Page 178 - IJOCTA-15-2

An investigation on the optimality condition of Caputo fractional time delay system
2
2
here (k + Bu) ∈ L (Q), z 0 ∈ L (Ω).
Z
∗
1
(B p(u)(v − u)dxdt + (Nu, v − u) 2 ≥ 0 Now ∀ϕ ∈ H (Ω), (46) is same as
L (Q)
Q 0
(40)
Z
Hence, the optimality condition is proved. C α
( D z(t) + (A(t) + F)z(t))ϕ(x)dxdt
0
t
Q
5. Fractional Neumann system Z
= (k + Bu)ϕdxdt
1
1
As H (Ω) ⊂ H (Ω) the bilinear form (6) can be Q
0
1
shown to be coercive in H (Ω), i.e. Z
− z 0t I 1−α ϕ(x, 0)dΓ,
T
Γ
1
π(z, z) ≥ c∥z∥ 2 , c > 0, ∀z ∈ H (Ω). (41) Z ∂ϕ
1
H (Ω)
+ u dΣ (49)
(1) Here, let us denote ∂z = Σ ∂ν
∂ν A(t) which is the same fractional partial differen-
∂z
Σ n a ∂x j cos(n, x j ) on Γ. tial equation
i,j=1 ij
(2) cos(n, x j ) corresponds to the jth direction
cosine of n. C α
( D z(t) + (A(t) + F)z(t)) = k(t) + Bu(t) (50)
(3) Here n is the normal on the boundary Γ 0 t
exterior to Ω. integrated against test function ϕ(x).
Lemma 4. If (41) holds, the ∃ a unique solution Using fractional Green’s lemma (2), to LHS
1
z ∈ H (Ω) which satisfies ∂ϕ(x,t)
of equation (49) and the fact that = 0, it
∂ν A
is shown that
C D z(t)+A(t)z(t)+Fz(t) = k(t)+Bu(t), in Q
α
0 t
(42) Z T Z
α
∂z C D z(t) + A(t)z(t) + Fz(t)dxdt
= u on Σ, (43) 0 t
∂ν A(t) 0 Z Ω
z(0) = z 0 (x), x ∈ Ω, (44) = − z(x, 0) t I T 1−α ϕ(x, 0)dx
Γ
15
Proof. Using Theorem (3.1) of [Lions, 1971] we Z T Z ∂z(x, t)
can show by the coercivity property (41) and Lax- + ∂ν ϕdΓdt
Milgram theorem 14 ∃ a unique solution z(t) ∈ 0 Γ
Z T Z
1
H (Ω) which satisfies C α
0 + z(x, t)( D ϕ(x, t)dxdt
t T
0 Ω
Z T Z
C
α
1
( D z(t), ϕ) 2 +ζ(t; z, ϕ) = M(ϕ), ∀ϕ ∈ H (Ω) ∗
0
0
L
t
(Q) + z(x, t)(A(t) + F) ϕ(x, t))dxdt
(45) 0 Ω
Z
This implies that there exists a unique solu- = (k + Bu)ϕdxdt (51)
1
tion z(t) ∈ H (Ω) which satisfies the following Q
0
1
∀ϕ ∈ H (Ω)
0 Now comparing both sides of Equation (51)
and using the fact that ϕ(x, T) = 0, in Ω, as well
C α as ϕ = 0 in Γ × [0, T], equations (43) -(44) are
( D z(t), ϕ) 2 + ((A(t) + F)z(t), ϕ) = M(ϕ),
L
t
0
(Q) inferred.
(46)
1
This is equivalent to ∀ϕ ∈ H (Ω)
0
5.1. Optimal control of fractional
Z
α
C
( D z(t) + (A(t) + F)z(t))ϕ(x)dxdt = M(ϕ), Neumann system
t
0
2
Q Let U = L (Ω) be the space of controls. The
(47) 1
state of variable z(u) ∈ H (Ω) satisfies ∀u ∈ U
It is variational form of Dirichlet fractional
the following equation
problem, with M(ϕ) being a continuous linear
1
form defined on H (Ω) as
0 α
0 D z(u)+A(t)z(u)+Fz(u) = k+Bu in Q (52)
t
Z
∂z(u)
M(ϕ) = (k + Bu)ϕdxdt − = u on Σ, (53)
Q ∂ν A
Z Z ∂ϕ z(x, 0; u) = z 0 (x), x ∈ Ω, (54)
z 0t I 1−α ϕ(x, 0)dΓ − u dΣ (48)
T Observability condition is given by
Γ Σ ∂ν
373