Page 174 - IJOCTA-15-2

An investigation on the optimality condition of Caputo fractional time delay system
the lifespan of dynamical systems by providing
improved control as it efficiently describes sys- C α
t
0 D z(t) + A(t)z(t) + c(x, t)z(x, t − ρ(t, z(t)))
tem dynamics, which in turn reduces wear and
= k(t) + Bu, x ∈ Ω, t ∈ [0, T]
tear.
Optimal control of infinite- dimensional Ca- z(x, t) = f(x, t), x ∈ Ω, t < 0
puto fractional systems is yet to see notable re- z(x, t) = 0, x ∈ Γ, t ∈ (0, T] (1)
sults on the infinite-dimensional maximum prin-
ciples, since it has not been extensively stud-
ied so far. The existing literature barely stud- where 0 < α < 1, Ω is a bounded (bdd) open
n
ied Pontryagin maximum principle for infinite- subset of R with a smooth boundary Γ of class of
2
dimensional Caputo fractional state-dependent C . For time T > 0. Let Q = Ω × (0, T) and Σ =
2
1
0
delay systems. Several forms of maximum Γ × (0, T). Initial condition z 0 ∈ H (Ω) ∩ H (Ω),
2
principles have been established for the finite- the control u ∈ L (Q) and the bounded lin-
2
2
dimensional optimal control problems. Those ear operator B ∈ L(L (Q), L (Q)). The func-
2
tion k(t) belongs to L (Q). The operator A(t) ∈
methodologies in finite- dimensional optimal con- 1 −1
0
trol are not applicable for infinite- dimensional L(H (Ω), H 0 (Ω)) is a second order self adjoint
fractional optimal control. Furthermore, tradi- operator given by
tional infinite-dimensional maximum principles
for non-fractional optimal control are also not ap- n ∂ ∂z(x, t)
A(t)z(x, t) = −Σ i,j=1 (c ij (x, t) )
plicable for fractional optimal control problems, ∂x i ∂x j
as fractional calculus framework is required to + c 0 (x, t)z(x, t) (2)
deal with the latter.
The framework of Hilbert spaces underscores
sufficient norm smoothness required for varia- Here, F is a delay operator defined on [0, T]
tional analysis. In this paper, a new perspec- as
tive on Pontryagin maximum principle is pro- Fz(t) = c(x, t)z(x, t − ρ(t, z(t)))
vided. Although some papers have studied max- Let
imum principle for the optimal control problems,
1
they are heavily dependent on the existence of S(0, T) := {z : z ∈ L (0, T; H (Ω)),
2
0
2
a C solution, which often fails to be the case. C D z(x, t) ∈ L (0, T; H −1 (Ω))}.
2
α
This paper presents both sufficient and neces- 0 t 0
sary conditions for optimal control, in contrast to
other approaches involving Mazur’s Lemma and f(x, t) belongs to S(0, T). Deviating argument
1
+
Balder’s theorem, which only offer sufficient con- is defined as ρ(t, z(t)) : [0, T]×H (Ω) → R ∪{0}.
0
ditions. See, 11,13 etc. Lastly, it is noteworthy It is a continuous function ∀t ∈ [0, T] with the
that the theorems in this paper can be extended property that 0 ≤ ρ(t, z(t)) ≤ t.
to broader kinds of fractional infinite dimensional Here the coefficients c(x, t), c 0 (x, t),
optimal control problems. c ij (x, t), i, j = 1, ...n, satisfy the following prop-
∞
In this paper, optimal control of an infinite- erties: c(x, t), c 0 (x, t), c ij (x, t) ∈ L (Ω) and
dimensional fractional problem with state- depen- c(x, t) ≥ a > 0, c 0 (x, t) ≥ a > 0, c ij (x, t)ξ i ξ j ≥
2
2
dent delay is studied. The state is driven by a left a(ξ + ... + ξ ), ξ ∈ R n
1 n
Caputo fractional evolution equation with a coer-
So now Equation (1)(1) can be rewritten as
cive operator in suitable Hilbert space. The objec-
below:
tive function is formulated by a running cost as a
α
function of both the state and control variables in- C D z(t) + A(t)z(t) + Fz(t) = k(t) + Bu(t),
0
t
volving a Hermitian operator. Subsequently, the x ∈ Ω, t ∈ [0, T] (3)
maximum principle, adjoint equation, and Hamil-
z(x, 0) = f(x, 0) = z 0 (x), x ∈ Ω, (4)
tonian maximization conditions are obtained via
duality and variational analysis. z(x, t) = 0, x ∈ Γ, t ∈ (0, T] (5)
2. Problem statement 3. Preliminaries
Here, the following fractional partial differential Definition 1. 14 For f : R + → R, fractional left
equations with state-dependent delay are consid- Riemann-Liouville integral of order α > 0 is de-
t
1
R
α
ered. fined as 0 I f(t) = Γ(α) 0 (t−s) α−1 f(s)ds, t > 0.
t
369