Page 180 - IJOCTA-15-2

An investigation on the optimality condition of Caputo fractional time delay system
6. Pontryagin maximum principle
(PMP) 1 2
H(t, z, u, p) = ∥(Cz(u) − ob d )∥ 2
L
2 Ω
This section establishes the necessary conditions
1
of optimal control for the system (3 − 5). Let ⟨Nu, u⟩ 2
L
2 Ω
1 +⟨p(t), (−A(t)z(t) − Fz(t) + k(t) + Bu(t))⟩
Z
2
J(u) = inf { (Cz(u) − ob d ) dxdt
u∈U 2
Q Therefore,
1
+ < Nu, u > 2 , } (71)
L (Q)
2 Z T Z
2
Let us define the value function V : U ad → R, V (u) = [ (Cz(u) − ob d ) dxdt
0 Ω
Z
+ Nu.udu
V (u) =
Ω
T
Z
α
1 + p(t)(− D z(t) − A(t)z(t)
C
⟨(Cz(u) − ob d ), (Cz(u) − ob d )⟩ 2 dt 0 t
L (Ω)
2
0 − Fz(t) + k(t) + Bu(t))]dt
T 1 Z T
Z Z
+ Nu.ududt (72) V (u) = [H(t, z, u, p)
0 Ω 2
0
α
− p(t)( D z(t))]dt (77)
t
0
V (u) =
Taking the first variation of J(u),
Z T
1
⟨(Cz(u) − ob d ), (Cz(u) − ob d )⟩ 2 dt
L (Ω)
2
0 Z T ∂H ∂H ∂H
Z T 1 δV (u) = {( δz + δu + δp)
+ < Nu, u > 2 dt (73) 0 ∂z ∂u ∂p
L
0 2 (Ω) C α C α
− ⟨δp, D z(s)⟩ − ⟨p, D δz(s)⟩}ds.
t
t
0
0
T 1
Z (78)
2
V (u) = ∥(Cz(u) − ob d )∥ dt Using integration by parts formula for frac-
0 2 tional order, 4
T 1
Z
+ < Nu, u > dt (74)
0 2 Z T C α
p(t) D δz(t)dt
Subject to the constraint, 0 t
0
Z T
α
C
C α = δz(t) D p(t)dt + p(x, T; u) 0 I 1−α δz(T)
t
0 D z(t) + A(t)z(t) + Fz(t) = k(t) + Bu(t), t T T
0
for t ∈ [0, T], + δz(0) t I 1−α p(x, 0; u) (79)
T
z(x, 0) = z 0 .
Substituting values and rearranging above
(75)
equation (79) we have,
α
where, C D z(t) represents Caputo fractional
t
0
derivative of order α ∈ (0, 1). Z T ∂H ∂H
C
α
δV (u) = {( − D p(s))δz + δu
t
T
0 ∂z ∂u
6.1. Fractional necessary optimality ∂H C α
+ ( − D z(s))δp}ds (80)
t
conditions ∂p t 0
To derive the necessary conditions for optimal + p(x, T; u) 0 I T 1−α δz(T)
control of the fractional problem (74) and (75), δz(0) t I T 1−α p(x, 0; u)
the fractional system (75) is incorporated in the
cost functional. This is augmented objective func- Now, for optimality conditions δV (u) = 0.
tion is defined as Therefore, setting all coefficients of δz, in equa-
tion (80) equals to zero we get,
T
Z
α
∗
∗
V (u) = [H(s, z, u, p) − ⟨p(s), C D z(s)⟩]ds C D p(t) = −A p(t) − F p(t)
α
t
0
0 t T
∗
(76) + C [(Cz(u) − ob d )], (81)
by introducing a Lagrange multiplier p(t) ∈ p(x, T; u) = 0 (82)
C([t 0 , t]; H), also referred to as the costate or ad- 1−α
t I p(x, 0; u) = 0, (83)
joint variable. The Hamiltonian H is defined as, T
375