is managing a portfolio of stocks, and they are employing a trading
strategy that involves making periodic decisions based on market conditions. The investor’s
goal is to achieve stability in their portfolio returns despite the inherent uncertainty and
randomness in the financial markets.

Let’s model this scenario using an impulsive fractional stochastic differential equation
(SFDE) to represent the investor’s portfolio value. Evaluate the following system

CDMa(r) +

x∑
g=1

ug
CDNga(r) = Ka(r) + V (r, a(r)) +H(r, a(r))dξ(r), (17)

r ∈ i = (0, 5] r 6= rt,

∆a(rt) = Jt(a(r
−
t )), t = 1, 2, ..., 7,

a′(0) = δ



M. Lavanya et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6820 16 of 20

Figure 4: Solution of the system (17) on [0, 5] for M = 1.5, K = −0.0834.

Figure 5: Solution of the system (17) on [0, 5] for M = 1.9, K = −0.0632.



M. Lavanya et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6820 17 of 20

Figure 6: Solution of the system (17) on [0, 5] for M = 1.2, K = −0.0543.

In this financial model:
1. Stochastic Component: The continuous stochastic processes involved in the port-

folio’s growth. The investor’s portfolio value is subject to random market fluctuations
(modeled by dξ(r)).

2. Impulsive Component: The impulsive trading decisions made by the investor.
Stochastic stability in this context means that, over time, despite the investor’s trading

decisions and the inherent randomness in the financial markets, the portfolio value tends
to remain within a certain bounded region. This stability is crucial for the investor’s
financial well-being and long-term success. The goal is to avoid excessive volatility and
losses that could jeopardize the portfolio. Achieving stochastic stability would mean that the
investor’s trading decisions are well-balanced, and the impact of these decisions, combined
with market fluctuations, results in a portfolio that grows over time or at least remains
within acceptable risk parameters. This stability is a key objective in financial modeling and
decision-making, where the balance between proactive trading actions and risk management
is critical.

Example 4. Stochastic stability in impulsive fractional stochastic differential equations
(IF-SDEs) can be illustrated with the example of a bouncing ball, which incorporates both
impulsive events and stochastic dynamics. This example demonstrates how stochastic
stability analysis can be applied to real-world physics problems.

Consider a ball dropped from a certain height above the ground. The ball experiences
random air resistance, which is modeled as a stochastic process due to the turbulent nature
of the surrounding air. Additionally, the ball undergoes impulsive collisions with the ground
when it bounces back up. The motion of the ball can be described by an IF-SDE.



M. Lavanya et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6820 18 of 20

The IF-SDE for the vertical position of the ball could be represented as follows:

CDMa(r) +

x∑
g=1

ug
CDNga(r) = Ka(r) + V (r, a(r)) +H(r, a(r))dξ(r), (18)

r ∈ i = (0, 10] r 6= rt,

∆a(rt) = Jt(a(r
−
t )), t = 1, 2, ..., 9,

a′(0) = δ

Figure 7: Solution of the system (18) on [0, 10] for M = 1.2, K = −0.0543.

In this example, stochastic stability analysis would involve studying how the behavior
of the ball evolves over time in the presence of both stochastic fluctuations (due to air
resistance) and impulsive events (bouncing off the ground). The goal would be to determine
whether, on average, the ball’s position remains bounded (i.e., it doesn’t bounce off into
infinity or hit the ground with infinite force), despite the stochastic and impulsive nature of
the system. By analyzing the properties of this IF-SDE and assessing the conditions under
which the ball’s position remains bounded, physicists can gain insights into the stochastic
stability of this real-life physical system. The results of such an analysis could be used
to design more robust systems or optimize the behavior of bouncing objects subjected to
random forces and collisions.

6. Conclusions

This work uses the Rosenblatt process along with a multi-term Caputo fractional
derivative to evaluate the stability and controllability of stochastic fractional impulsive
differential equations. Using the fixed point theory of the Banach contraction mapping



M. Lavanya et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6820 19 of 20

principle, an attempt was made to construct acceptable criteria for stochastic stability
and controllability analysis. The full study has been provided with examples. Last but
not least, it is possible to investigate the dynamical behaviour of numerous real-world
objects by combining the multi-term Caputo fractional derivative with the Rosenblatt
process stochastic differential equation. As a direction for future research, the present
analysis can be extended to multi-term stochastic fractional differential inclusions, where
the control functions belong to a set-valued map. This would allow us to investigate more
general systems with uncertainties and constraints. Potential applications of such models
can be explored in engineering systems with random perturbations, financial mathematics
for modeling market volatility, and biological systems where impulsive effects and memory
play a significant role. Further studies may also focus on developing numerical algorithms
to approximate the controllability and stability conditions for such inclusions.

Acknowledgements

The first author is supported by RUSA-Phase 2.0 grant sanctioned vide letter No.F
24-51/2014-U, Policy (TN Multi- Gen), Dept. of Edn. Govt. of India, Dt.09.10.2018. The
second author would like to thank the Department of Science and technology, Government
of India, under the scheme FIST(SR/FST/MSI/2023/131).

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