# The Constants in A Posteriori Error Indicator for State-Constrained Optimal Control Problems with Spectral Methods

Authors: Jianwei Zhou
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We employ Legendre-Galerkin spectral methods to solve
state-constrained optimal control problems. The constraint on the
state variable is an integration form. We choose one-dimensional
case to illustrate the techniques. Meanwhile, we investigate the
explicit formulae of constants within a posteriori error indicator.


---

## Body

## 1. Introduction

Spectral methods provide higher accurate approximations with a relatively small number of unknowns and play increasingly important roles in design optimization, engineering design, and other scientific and engineering computations. Gottlieb and Orszag [1] summarized the theories and applications of spectral methods. There have been extensive researches on finite element methods for optimal control problems, most of which focus on control-constrained problems; see [2–5]. The authors studied the optimal control problems with the control constraint with spectral methods in [6]. In applications of engineering, one cares more about how to control the average value or L
2-norms of the state variable. The authors [7] discussed state-constrained optimal control problems with finite element methods. However, there are few work on the state-constrained optimal control problems with spectral methods.In order to get a numerical solution with acceptable accuracy, spectral methods only increase the degree of basis when the error indicator is larger than the a posteriori error indicator, while the finite element methods refine meshes (see [8, 9]). There have been lots of papers on the a posteriori error estimates for h-version finite element methods but not for spectral methods. Guo [10] got a reliable and efficient error indicator for p-version finite element method in one dimension with a certain weight. The authors [11] deduced a simple error indicator for spectral Galerkin methods. In [12], the authors investigated Legendre-Galerkin spectral method for optimal control problems with integral constraint on state. It is difficult to obtain optimal a posteriori error indicators. Thus, if one gets the constants within upper bound a posteriori error estimates, it is easy to ensure the degree of polynomials to get an acceptable accuracy.In this paper, we employ Legendre-Galerkin spectral methods to solve optimal control problems with state-constrained case and calculate constants in upper bound of the a posteriori error indicator, which can be used to decide the least unknowns for acceptable accuracy. With the help of auxiliary systems, we investigate explicit formulae of the constants in the a posteriori error indicator.The outline of this paper is as follows. In Section2, the model problem and its Legendre-Galerkin spectral approximations are listed. In Section 3, the constants within the a posteriori error indicator are investigated in detail and the explicit formulae are obtained. The conclusions are given in Section 4.


## 2. A Model Problem and Its Legendre-Galerkin Spectral Approximations

Throughout this paper we adopt the standard notations of Sobolev spaces [13]. Let H
m
(
I
) be a Sobolev space on I
=
(
-
1,1
),  L
2
(
I
)
=
H
0
(
I
) and H
0
1
(
I
)
=
{
v
∈
H
1
(
I
)
:
v
=
0
on
⁡
∂
I
}, and the corresponding norms are denoted by ∥
·
∥
m
,
∥
·
∥
0, and ∥
·
∥
0,1, respectively. This work focuses on the Legendre polynomials, which are orthogonal polynomials on [−1,1].We concern the following distributed convex optimal control problems with integral constraint on state:(1)
(
OCP
)
{
min
⁡
y
∈
K
J
(
u
,
y
)
=
1
2
∫
I
‍
(
y
-
y
d
)
2
+
α
2
∫
I
‍
u
2
,
s
.
t
.
-
y
′′
=
u
in
I
,
y
=
0
on
⁡
∂
I
,
y
∈
K
,
where u
∈
U
=
L
2
(
I
) is the control variable, y
∈
K
=
{
w
:
∫
I
‍
w
≥
γ
}
⊂
H
0
1
(
I
)
≜
V is the state, and y
d
∈
L
2
(
I
) is the observation.In order to assure the existence and regularity of the solution, we assume thatα is a given positive constant and y
d is an infinitely smooth function. It is well known that the problem (OCP) has a unique solution (see [3]).We give some basic notations which will be used in the sequel. Let(2)
(
v
,
w
)
=
∫
I
‍
v
w
,
∀
v
,
w
∈
L
2
(
I
)
,
a
(
v
,
w
)
=
∫
I
‍
v
′
w
′
,
∀
v
,
w
∈
H
0
1
(
I
)
.
Hence, the state equation reduces to
(3)
a
(
y
,
w
)
=
(
u
,
w
)
,
∀
w
∈
H
0
1
(
I
)
.
Then (
OCP
) can be rewritten as finding (
u
,
y
) such that
(4)
(
P
)
{
min
⁡
y
∈
K
J
(
u
,
y
)
=
1
2
∫
I
‍
(
y
-
y
d
)
2
+
α
2
∫
I
‍
u
2
,
s
.
t
.
a
(
y
(
u
)
,
w
)
=
(
u
,
w
)
,
∀
w
∈
V
.
We recall the following optimal conditions of (
P
) (for details please refer to [7]).Lemma 1.
