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U. Arora, S. Singh, V. Vijayakumar, A. Shukla / IJOCTA, Vol.15, No.3, pp.483-492 (2025)
2
and Y = V = L (Ω). The control term Conflict of interest
B(ϱ, v(ϱ)) = v(ϱ) is linear. Also,
Anurag Shukla is an Editorial Board Member of
1
2
F(ϱ, y(ϱ), z(ϱ)) = [sin y(ϱ) + sin z(ϱ)] this journal, but was not in any way involved in
2 the editorial and peer-review process conducted
and G(ϱ, α, y(α)) = 1 [cos y(α)], both are Lips- for this paper, directly or indirectly. Separately,
2
chitz continuous and satisfy all the assumptions. other authors declared that they have no known
Therefore, using the results of Case-I of Section competing financial interests or personal relation-
2, we get that (20) is T-controllable. ships that could have influenced the work re-
ported in this paper.
Example 3. Consider the nonlinear frac-
tional integro-differential system represented in
Author contributions
the sense of a Caputo fractional derivative
Conceptualization: Urvashi Arora, Anurag
c ν 2
D y(ϱ) =4y(ϱ) + v (ϱ) + 10 cos y(ϱ) Shukla
ϱ
ϱ Formal analysis: Urvashi Arora, V. Vijayakumar
Z
2
+ 9 (ϱ + α + sin y(ϱ))dα, Investigation: Sachin Singh, Urvashi Arora
0 (23) Methodology: Anurag Shukla, V. Vijayakumar
ϱ ∈ [0, 1],
Writing -original draft: Urvashi Arora, Anurag
y(0) = y 0 , y(0) = η 0 . Shukla
where ν ∈ (1, 2]. The control function Writing-review & editing: V. Vijayakumar,
2 Sachin Singh, Anurag Shukla
Bv(ϱ) = v (ϱ) is continuous on [0, 1] × R
and coercive. Further, the nonlinear functions
ϱ
Z
G ϱ, y(ϱ), g(ϱ, α, y(α))dα = 10 cos y(ϱ) + Availability of data
0
Z ϱ Not applicable.
2
9 (ϱ + α + sin y(ϱ))dα is Lipschitz with Lip-
0
schitz constants δ 1 and δ 2 . All the hypotheses References
of Theorem 1 are satisfied. Therefore, the non-
linear fractional integro-differential system (3.1)- 1. Arora U, Sukavanam N. Controllability of frac-
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and y 0 , η 0 ∈ R. integral contractor. IMA J Math Control Infor-
mat. 2019;36(1):271-283.
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