Page 181 - IJOCTA-15-2

S. Das / IJOCTA, Vol.15, No.2, pp.368-378 (2025)
Then setting all coefficients of δu in Equation
+
1
(80) equals to zero we get, where as ρ(t, z(t)) : [0, T]×H (Ω) → R ∪{0}.
0
It is a continuous function ∀t ∈ [0, T] with the
∗
Nu + (B p(t)) = 0. (84) property that 0 ≤ ρ(t, z(t)) ≤ t.
Then the above system can be reformulated
Then setting all coefficients of δp in equation
by defining the suitable operator
(80) equals to zero, we get,
Fz(t) = c(x, t)z(x, t − ρ(t, z(t)))
C α
t D z(t) = −Az(t) − Fz(t) as below, hlwith c(x, t) as identity operator.
T
+ k(t) + Bu(t) (85)
1
2
1
For every t ∈ [0, T] and z 0 ∈ H (Ω), the C D z(u, t)+ △ z(u, t) + Fz(u, t)
0 0 t
minimum of J(u) i.e. V (t, z 0 ) is achieved; i.e; = K(t) + Bu(t)in Q
∗
∗
2
2
1
∃ (z , u ) ∈ L ([0, T]; H (Ω)) × L ([0, T]; U) s.t. z(u) = 0 on Γ,
0
z(x, 0; u) = z 0 (x), x ∈ Ω,
V (t, z(t)) =
(90)
T
Z
1 2
⟨(Cz(u) − ob d ), (Cz(u) − ob d )⟩ 2 dt Cost function is defined for all ob d ∈ L (Q)
L (Ω)
2
0 as
Z T Z 1
+ Nu.ududt (86) Z
0 Ω 2 1 2 1
J(u) = (Cz(u)−ob d ) dxdt+ (Nu, u) 2
L (Q)),
∗
∗
∗
α ∗
C D z (t) + Az (t) + Fz (t) = k(t) + Bu (t) 2 Q 2
t 0 t (91)
in [0, T]; where N ∈ L(L (Q), L (Q)), N is Hermitian
2
2
∗
z (0) = z 0 . (87) positive definite, i.e.
According to the obtained optimality condi-
2
∗
tions, ∃ p ∈ C([0, T]; H) that satisfies (Nu, u) 2 ≥ d∥u∥ 2 , d > 0. (92)
L (Q)
L (Q)
Then one can characterize the optimal control
C α ∗ ∗ ∗ ∗ u from the system (90) and the its corresponding
t D p (t) = −A p(t) − F p (t) (88)
T
∗
∗
+ C [(Cz (u) − ob d )], adjoint system
∗
p (x, T; u) = 0, 1
C D p(u)+ △ p(u) + F p(u)
∗
2
t I T 1−α ∗ t T ∗
p (x, 0; u) = 0,
∗
Nu + (B p(t)) = 0. = C (Cz(u) − ob d ) in Q
p(x, T; u) = 0, x ∈ Ω,
Considering solutions of (87) in a mild sense, (93)
∗
with z 0 ∈ D(A) and z being H¨older continuous
∗
on [0, T] then it can be deduced that z is a mild along with the optimality condition is given
16
solution of (87) by applying corollary 3.3 of. ] by
∗
Moreover p is a mild solution of (88) on [0, T].
∗
Therefore, u is a H¨older continuous function in Z ∗
(B p(u) + Nu)(v − u)dxdt ≥ 0 ∀v ∈ U ad (94)
each [0, t] ⊂ [0, T]. Q
1
Thus, for every t ∈ [0, T] and z 0 ∈ H (Ω), where the adjoint state is denoted by p(u).
0
∗
∗
the minimum of J(u) is achieved; ∃ (z , u ) ∈
1
2
2
L ([0, T]; H (Ω)) × L ([0, T]; U). 8. Conclusion
0
7. Example In this article, an optimal control of a fractional
delay differential system is established using the
Consider the optimal control problem with the variational principle as its central framework. Pri-
state constraint equation defined as, marily, fractional Green’s theorem is used to find
optimal control of both Dirichlet and Neumann
1
C 2 control problem. Pontryagin’s Maximum princi-
0 D z(u, t)+ △ z(u, t) + z(u, t − ρ(t, z))
t
ple is also illustrated using a variational method
= K(t) + Bu(t) in Q
for the fractional delay differential control sys-
z(x, t) = f(x, t), x ∈ Ω, t < 0 tem. This study concludes with the derivation
z(x, t) = 0, x ∈ Γ, t ∈ (0, T] (89) of optimality systems for the fractional control
376