The pair(
u
,
y
)
∈
U
×
V is the optimal solution of (
P
) if and only if there exists a unique pair (
p
,
λ
)
∈
V
×
R
-
1
(
R
-
1
≜
{
c
∈
R
1
;
c
≤
0
}
) such that
(5)
(
O
C
P
-
O
P
T
)
{
a
(
y
,
w
)
=
(
u
,
w
)
,
∀
w
∈
V
,
a
(
q
,
p
)
=
(
y
-
y
d
,
q
)
+
λ
(
1
,
q
)
,
∀
q
∈
V
,
λ
(
w
-
y
)
≤
0
,
∀
w
∈
K
,
p
+
α
u
=
0
.LetP
N
(
I
)
=
{
polynomials
of
degree
≤
N
on
⁡
I
} and let V
N
=
P
N
∩
H
0
1
(
I
). One prefers to choose appropriate bases of V
N such that the resulting linear system is as simple as possible. We denote by {
L
j
}
j
=
0
N the Legendre polynomial and employ the following basis functions (see [14]):
(6)
U
N
=
span
⁡
{
L
0
(
x
)
,
L
1
(
x
)
,
…
,
L
N
(
x
)
}
,
V
N
=
span
⁡
{
ϕ
0
(
x
)
,
ϕ
1
(
x
)
,
…
,
ϕ
N
-
2
(
x
)
}
,
where
(7)
ϕ
i
(
x
)
=
c
i
(
L
i
(
x
)
-
L
i
+
2
(
x
)
)
,
c
i
=
1
4
i
+
6
.
For 1
≤
j
,
k
≤
N
-
2, we denote a
j
k
=
a
(
ϕ
k
(
x
)
,
ϕ
j
(
x
)
) and b
j
k
=
(
ϕ
k
(
x
)
,
ϕ
j
(
x
)
). By simple calculations, these coefficients satisfy
(8)
a
j
k
=
{
1
,
k
=
j
,
0
,
k
≠
j
,
b
j
k
=
b
k
j
=
{
c
k
c
j
(
2
2
j
+
1
+
2
2
j
+
5
)
,
k
=
j
,
-
c
k
c
j
2
2
k
+
1
,
k
=
j
+
2
,
0
,
otherwise
.
Then Legendre-Galerkin spectral approximations of (
OCP
) can be read as finding (
u
N
,
y
N
) such that
(9)
(
P
N
)
{
min
⁡
y
N
∈
K
J
(
u
N
,
y
N
)
=
1
2
∫
I
‍
(
y
N
-
y
d
)
2
+
α
2
∫
I
‍
u
N
2
,
s
.
t
.
a
(
y
N
,
w
N
)
=
(
u
N
,
w
N
)
,
∀
w
N
∈
V
N
.
The Legendre-Galerkin spectral approximations of (5) can be read as follows.Theorem 2.
The pair(
u
N
,
y
N
)
∈
U
N
×
V
N is the optimal solution of (
P
N
) if and only if there exists a unique pair (
p
N
,
λ
N
)
∈
V
N
×
R
-
1 such that
(10)
(
O
C
P
-
O
P
T
)
N
{
a
(
y
N
,
v
N
)
=
(
u
N
,
v
N
)
,
∀
v
N
∈
V
N
,
a
(
q
N
,
p
N
)
=
(
y
N
-
y
d
,
q
N
)
+
λ
N
(
1
,
q
N
)
,
∀
q
N
∈
V
N
,
λ
N
(
1
,
w
N
-
y
N
)
≤
0
,
∀
w
N
∈
K
N
,
α
u
N
+
p
N
=
0
.


## 3. Constants within the A Posteriori Error Estimates

In this section, we calculate all constants within the a posteriori error estimates. Here, we analyze the constant in the Poincaré inequality.For allv
∈
W
0
1
,
p
(
I
), 1
≤
p
<
∞, we recall the Poincaré inequality with L
2-norm as (see [15])
(11)
∥
v
∥
0
≤
|
I
|
2
∥
v
′
∥
0
.Now, we are at the point to investigate all constants in detail. We introduce an auxiliary statey
(
u
N
)
∈
H
0
1
(
I
), which satisfies
(12)
a
(
y
(
u
N
)
,
w
)
=
(
u
N
,
w
)
,
∀
w
∈
H
0
1
(
I
)
.
Subtracting (12) from (3), we get
(13)
a
(
y
-
y
(
u
N
)
,
w
)
=
(
u
-
u
N
,
w
)
,
∀
w
∈
H
0
1
(
I
)
.
Let w
=
y
(
u
N
)
-
y
∈
H
0
1
(
I
). It is clear that
(14)
a
(
y
(
u
N
)
-
y
,
y
(
u
N
)
-
y
)
=
(
u
N
-
u
,
y
(
u
N
)
-
y
)
.
