Regional gradient optimal control problem governed by a distributed bilinear systems

TELKOMNIKA, Vol.17, No.4, August 2019, pp.1957~1965
ISSN: 1693-6930, accredited First Grade by Kemenristekdikti, Decree No: 21/E/KPT/2018
DOI: 10.12928/TELKOMNIKA.v17i4.11275  1957
Received September 23, 2018; Revised March 28, 2019; Accepted April 16, 2019
Regional gradient optimal control problem governed by
a distributed bilinear systems
Maawiya Ould Sidi
*
, Sid Ahmed Beinane
Mathematics Department, College of Science, Jouf University,
P.O. Box: 2014, Sakaka, Saudi Arabia
*Corresponding author, e-mail: maawiya81@gmail.com, beinane06@gmail.com
Abstract
This paper gives an extension of previous work on gradient optimal control of distributed
parabolic systems to the case of distributed bilinear systems which are a type of nonlinear systems. We
introduce the notion of flux optimal control of distributed bilinear systems. The idea is trying to achieve a
neighborhood of the gradient state of the considered system by minimizing a nonlinear quadratic cost.
Using optimization techniques, a method showing how to reach a desired flux at a final time, only on
internal subregion of the system domain will be proposed. The proposed simulation illustrates
the theoretical approach by commanding the heat bilinear equation flux to a desired profile.
Keywords: bilinear systems, gradient state, optimal control problem, simulations
Copyright © 2019 Universitas Ahmad Dahlan. All rights reserved.
1. Introduction
Infinite dimensional Bilinear systems analysis can formulate many real problems see
Lions [1, 2]. The controllability is among the most important analysis notions, within there are
many concepts as exact controllability, approximate controllability, regional controllability and so
on. In [3], Ball, Marsden, and Slemrod, discussed the controllability for distributed Bilinear
systems. Bradly, Lenhart, and Yong, in [4] treated the optimal control of the Velocity term in a
Kirchhoff plate equation. A very important applications of optimal control problems, refereeing to
the optimal control problem in which the interest state is specified only on ω, a subregion of
the system domain. El Jai, Pritchard, Simon, and Zerrik in [5], give an example where the
control is required to achieve the temperature at a level in a specified subregion of the furnace.
Many interesting results were developed in the case of parabolic and hyperbolic
systems, we cite in particular the result proving that there exist a systems which are not
controllable to the whole domain but controllable to a subregion see Zerrik, Kamal [6].
These results were generalized by Zerrik, Kamal in [7] and Zerrik et al in [8] to the case called
boundary controllability refereeing to the subregion ω located on the boundary of system
domain. The gradient optimal control concerns the optimal control of the gradient state to a
subregion of the system domain. The readers can obtain very interest contributions in this field,
particularly characterizations of the optimal control that achieves regional gradient controllability
by Zerrik et all in [9] and Kamal et al [10] in the case of parabolic linear systems, and Ould
Beinane et all [11] in the case of semi linear systems.
For bilinear distributed systems, the notion of regional optimal control is introduced by
Zerrik and Ould Sidi in [12-14], showing the existence of an optimal control by a minimizing
sequence, solution of the quadratic cost control problem which involves the minimization of
the norm control and the final state error and deriving a characterization for optimal control,
using the solution of an optimality system. Thereafter, Zerrik. and El Kabouss in [15] gives an
extension of previous regional optimal control works to the case of a spatiotemporal damping.
El Harraki and Boutoulout in [16] studied the controllability of the wave equation with
multiplicative controls. Zine and Ould Sidi in [17, 18] and Zine in [19] treated the regional optimal
control problems governed by bilinear hyperbolic distributed systems.
This paper discuss an extension of the previous results on the regional optimal control
of distributed systems (linear, semi linear and bilinear) to the case of gradient optimal control of
bilinear parabolic system, which constitutes an important progress in system theory.
In particular, we treat the problem of regional gradient optimal control of bilinear systems using

 ISSN: 1693-6930
TELKOMNIKA Vol. 17, No. 4, August 2019: 1957-1965
1958
a quadratic nonlinear method. We show the existence of an optimal control solution of
the considered problem. Using the optimization techniques, we give a characterization for
the optimal control. Numerical simulations are established illustrating successfully
the theoretical approach.
