Page 177 - IJOCTA-15-2

S. Das / IJOCTA, Vol.15, No.2, pp.368-378 (2025)
4.1. Optimal control of Dirichlet
fractional system C α ∗ ∗ ∗
t D p(u)+A (t)p(u)+F p(u) = C (Cz(u)−ob d )
T
2
Let U = L (Ω) be defined as the space of controls. (34)
1
The state of the system z(u) ∈ H (Ω) satisfies the in Q
following ∀u ∈ U p(x, T; u) = 0, x ∈ Ω. (35)
Multiplying (34) by z(v)−z(u) and by apply-
α
C D z(u)+A(t)z(u)+Fz(u) = k+Bu in Q (23) ing Green’s formula, also considering conditions
0 t
in (23) and (24) we get
z(u) = 0 on Γ, (24)
z(x, 0; u) = z 0 (x), x ∈ Ω, (25) Z
∗
Observability condition is given by C (Cz(u) − ob d )(z(v) − z(u))dxdt
Q
Z
∗
ob(u) = Cz(u) (26) = ( D p(u) + A (t)p(u)
C
α
T
t
2
2
s.t. C ∈ L(L (Q), L (Q)). Q
∗
+F p(u))(z(v) − z(u))dxdt
2
Cost function is defined for all ob d ∈ L (Q) Z 1−α +
= t I p(x, 0)(z(v; x, 0 )dx
as T
Ω
Z
+
Z = − t I 1−α p(x, 0)z(u; x, 0 ))dx
1 2 1 T
J(u) = (Cz(u)−ob d ) dxdt+ (Nu, u) 2 Ω
L (Q)),
2 Q 2 Z ∂z(v) ∂z(u)
(27) + p(u)( − )dΓ
2
2
where N ∈ L(L (Q), L (Q)), N is Hermitian Γ ∂ν A(t) ∂ν A(t)
Z
positive definite, i.e., ∂p(u)
− (z(v) − z(u))dΓ
Γ ∂ν A ∗
Z
2
α
(Nu, u) 2 ≥ d∥u∥ 2 , d > 0. (28) + p(u)( 0 D + A(t) + F)(z(v) − z(u))dxdt
L (Q)
L (Q)
t
U ad be the set of admissible controls, which is Q
convex and closed subset of U. (36)
By Equation (3), we know
The control problem is to find u ∈ U ad such
J(u)
that J(u) = inf u∈U ad C α
( D t + A(t) + F)(z(v) − z(u))
0
Theorem 1. Suppose the cost function is defined = Bv − Bu (37)
by (27). If (28) holds, then optimal control u is
characterized by (23), (24), (25) along with So
Z
∗
C (Cz(u) − ob d )(z(v) − z(u))dxdt
C α ∗ ∗ ∗ Q
t D p(u)+A (t)p(u)+F p(u) = C (Cz(u)−ob d ) Z
T
(29) = p(u)(Bv − Bu)dxdt. (38)
in Q Q
p(x, T; u) = 0, x ∈ Ω, (30) which gives
and optimality condition is given by
Z
(Cz(u) − ob d )(Cz(v) − Cz(u))dxdt
Z
∗
(B p(u) + Nu)(v − u)dxdt ≥ 0 ∀v ∈ U ad (31) Q Z
Q ∗
= B p(u)(v − u)dxdt. (39)
where the adjoint state is denoted by p(u) Q
Proof. We know that the control u ∈ U ad is op- From Equation (33) we have
timal iff
L (Q)
′
J (u)(v − u) ≥ 0 ∀v ∈ U ad (32) (Cz(u)−ob d , Cz(v)−Cz(u)) 2 +(Nu, v−u) U ≥ 0
This implies
i.e.
(Cz(u)−ob d , Cz(v)−Cz(u)) 2 +(Nu, v−u) U ≥ 0 Z
L (Q)
∗
(33) (B p(u) + Nu)(v − u)dxdt ≥ 0, ∀v ∈ U ad
The adjoint state p satisfies the adjoint prob- Q
lem Thus, by substituting (39) in (33) we obtain
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