Then
(15)
∥
(
y
(
u
N
)
-
y
)
′
∥
0
2
≤
∥
u
N
-
u
∥
0
∥
y
(
u
N
)
-
y
∥
0
≤
|
I
|
2
∥
u
N
-
u
∥
0
∥
(
y
(
u
N
)
-
y
)
′
∥
0
,
which means
(16)
∥
(
y
(
u
N
)
-
y
)
′
∥
0
≤
|
I
|
2
∥
u
N
-
u
∥
0
.
Hence
(17)
∥
y
(
u
N
)
-
y
∥
1
≤
(
∥
(
y
(
u
N
)
-
y
)
′
∥
0
2
+
(
|
I
|
2
)
2
∥
(
y
(
u
N
)
-
y
)
′
∥
0
2
)
1
/
2
=
(
1
+
(
|
I
|
2
)
2
)
1
/
2
∥
(
y
(
u
N
)
-
y
)
′
∥
0
.
Then
(18)
∥
y
(
u
N
)
-
y
∥
1
≤
(
1
+
(
|
I
|
2
)
2
)
1
/
2
|
I
|
2
∥
u
N
-
u
∥
0
.
We denote by c
1 the constant in (18); that is,
(19)
c
1
=
(
1
+
(
|
I
|
2
)
2
)
1
/
2
|
I
|
2
.Similarly, we introduce an auxiliary statep
(
u
N
)
∈
H
1
(
I
), which satisfies
(20)
a
(
q
,
p
(
u
N
)
)
=
(
y
(
u
N
)
-
y
d
,
q
)
+
λ
(
1
,
q
)
,
∀
q
∈
H
0
1
(
I
)
.
Subtracting (20) from the continuous systems (5), we get
(21)
a
(
p
-
p
(
u
N
)
,
w
)
=
(
y
-
y
(
u
N
)
,
w
)
+
(
λ
-
λ
N
)
(
1
,
w
)
,
vvvvvvvvvvvvvvvvvvvvvv
∀
w
∈
H
0
1
(
I
)
.We selectφ
∈
C
0
∞
(
I
) which satisfies φ
¯
=
1, where φ
¯
≜
∫
I
‍
φ
/
|
I
|
=
1 denotes the integral average on I of the function φ and ∥
φ
∥
1
≤
C
φ. Obviously, p
-
p
(
u
N
)
¯
φ
∈
C
0
∞
(
I
). In fact, p
-
p
(
u
N
)
-
p
-
p
(
u
N
)
¯
φ
∈
H
0
1
(
I
). Then there hold
(22)
∥
p
-
p
(
u
N
)
∥
1
2
≤
(
1
+
(
|
I
|
2
)
2
)
|
p
-
p
(
u
N
)
|
1
2
=
(
1
+
(
|
I
|
2
)
2
)
a
(
p
-
p
(
u
N
)
,
p
-
p
(
u
N
)
)
=
(
1
+
(
|
I
|
2
)
2
)
×
{
a
(
p
-
p
(
u
N
)
¯
φ
,
p
-
p
(
u
N
)
)
+
(
y
-
y
(
u
N
)
,
p
-
p
(
u
N
)
-
p
-
p
(
u
N
)
¯
φ
)
}
≤
(
1
+
(
|
I
|
2
)
2
)
×
{
|
p
-
p
(
u
N
)
¯
|
·
(
ϵ
1
2
|
φ
|
1
2
+
1
2
ϵ
1
|
p
-
p
(
u
N
)
|
1
2
)
+
ϵ
2
2
∥
y
-
y
(
u
N
)
∥
0
2
+
1
ϵ
2
(
∥
p
-
p
(
u
N
)
∥
0
2
+
|
p
-
p
(
u
N
)
¯
|
2
·
∥
φ
∥
0
2
)
}
,
where we used the generalized Schwarz inequality, continuous systems (5), and auxiliary equation (20). Let
(23)
(
1
+
(
|
I
|
2
)
2
)
|
p
-
p
(
u
N
)
¯
|
2
ϵ
1
=
(
1
+
(
|
I
|
2
)
2
)
1
ϵ
2
=
1
2
.
Then
(24)
ϵ
1
=
(
1
+
(
|
I
|
2
)
2
)
|
p
-
p
(
u
N
)
¯
|
,
ϵ
2
=
2
(
1
+
(
|
I
|
2
)
2
)
.
It is clear that (22) reduces to
(25)
∥
p
-
p
(
u
N
)
∥
1
2
≤
(
1
+
(
|
I
|
2
)
2
)
×
{
1
+
(
|
I
|
/
2
)
2
2
|
p
-
p
(
u
N
)
¯
|
2
1
+
(
|
I
|
/
2
)
2
2
|
p
-
p
(
u
N
)
¯
|
2
·
|
φ
|
1
2
+
|
p
-
p
(
u
N
)
¯
|
2
2
(
1
+
(
|
I
|
/
2
)
2
)
∥
φ
∥
0
2
+
(
1
+
(
|
I
|
2
)
2
)
∥
y
-
y
(
u
N
)
∥
0
2
1
+
(
|
I
|
/
2
)
2
2
|
p
-
p
(
u
N
)
¯
|
2
·
|
φ
|
1
2
+
|
p
-
p
(
u
N
)
¯
|
2
2
(
1
+
(
|
I
|
/
2
)
2
)
}
+
1
2
∥
p
-
p
(
u
N
)
∥
1
2
.