2. Preliminary
We consider the following equation, which is governed by a heat bi-linear system
{
𝜕𝑢
𝜕𝑡
+ Δ𝑢 = 𝑄(𝑡)𝑢 Π
𝑢(𝑥, 0) = 𝑢0(𝑥) Ω
𝑢 = 0 Σ
(1)
where Ω is an open bounded domain in 𝐼𝑅 𝑛
(𝑛 = 1,2,3), with a regular boundary 𝜕Ω.
For 𝑇 > 0, Π = Ω ×]0, 𝑇[, Σ = 𝜕Ω ×]0, 𝑇[ and 𝑄 ∈ 𝐿2
(0, 𝑇) is the control function.
The Laplace operator Δ generates the strongly continuous semi-group (𝑆(𝑡)) 𝑡≥0 on the state
space 𝐿2
(Ω) such that
𝑆(𝑡)𝑢 𝑄(𝑡) = ∑ ‍+∞
𝑛=1 𝑒 𝜆 𝑛 𝑡
< 𝑢 𝑄(𝑡), 𝜙 𝑛 > 𝜙 𝑛. (2)
where 𝜆 𝑛 is the eigenvalues of Δ and 𝜙 𝑛 its associate eigenfunctions. For a given 𝑢0 ∈ 𝐻1
(Ω),
the system (1) may be written as:
𝑢(𝑡) = 𝑆(𝑡)𝑢0 + ∫ ‍
𝑡
0
𝑆(𝑡 − 𝑠)𝑄(𝑠)𝑢(𝑠)𝑑𝑠. (3)
and solutions of (3) are often called mild solutions of (1). The existence of a unique solution
𝑢 𝑄(𝑥, 𝑡) in 𝐿2
(0, 𝑇; 𝐻0
1
(Ω)) satisfying (3) follows from standard results in [20, 21]. For 𝜔 ∈ Ω,
we define the restriction operator to 𝜔 by:
𝜒 𝜔: (𝐿2
(Ω)) 𝑛
⟶ (𝐿2
(𝜔)) 𝑛
‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍𝑢 ⟶ 𝜒 𝜔 𝑢 = 𝑢| 𝜔
and 𝜒 𝜔
∗
its adjoint given by
𝜒 𝜔
∗
𝑢 = {
𝑢‍‍𝑖𝑛Ω
0 ∈ Ω𝜔
and
𝜒̃ 𝜔: (𝐿2
(Ω)) ⟶ (𝐿2
(𝜔))
‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍𝑢 ⟶ 𝜒̃ 𝜔 𝑢 = 𝑢| 𝜔
let ∇ the operator defined by
∇: 𝐻1
(Ω) ⟶ (𝐿2
(Ω)) 𝑛
‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍‍𝑢 ⟶ ∇𝑢 = (
𝜕𝑢
𝜕𝑥1
, . . . . ,
𝜕𝑢
𝜕𝑥 𝑛
)
with adjoint ∇∗
.
2.1. Definition
The system (1) is said to be weakly regionally gradient controllable on 𝜔 ⊂ Ω if for all
𝑔 𝑑
∈ (𝐿2
(𝜔)) 𝑛
and 𝜀 > 0 there exists a control 𝑄 ∈ 𝐿2
[0, 𝑇] such that
||𝜒 𝜔∇𝑢 𝑄(𝑇) − 𝑔 𝑑
||(𝐿2(𝜔)) 𝑛 ≤ 𝜀
where 𝑔 𝑑
= (𝑦1
𝑑
, . . . . 𝑦𝑛
𝑑
) is the gradient of the desired state in the space 𝐿2
(𝜔).

