Page 116 - IJOCTA-15-3

U. Arora, S. Singh, V. Vijayakumar, A. Shukla / IJOCTA, Vol.15, No.3, pp.483-492 (2025)
therefore u is measurable. So, we can extract Definition 10. [43] “A function x(·) ∈ C(I; Y)
v(ϱ) from B(ϱ, v(ϱ)) so that for this control, the is said to be the mild solution of (11)-(12) if it
mild solution of the considered system equals to satisfies
the prescribed trajectory z(ϱ), which concludes Z ϱ
T-controllability of (1)-(2). y(ϱ) =C ν (ϱ)y 0 + S ν (ϱ)z 0 + P ν (ϱ − α)B(α, v(α))dα
0
Z ϱ Z ϱ
3. Fractional differential systems of + P ν (ϱ − α)G α, x(s), g(ϱ, α, y(α))dα }ds
order ν ∈ (1, 2] 0 0
(15)
We are primarily focusing on the trajectory con-
trollability of the subsequent fractional semilinear where S ν (ϱ) and P ν (ϱ) are the fractional SF
integrodifferential system of order ν ∈ (1, 2] and fractional R-L family, respectively, associ-
ated with strongly continuous CF C ν (ϱ) as defined
c ν
D y(ϱ) =Ay(ϱ) + B(ϱ, v(ϱ)) above.”
ϱ
ϱ
Z
+ G ϱ, y(ϱ), g(ϱ, α, y(α))dα , Assume that P be the set of all functions de-
0 ′
fined on I with z(0) = y 0 , z (0) = z 0 , z(b) = y 1
ϱ ∈ I, ν ∈ (1, 2], c ν
and the fractional derivative D z exists almost
ϱ
(11) everywhere for ν ∈ (1, 2]. Note that P be the set
′
y(0) = y 0 , y (0) = z 0 . (12)
of all feasible trajectories for (11)-(12).
Here, A is the infinitesimal generator of We prove the trajectory controllability of the
the strongly continuous cosine family (CF) system (11)-(12) in two cases.
{C ν (ϱ), t ≥ 0} and y 0 , z 0 ∈ Y. Case-I: If the control appears linearly:
To define the mild solution of (11)-(12), we Consider the system (11)-(12) with the con-
assume the subsequent system trol appears linearly, i.e. , B(ϱ, v(ϱ)) = b(ϱ)v(ϱ),
c ν ′ where b : IR and u : IV, then the system (11)-(12)
D y(ϱ) = Ay(ϱ), y(0) = η y (0) = 0. (13)
ϱ
becomes
In the above, ν ∈ (1, 2]; A : D(A) ⊆ Y → Y is
ν
c D y(ϱ) = Ay(ϱ) + b(ϱ)v(ϱ)
a closed and densely defined operator in Y. Tak- ϱ
Z ϱ
ing Riemann fractional integral of order ν on both
+ G ϱ, y(ϱ), g(ϱ, α, y(α))dα , (16)
sides of (13), one can attain
0
1 Z ϱ ϱ ∈ I, ν ∈ (1, 2],
y(ϱ) = η + (ϱ − α) ν−1 Ax(α)dα (14)
Γ(ν) 0
′
Definition 7. [43] “Let ν ∈ (1, 2]. A family y(0) = y 0 and y (0) = z 0 . (17)
{C ν (ϱ)} t≥0 ⊂ L(Y) is called a solution operator Assumptions [III].
(or a strongly continuous ν-order fractional CF)
for (13) if the following conditions are satisfied: (1) A generates a strongly continuous CF
{C ν (ϱ) : t ≥ 0} on Y.
(1) C ν (ϱ) is strongly continuous for t ≥ 0 and
(2) b(ϱ) does not disappear on I.
C ν (0) = I;
(3) G and h are Lipschitz continuous in a sim-
(2) C ν (ϱ)D(A) ⊂ D(A) and AC ν (ϱ)η =
ilar manner as in Assumptions [I] of Sec-
C ν (ϱ)Aη for all η ∈ D(A), t ≥ 0;
tion 2.
(3) C ν (ϱ)η is a solution of (13),∀ η ∈ D(A),
ϱ ≥ 0. By referring these two assumptions, we
A is called infinitesimal generator of C ν (ϱ). The may create the control explicitly to verify T-
strongly continuous ν-order fractional CF is also controllability of (16)-(17). For verifying this, we
called ν-order CF.” continue in the following way:
For every v ∈ L 2 (I, V), the existence and
Definition 8. The sine family (SF) S ν : uniqueness of (16)-(17) follows from Assumptions
[0, ∞) → L(Y) connected with C ν is presented as [III] by employing the Lipschitz continuity of
ϱ functions G and h.
Z
S ν (ϱ) = C ν (α)dα, ϱ ≥ 0. Assume that z(ϱ) be the given trajectory in
0
P. We present v(ϱ) as
Definition 9. The fractional R-L family P ν :
Z ϱ
[0, ∞) → L(Y) connected with C ν is presented as c D z(ϱ) − Az(ϱ) − G ϱ, z(ϱ), g(ϱ, α, z(α))dα
ν
ϱ
P ν (ϱ) = J ϱ ν−1 C ν (ϱ). v(ϱ) = b(ϱ) 0
488