Then
(26)
∥
p
-
p
(
u
N
)
∥
1
2
≤
2
(
1
+
(
|
I
|
2
)
2
)
×
{
(
1
+
(
|
I
|
2
)
2
)
|
p
-
p
(
u
N
)
¯
|
2
·
∥
φ
∥
1
2
+
(
1
+
(
|
I
|
2
)
2
)
∥
y
-
y
(
u
N
)
∥
0
2
}
=
2
(
1
+
(
|
I
|
2
)
2
)
2
{
|
p
-
p
(
u
N
)
¯
|
2
·
∥
φ
∥
1
2
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
+
∥
y
-
y
(
u
N
)
∥
0
2
|
p
-
p
(
u
N
)
¯
|
2
}
,
where we used (
1
+
(
|
I
|
/
2
)
2
)
|
p
-
p
(
u
N
)
¯
|
2
≥
(
1
/
(
1
+
(
|
I
|
/
2
)
2
)
)
|
p
-
p
(
u
N
)
¯
|
2.Hence(27)
∥
p
-
p
(
u
N
)
∥
1
2
≤
2
(
1
+
(
|
I
|
2
)
2
)
2
×
{
(
1
+
(
|
I
|
2
)
2
)
∥
φ
∥
1
2
·
1
|
I
|
(
α
∥
u
-
u
N
∥
0
+
∥
p
N
-
p
(
u
N
)
∥
0
)
2
v
v
v
v
v
v
v
+
(
1
+
(
|
I
|
2
)
2
)
(
|
I
|
2
)
2
∥
u
-
u
N
∥
0
2
}
=
2
(
1
+
(
|
I
|
2
)
2
)
2
×
{
[
2
∥
φ
∥
1
2
α
2
|
I
|
+
(
1
+
(
|
I
|
2
)
2
)
(
|
I
|
2
)
2
]
z
z
z
z
z
z
z
×
∥
u
-
u
N
∥
0
2
+
2
∥
φ
∥
1
2
|
I
|
∥
p
N
-
p
(
u
N
)
∥
0
2
[
2
∥
φ
∥
1
2
α
2
|
I
|
+
(
1
+
(
|
I
|
2
)
2
)
(
|
I
|
2
)
2
]
}
=
2
(
1
+
(
|
I
|
2
)
2
)
2
max
⁡
⁡
{
2
∥
φ
∥
1
2
α
2
|
I
|
+
c
1
2
,
2
∥
φ
∥
1
2
|
I
|
}
×
{
∥
u
-
u
N
∥
0
2
+
∥
p
N
-
p
(
u
N
)
∥
0
2
}
,
which means that
(28)
∥
p
-
p
(
u
N
)
∥
1
≤
(
2
(
1
+
(
|
I
|
2
)
2
)
2
×
max
⁡
⁡
{
2
C
φ
2
α
2
|
I
|
+
(
1
+
(
|
I
|
2
)
2
)
(
|
I
|
2
)
2
,
2
C
φ
2
|
I
|
}
)
1
/
2
×
{
∥
u
-
u
N
∥
0
+
∥
p
N
-
p
(
u
N
)
∥
0
}
.
Denote by c
2 the constant in (28). With simple calculation, we have
(29)
c
2
=
2
(
1
+
(
|
I
|
2
)
2
)
2
max
⁡
⁡
{
2
C
φ
2
α
2
|
I
|
+
c
1
2
,
2
C
φ
2
|
I
|
}
.We selectφ
∈
C
0
∞
(
I
) which satisfies φ
¯
=
1 and ∥
φ
∥
1
≤
C
φ. For instance, φ
=
(
3
/
2
)
(
1
-
x
2
), which satisfies
(30)
∥
φ
∥
1
=
3
2
74
15
≜
C
φ
.
Meanwhile,
(31)
(
λ
-
λ
N
,
(
λ
-
λ
N
)
φ
)
=
(
λ
-
λ
N
)
2
∫
I
φ
‍
=
(
λ
-
λ
N
)
2
|
I
|
=
∥
λ
-
λ
N
∥
0
2
.