TELKOMNIKA ISSN: 1693-6930 
Regional gradient optimal control problem governed… (Maawiya Ould Sidi)
1959
Our main objective is to solve the regional gradient quadratic control problem governed by the
bi-linear distributed in (1)
min
𝑄‍‍∈𝐿2([0,𝑇])
𝐽𝜀(𝑄). (4)
where the gradient quadratic cost 𝐽𝜀 is defined for 𝜀 > 0 by
𝐽𝜀(𝑄) =
1
2
∥ 𝜒 𝜔∇𝑢(𝑇) − 𝑔 𝑑
∥(𝐿2(𝜔)) 𝑛
2
+ 𝜀 ∥ 𝑄(𝑡) ∥ 𝐿2([0,𝑇])
2
=
1
2
∑ ‍𝑛
𝑖=1 ∥ 𝜒̃ 𝜔
𝜕𝑢(𝑇)
𝜕𝑥 𝑖
− 𝑦𝑖
𝑑
∥ 𝐿2(𝜔)
2
+ 𝜀 ∥ 𝑄(𝑡) ∥ 𝐿2([0,𝑇])
2
(5)
Quadratic optimal control problem governed by bilinear systems aims in general to steer
the state of a considered system to a desired profile. Many references use quadratic cost us (5),
we cite for example Addou and Benbrik in [22], Bradly and Lenhart in [23], Bradly et all in [4],
and Lenhart [24]. In applications there are many motivations of the regional gradient optimal
control problems governed by bilinear systems, for example in thermal isolation problems it
happens that the control is maintained to reducing the gradient temperature before the brick
see [6-8]. The original goal of this paper is to street the gradient state of the bilinear system (1)
to the desired state 𝑔 𝑑
(𝑥) by minimizing objective functional (5), and characterize an optimal
control 𝑄∗
∈ 𝐿2
(0, 𝑇) such that 𝐽𝜀(𝑄∗
) = min 𝑄‍‍∈‍‍𝐿2(0,𝑇) 𝐽𝜀(𝑄).
3. Existence of Solution
Firstly, we prove our main theorem of this section.
3.1. Theorem
There exists a pair (𝑢̅, 𝑄∗
) ∈ 𝐶([0, 𝑇]; 𝐻0
1
(Ω)) × 𝐿2
([0, 𝑇]), such that 𝑢̅ is the unique
solution of
{
𝜕𝑢(𝑥,𝑡)
𝜕𝑡
= −Δ𝑢(𝑥, 𝑡) + 𝑄∗
(𝑡)𝑢(𝑥, 𝑡) Π
𝑢(𝑥, 0) = 𝑢0(𝑥) Ω (6)
and 𝑄∗
is solution of the problem (4).
Proof. The set {𝐽𝜀(𝑄)|‍‍𝑄 ∈ 𝐿2
([0, 𝑇])} is a nonempty set of 𝐼𝑅+
, then it admit a lower
bounded. We choose (𝑄 𝑛) 𝑛 a minimizing sequence such that
𝐽∗
= lim
𝑛→+∞
𝐽(𝑄 𝑛) = inf
𝑄∈𝐿2([0,𝑇])
𝐽𝜀(𝑄)
𝐽𝜀(𝑄 𝑛) is then bounded, it follows that ||𝑄 𝑛|| 𝐿2([0,𝑇]) ‍‍≤ ‍‍𝑀, for a positive constant 𝑀. Using
𝑢 𝑄(𝑥, 𝑡) the weak solution of (1) in 𝑊 = 𝐿2
(0, 𝑇; 𝐻0
1
(Ω)), and from (3), we
deduce that
