Page 179 - IJOCTA-15-2

S. Das / IJOCTA, Vol.15, No.2, pp.368-378 (2025)

ob(u) = Cz(u) (55) Z
∗
C (Cz(u) − ob d )(z(v) − z(u))dxdt
2
2
s.t. C ∈ L(L (Q), L (Q)). Q
Z
∗
α
= ( t D p(u) + A (t)p(u)
2
T
Cost function is defined for all ob d ∈ L (Q) Q
as +F p(u))(z(v) − z(u))dxdt
∗
Z
+
1 Z 1 = t I T 1−α p(x, 0)(z(v; x, 0 )dx
2
J(u) = (Cz(u)−ob d ) dxdt+ (Nu, u) 2 Ω
L (Q)),
2 Q 2 Z
+
(56) = − t I T 1−α p(x, 0)z(u; x, 0 ))dx
2
2
where N ∈ L(L (Q), L (Q)), N is Hermitian Z Ω ∂z(v) ∂z(u)
positive definite, i.e., + p(u)( − )dΓ
Γ ∂ν A(t) ∂ν A(t)
Z ∂p(u)
2 −
(Nu, u) 2 ≥ d∥u∥ 2 , d > 0. (57) (z(v) − z(u))dΓ
L (Q)
L (Q)
Γ ∂ν A ∗
U ad be the set of admissible controls, which is Z C α
+ p(u)( D + A(t) + F)(z(v) − z(u))dxdt
convex and closed subset of U. 0 t
Q
(66)
The control problem is to find u ∈ U ad such
J(u) By Equation (52), we know
that J(u) = inf u∈U ad
Theorem 2. Let the cost function be defined by C α
(56). If (57) holds then optimal control u is char- ( D t + A(t) + F)(z(v) − z(u))
0
acterized by (52), (53), (54) along with = Bv − Bu (67)
So
C α ∗ ∗ ∗ Z
∗
T
t D p(u)+A (t)p(u)+F p(u) = C (Cz(u)−ob d ) C (Cz(u) − ob d )(z(v) − z(u))dxdt
(58) Q
in Q Z
∂p(u) = p(u)(Bv − Bu)dxdt. (68)
= u on Σ, (59) Q
∂ν A ∗
p(x, T; u) = 0, x ∈ Ω, (60) which gives
and optimality condition is
Z
(Cz(u) − ob d )(Cz(v) − Cz(u))dxdt
Z
∗
(B p(u) + Nu)(v − u)dΣ ≥ 0 ∀v ∈ U ad (61) Q
Z
Σ ∗
= B p(u)(v − u)dxdt. (69)
where p(u) denotes the adjoint state Q
From Equation (63) we have
Proof. The control u ∈ U ad is optimal iff
′
J (u)(v − u) ≥ 0 ∀v ∈ U ad (62)
i.e. (Cz(u)−ob d , Cz(v)−Cz(u)) 2 +(Nu, v−u) U ≥ 0
L (Q)
(Cz(u) − ob d , Cz(v) − Cz(u)) 2 This implies
L (Q)
+(Nu, v − u) U ≥ 0 (63)
Z
The adjoint state p satisfies the adjoint prob- ∗
(B p(u) + Nu)(v − u)dxdt ≥ 0, ∀v ∈ U ad
lem Q
C α ∗ ∗ ∗
t D p(u)+A (t)p(u)+F p(u) = C (Cz(u)−ob d ) Thus, by substituting (69) in (63) we obtain
T
(64)
in Q
Z
p(x, T; u) = 0, x ∈ Ω, (65) ∗
(B p(u)(v − u)dxdt + (Nu, v − u) 2 ≥ 0
L (Q)
Multiplying (64) by z(v) − z(u) and by us- Q
(70)
ing Green’s formula, also considering conditions
in (52) and (53) we get Hence, the optimality condition is proved.
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