Hence
(32)
∥
λ
-
λ
N
∥
0
2
=
(
λ
-
λ
N
,
(
λ
-
λ
N
)
φ
)
=
a
(
(
λ
-
λ
N
)
φ
,
p
-
p
(
u
N
)
)
-
(
y
-
y
(
u
N
)
,
(
λ
-
λ
N
)
φ
)
≤
|
λ
-
λ
N
|
{
∥
φ
′
∥
0
·
∥
(
p
-
p
(
u
N
)
)
′
∥
0
v
v
v
v
v
v
v
v
v
v
v
v
i
+
∥
φ
∥
0
·
∥
y
-
y
(
u
N
)
∥
0
∥
(
p
-
p
(
u
N
)
)
′
∥
0
}
≤
ϵ
1
2
|
λ
-
λ
N
|
2
·
∥
φ
′
∥
0
2
+
1
2
ϵ
1
∥
(
p
-
p
(
u
N
)
)
′
∥
0
2
+
ϵ
2
2
|
λ
-
λ
N
|
2
·
∥
φ
∥
0
2
+
1
2
ϵ
2
∥
y
-
y
(
u
N
)
∥
0
2
=
ϵ
1
2
|
I
|
∥
λ
-
λ
N
∥
0
2
∥
φ
∥
1
2
+
1
2
ϵ
1
(
∥
(
p
-
p
(
u
N
)
)
′
∥
0
2
+
∥
y
-
y
(
u
N
)
∥
0
2
)
,
where ϵ
1
=
ϵ
2
=
|
I
|
/
∥
φ
∥
1
2.Thus(33)
∥
λ
-
λ
N
∥
0
2
≤
∥
φ
∥
1
2
|
I
|
{
∥
(
p
-
p
(
u
N
)
)
′
∥
0
2
+
∥
y
-
y
(
u
N
)
∥
0
2
}
≤
∥
φ
∥
1
2
|
I
|
{
∥
p
-
p
(
u
N
)
∥
1
2
+
∥
y
-
y
(
u
N
)
∥
0
2
}
.
With the constant c
2, we infer that
(34)
∥
λ
-
λ
N
∥
0
2
≤
C
φ
2
|
I
|
{
4
(
1
+
(
|
I
|
2
)
2
)
2
v
v
v
v
v
v
v
v
v
v
×
max
⁡
⁡
{
2
C
φ
2
α
2
|
I
|
+
(
1
+
(
|
I
|
2
)
2
)
(
|
I
|
2
)
2
,
2
C
φ
2
|
I
|
}
v
v
v
v
v
v
v
i
v
+
(
1
+
(
|
I
|
2
)
2
)
(
|
I
|
2
)
2
}
×
{
∥
p
N
-
p
(
u
N
)
∥
0
2
+
∥
u
-
u
N
∥
0
2
}
=
C
φ
2
|
I
|
{
2
c
2
2
+
c
1
2
}
{
∥
p
N
-
p
(
u
N
)
∥
0
2
+
∥
u
-
u
N
∥
0
2
}
.
Then
(35)
∥
λ
-
λ
N
∥
0
≤
C
φ
2
|
I
|
{
2
c
2
2
+
c
1
2
}
{
∥
p
N
-
p
(
u
N
)
∥
0
+
∥
u
-
u
N
∥
0
}
.
We denote by c
3 the constant in (35); that is,
(36)
c
3
=
C
φ
2
|
I
|
{
2
c
2
2
+
c
1
2
}
.We calculate the error ofu in L
2-norm as follows:
(37)
∥
u
-
u
N
∥
0
2
≤
(
p
N
-
p
(
u
N
)
,
u
-
u
N
)
-
(
λ
-
λ
N
,
y
N
-
y
(
u
N
)
)
≤
ϵ
1
2
∥
p
N
-
p
(
u
N
)
∥
0
2
+
1
2
ϵ
1
∥
u
-
u
N
∥
0
2
+
1
2
ϵ
2
∥
y
N
-
y
(
u
N
)
∥
0
2
+
ϵ
2
·
c
3
2
{
∥
u
-
u
N
∥
0
2
+
∥
p
N
-
p
(
u
N
)
∥
0
2
}
=
(
ϵ
1
2
+
ϵ
2
·
c
3
2
)
∥
p
N
-
p
(
u
N
)
∥
0
2
+
(
1
2
ϵ
1
+
ϵ
2
·
c
3
2
)
∥
u
-
u
N
∥
0
2
+
1
2
ϵ
2
∥
y
N
-
y
(
u
N
)
∥
0
2
.
Provided that 1
/
2
ϵ
1
+
ϵ
2
·
c
3
2
=
1
/
2, we get
(38)
∥
u
-
u
N
∥
0
2
≤
2
(
ϵ
1
2
+
ϵ
2
·
c
3
2
)
∥
p
N
-
p
(
u
N
)
∥
0
2
+
1
ϵ
2
∥
y
N
-
y
(
u
N
)
∥
0
2
=
max
⁡
⁡
{
ϵ
1
+
2
ϵ
2
·
c
3
2
,
1
ϵ
2
}
{
∥
p
N
-
p
(
u
N
)
∥
0
2
vvvvvvvvvvvvvvvvvvvv
.
vvv
+
∥
y
N
-
y
(
u
N
)
∥
0
2
}
.
Considering the item max
⁡
{
ϵ
1
+
2
ϵ
2
·
c
3
2
,
1
/
ϵ
2
} with the constraint 1
/
2
ϵ
1
+
ϵ
2
·
c
3
2
=
1
/
2, we get
(39)
F
(
ϵ
1
)
=
max
⁡
{
ϵ
1
+
1
-
1
ϵ
1
,
2
c
3
2
ϵ
1
ϵ
1
-
1
}
.
In fact, for ∀
ϵ
1
>
1, the derivation of the following function
(40)
f
(
ϵ
1
)
=
ϵ
1
+
1
-
1
ϵ
1
-
2
c
3
2
ϵ
1
ϵ
1
-
1
is
(41)
f
′
(
ϵ
1
)
=
1
+
1
ϵ
1
2
+
2
c
3
2
(
ϵ
1
-
1
)
2
≥
0
.
Then we have ϵ
1
0
=
2
c
3
2
+
ϵ
>
1 and
(42)
lim
⁡
ϵ
→
0
f
(
ϵ
1
0
)
=
0
.
Now, we are at the point to investigate
(43)
min
⁡
F
(
ϵ
1
)
.
If ϵ
1
=
ϵ
1
0, we get
(44)
min
⁡
F
(
ϵ
1
)
=
ϵ
1
0
+
1
-
1
ϵ
1
0
≤
2
c
3
2
+
ϵ
+
1
.
If 1
<
ϵ
1
<
ϵ
1
0, f
(
ϵ
1
)
<
0, we obtain
(45)
F
(
ϵ
1
)
=
2
c
3
2
ϵ
1
ϵ
1
-
1
,
F
′
(
ϵ
1
)
=
-
2
c
3
2
(
ϵ
1
-
1
)
2
<
0
.
Then
(46)
lim
⁡
ϵ
′
→
0
min
⁡
F
(
ϵ
1
)
=
2
c
3
2
(
1
+
1
2
c
3
2
+
ϵ
-
ϵ
′
-
1
)
<
4
c
3
2
.
If ϵ
1
>
ϵ
1
0,  f
(
ϵ
1
)
>
0, we infer that
(47)
F
(
ϵ
1
)
=
ϵ
1
+
1
-
1
ϵ
1
,
F
′
(
ϵ
1
)
=
1
+
1
ϵ
1
2
>
0
.
Then there hold
(48)
lim
⁡
ϵ
′′
→
0
min
⁡
F
(
ϵ
1
)
=
2
c
3
2
+
ϵ
+
ϵ
′′
+
1
-
1
2
c
3
2
+
ϵ
+
ϵ
′′
<
2
c
3
2
+
1
.
Combining the above discussions, we deduce that
(49)
min
⁡
{
max
⁡
{
ϵ
1
+
2
ϵ
2
·
c
3
2
,
1
ϵ
2
}
}
<
2
c
3
2
+
1
.
Obviously,
(50)
∥
u
-
u
N
∥
0
≤
2
c
3
2
+
1
{
∥
p
N
-
p
(
u
N
)
∥
0
+
∥
y
N
-
y
(
u
N
)
∥
0
}
.
We denote by c
4 the constant in (50); that is,
(51)
c
4
=
2
c
3
2
+
1
.For anyv
∈
L
2
(
I
), we define a projection operator P
N
:
L
2
(
I
)
→
V
N, which satisfies
(52)
(
P
N
v
-
v
,
w
N
)
=
0
,
∀
w
N
∈
U
N
.Lemma 3.
For allv
∈
H
σ
(
I
)
(
σ
≥
0
), one has
(53)
∥
P
N
v
-
v
∥
0
≤
c
5
N
-
σ
∥
v
∥
σ
,
where c
5
=
2
2.Proof.
Firstly, assuming thatσ
=
2
p
(
p
≥
1
) is integer, we define a differential operator as
(54)
A
=
d
d
x
(
(
1
-
x
2
)
d
d
x
)
.
From the fact that
(55)
d
d
x
(
(
1
-
x
2
)
d
L
k
d
x
)
+
k
(
k
+
1
)
L
k
=
0
,
it is easy to get
(56)
v
^
k
=
(
k
+
1
2
)
(
v
,
L
k
)
=
k
+
1
/
2
k
(
k
+
1
)
∫
-
1
1
‍
A
L
k
(
x
)
v
(
x
)
d
x
=
-
k
+
1
/
2
k
(
k
+
1
)
∫
-
1
1
‍
A
v
(
x
)
L
k
(
x
)
d
x
=
-
k
+
1
/
2
k
(
k
+
1
)
(
A
v
(
x
)
,
L
k
(
x
)
)
.
By iterations, we obtain
(57)
v
^
k
=
(
-
1
k
(
k
+
1
)
)
p
(
k
+
1
2
)
(
A
p
v
,
L
k
)
.
Secondly, for all v
∈
H
2
p
(
I
), we note that A
p
v
=
∑
i
=
0
∞
‍
α
i
L
i
(
x
) and
(58)
(
A
p
v
(
x
)
,
L
k
(
x
)
)
=
α
i
(
k
+
1
2
)
-
1
,
∥
A
p
v
∥
0
2
=
∑
k
=
0
∞
‍
|
α
k
|
2
(
k
+
1
2
)
-
1
.
Hence
(59)
∥
P
N
v
-
v
∥
0
2
=
∑
k
=
N
+
1
∞
(
k
+
1
2
)
-
1
|
v
^
k
|
2
‍
=
∑
k
=
N
+
1
∞
(
1
k
(
k
+
1
)
)
2
p
(
k
+
1
2
)
‍
|
A
p
v
,
L
k
|
2
≤
N
-
4
p
∑
k
=
N
+
1
∞
(
k
+
1
2
)
|
α
k
|
2
‍
(
k
+
1
2
)
-
2
≤
N
-
4
p
∥
A
p
v
∥
0
2
.
Finally, there hold
(60)
|
A
v
|
2
=
|
d
d
x
(
(
1
-
x
2
)
d
v
d
x
)
|
2
=
|
(
1
-
x
2
)
v
′′
-
2
x
v
′
|
2
≤
(
|
v
′′
|
+
2
|
v
′
|
)
2
≤
2
v
′′
2
+
8
v
′
2
≤
8
{
v
′′
2
+
v
′
2
+
v
2
}
,
which means that
(61)
∥
A
v
∥
0
2
≤
8
∥
v
∥
2
2
.
Let p
=
1
,
σ
=
2. It is clear that
(62)
∥
P
N
v
-
v
∥
0
≤
8
N
-
2
∥
v
∥
2
.
This completes the proof.Now, we are at the point to calculate the constant for∥
y
N
-
y
(
u
N
)
∥
1
+
∥
p
N
-
p
(
u
N
)
∥
1. Similarly, let E
p
=
p
N
-
p
(
u
N
) and let E
I
p
=
P
N
E
p
∈
V
N. Then
(63)
∥
p
N
-
p
(
u
N
)
∥
1
2
≤
(
1
+
(
|
I
|
2
)
2
)
(
a
(
E
p
,
E
p
-
E
I
p
)
+
(
y
(
u
N
)
-
y
N
,
E
I
p
)
)
=
(
1
+
(
|
I
|
2
)
2
)
(
a
(
p
(
u
N
)
-
p
N
,
E
p
-
E
I
p
)
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
+
(
y
(
u
N
)
-
y
N
,
E
I
p
)
)
=
(
1
+
(
|
I
|
2
)
2
)
×
(
(
-
p
′′
(
u
N
)
,
E
p
-
E
I
p
)
v
v
v
v
.
i
v
v
+
(
p
N
′′
,
E
p
-
E
I
p
)
+
(
y
(
u
N
)
-
y
N
,
E
I
p
)
)
=
(
1
+
(
|
I
|
2
)
2
)
(
(
y
N
-
y
d
+
λ
N
+
p
N
′′
,
E
p
-
E
I
p
)
v
v
v
v
v
v
v
v
v
v
v
v
v
i
v
v
v
v
+
(
y
(
u
N
)
-
y
N
,
E
p
)
(
y
N
-
y
d
+
λ
N
+
p
N
′′
,
E
p
-
E
I
p
)
)
≤
(
1
+
(
|
I
|
2
)
2
)
∥
E
p
∥
1
{
c
5
N
-
1
∥
y
N
-
y
d
+
λ
N
+
p
N
′′
∥
0
vvvvvvvvvvvvvvv
.
vvivvv
.
+
∥
y
N
-
y
(
u
N
)
∥
0
N
-
1
∥
y
N
-
y
d
+
λ
N
+
p
N
′′
∥
0
}
,
which means that
(64)
∥
p
N
-
p
(
u
N
)
∥
1
≤
(
1
+
(
|
I
|
2
)
2
)
{
c
5
N
-
1
∥
y
N
-
y
d
+
λ
N
+
p
N
′′
∥
0
v
v
v
v
v
v
v
v
v
v
v
v
v
v
i
v
v
v
+
∥
y
N
-
y
(
u
N
)
∥
0
∥
y
N
-
y
d
+
λ
N
+
p
′′
N
∥
0
}
.
Likewise, let E
y
=
y
N
-
y
(
u
N
) and let E
I
y
=
P
1
,
N
0
E
y
∈
V
N. Then
(65)
∥
y
N
-
y
(
u
N
)
∥
1
2
=
∥
E
y
∥
1
2
≤
(
1
+
(
|
I
|
2
)
2
)
a
(
E
y
,
E
y
)
=
(
1
+
(
|
I
|
2
)
2
)
a
(
E
y
-
E
I
y
,
E
y
)
=
(
1
+
(
|
I
|
2
)
2
)
(
-
(
u
N
+
y
N
′′
)
,
E
y
-
E
I
y
)
≤
(
1
+
(
|
I
|
2
)
2
)
c
5
N
-
1
∥
u
N
+
y
N
′′
∥
0
·
∥
E
y
∥
1
,
which is equivalent to
(66)
∥
y
N
-
y
(
u
N
)
∥
1
≤
(
1
+
(
|
I
|
2
)
2
)
c
5
N
-
1
∥
u
N
+
y
N
′′
∥
0
.
Hence
(67)
∥
y
N
-
y
(
u
N
)
∥
1
+
∥
p
N
-
p
(
u
N
)
∥
1
≤
c
6
η
,
where
(68)
c
6
=
(
1
+
(
|
I
|
2
)
2
)
2
c
5
,
η
=
N
-
1
∥
y
N
-
y
d
+
λ
N
+
p
N
′′
∥
0
+
N
-
1
∥
u
N
+
y
N
′′
∥
0
.
Combining the above analyses, we get that
(69)
∥
u
-
u
N
∥
0
+
∥
y
-
y
N
∥
1
+
∥
p
-
p
N
∥
1
+
∥
λ
-
λ
N
∥
0
≤
∥
u
-
u
N
∥
0
+
∥
y
-
y
(
u
N
)
∥
1
+
∥
y
N
-
y
(
u
N
)
∥
1
+
∥
p
-
p
(
u
N
)
∥
1
+
∥
p
N
-
p
(
u
N
)
∥
1
+
∥
λ
-
λ
N
∥
0
≤
∥
y
N
-
y
(
u
N
)
∥
1
+
∥
p
N
-
p
(
u
N
)
∥
1
+
c
4
{
∥
p
N
-
p
(
u
N
)
∥
0
+
∥
y
N
-
y
(
u
N
)
∥
0
}
+
c
1
c
4
{
∥
p
N
-
p
(
u
N
)
∥
0
+
∥
y
N
-
y
(
u
N
)
∥
0
}
+
c
2
{
∥
p
N
-
p
(
u
N
)
∥
0
vvvivvvv
+
c
4
(
∥
p
N
-
p
(
u
N
)
∥
0
+
∥
y
N
-
y
(
u
N
)
∥
0
)
}
+
c
3
{
∥
p
N
-
p
(
u
N
)
∥
0
,
I
vvvivvvv
+
c
4
(
∥
p
N
-
p
(
u
N
)
∥
0
+
∥
y
N
-
y
(
u
N
)
∥
0
)
}
≤
∥
p
N
-
p
(
u
N
)
∥
1
{
c
4
+
1
+
c
1
c
4
+
c
2
(
c
4
+
1
)
+
c
3
(
c
4
+
1
)
}
+
∥
y
N
-
y
(
u
N
)
∥
1
{
c
4
+
1
+
c
1
c
4
+
c
2
c
4
+
c
3
c
4
}
≤
{
c
4
+
1
+
c
1
c
4
+
c
2
(
c
4
+
1
)
+
c
3
(
c
4
+
1
)
}
vvvvi
×
{
∥
p
N
-
p
(
u
N
)
∥
1
+
∥
y
N
-
y
(
u
N
)
∥
1
}
≤
{
c
4
+
1
+
c
1
c
4
+
c
2
(
c
4
+
1
)
+
c
3
(
c
4
+
1
)
}
c
6
η
,
which means that
(70)
∥
u
-
u
N
∥
0
+
∥
y
-
y
N
∥
1
+
∥
p
-
p
N
∥
1
+
∥
λ
-
λ
N
∥
0
≤
C
η
,
where
(71)
C
=
{
1
+
c
4
+
c
1
c
4
+
c
2
(
c
4
+
1
)
+
c
3
(
c
4
+
1
)
}
c
6
,
c
1
=
(
1
+
(
|
I
|
2
)
2
)
1
/
2
|
I
|
2
,
c
2
=
2
(
1
+
(
|
I
|
2
)
2
)
2
max
⁡
⁡
{
2
C
φ
2
α
2
|
I
|
+
c
1
2
,
2
C
φ
2
|
I
|
}
,
c
3
=
C
φ
2
|
I
|
{
2
c
2
2
+
c
1
2
}
,
c
4
=
2
c
3
2
+
1
,
c
5
=
2
,
c
6
=
(
1
+
(
|
I
|
2
)
2
)
2
c
5
.


## 4. Conclusions

This paper discusses the explicit formulae of constants within upper bound of the a posteriori error estimate for optimal control problems with Legendre-Galerkin spectral methods in one dimension. Thus, with those formulae, it is easy to choose a suitable degree of polynomials to obtain an acceptable accuracy. In the future, we will study the corresponding constants in lower bound of the a posteriori error indicator. Meanwhile, the corresponding constants in a two-dimensional domain will be investigated.



---

*Source: 946026-2014-05-19.xml*