||𝑢 𝑄(𝑡)|| 𝑊 ≤ (||𝑢0|| 𝐿2(Ω) + ∫ ‍
𝑡
0
|𝑄(𝑠)|||𝑢 𝑄(𝑠)|| 𝑊 𝑑𝑠)
by Gronwall inequality, we have
||𝑢 𝑄(𝑡)|| 𝑊 ≤ 𝐶exp(𝑀𝑇) (7)
where 𝐶 = ||𝑢0|| 𝐿2(Ω), and from there, we deduce that 𝑢 𝑛(𝑥, 𝑡) = 𝑢 𝑄 𝑛
(𝑥, 𝑡) is bounded. From the
bounds of 𝑄 𝑛 and 𝑢 𝑛(𝑥, 𝑡) follows
||Δ(𝑦𝑛)|| 𝑊 ‍‍≤‍‍ 𝑀1, ||𝑄 𝑛(𝑢 𝑛)|| 𝑊 ‍‍≤‍‍ 𝑀2,‍‍‍𝑎𝑛𝑑‍‍‍||𝑢 𝑛′|| 𝑊 ‍‍≤‍‍ 𝑀3

 ISSN: 1693-6930
1960
where 𝑀1, 𝑀2 and 𝑀3 are three positive constants. From the priori estimates, we can extract a
subsequences such as:
𝑄 𝑛 ⇀ 𝑄∗
‍𝑤𝑒𝑎𝑘𝑙𝑦‍‍𝑖𝑛‍‍‍ 𝐿2
(0, 𝑇)
𝑢 𝑛 ⇀ 𝑢̅ ‍𝑤𝑒𝑎𝑘𝑙𝑦‍‍𝑖𝑛‍‍‍ 𝑊
Δ𝑢 𝑛 ⇀ 𝜒 ‍𝑤𝑒𝑎𝑘𝑙𝑦‍‍𝑖𝑛‍‍‍ 𝑊
𝑢 𝑛(𝑄 𝑛) ⇀ Λ ‍𝑤𝑒𝑎𝑘𝑙𝑦‍‍𝑖𝑛‍‍‍ 𝑊
𝑢 𝑛′ ⇀ Ψ ‍𝑤𝑒𝑎𝑘𝑙𝑦‍‍𝑖𝑛‍‍‍ 𝑊
(8)
by classical argument, we check that 𝑢̅(0) = 𝑢0, then by limit as 𝑛 ⟶ ∞ the system (6) gives
𝑢̅′ = Ψ , Δ𝑢̅ = 𝜒 and 𝑄∗
𝑢̅ = Λ. Furthermore 𝑢̅ = 𝑢(𝑄∗
). To prove that 𝑄∗
is optimal, we use the
lower semi continuity of 𝐽𝜀(𝑄), and applying Fatou’s Lemma we have
𝐽(𝑄∗
) =
1
2
inf
𝑛
∑ ‍𝑛
𝑖=1 ∫ ‍𝜔
(𝜒̃ 𝜔
𝜕𝑢 𝑛
𝜕𝑥 𝑖
− 𝑦𝑖
𝑑
)2
𝑑𝑥 + 𝜀 ∫ ‍
𝑇
0
𝑄 𝑛
2
(𝑡)𝑑𝑡
≤ lim
𝑛⟶∞
𝐽𝜀(𝑄 𝑛) = inf
𝑄
𝐽𝜀(𝑄)
(9)
which prove that 𝑄∗
is optimal for the problem (4).
4. Characterization of Solution
To formulate an explicit solution of the optimal problem (4), we propose an adjoint
equation by differentiating the quadratic cost 𝐽𝜀(𝑄) respecting to Q. Next Lemma study the
differential of 𝑄 ⟶ 𝑢(𝑄) with respect to 𝑄.
4.1. Lemma
The function
𝐿2
(0, 𝑇) ⟶ 𝐶([0, 𝑇]; 𝐻1
(Ω)),
‍‍‍‍‍‍‍‍𝑄 ⟶ 𝑢(𝑄)
solution of (6) is differentiable and its differential 𝜓 verify the system
{
𝜕𝜓(𝑥,𝑡)
𝜕𝑡
= −Δ𝜓(𝑥, 𝑡) + 𝑄∗
(𝑡)𝜓(𝑥, 𝑡) + ℎ(𝑡)𝑢̅(𝑥, 𝑡) Π
𝜓(𝑥, 0) = 𝜓0(𝑥) = 0 Ω (10)
with 𝑢̅ = 𝑢(𝑄∗
), ℎ ∈ 𝐿2
([0, 𝑇]), and 𝑑(𝑢(𝑄∗
))ℎ is the differential of 𝑄 → 𝑢(𝑄) respecting 𝑄∗
.
Proof. Since 𝜓 is solution of the (10), we have
||𝜓|| 𝑊 ≤ 𝑘1||𝑢̅|| 𝐿∞(0,𝑇;𝐻0
1(Ω))||ℎ|| 𝐿2([0,𝑇])
and
||𝜓′|| 𝑊 ≤ 𝑘2||𝑢̅|| 𝐿∞(0,𝑇;𝐻0
1(Ω))||ℎ|| 𝐿2([0,𝑇])
consequently,
||𝜓|| 𝐶([0,𝑇];𝐻0
1(Ω)) ≤ 𝑘3||ℎ|| 𝐿2([0,𝑇])
we deduce that ℎ ∈ 𝐿2
([0, 𝑇]) → 𝜓 ∈ 𝐶((0, 𝑇); 𝐻0
1
(Ω)) is bounded, see [13]. Put 𝑢ℎ = 𝑢(𝑄∗
+ ℎ)
and 𝜑 = 𝑢ℎ − 𝑢̅, then 𝜑 verify
{
𝜕𝜑(𝑥,𝑡)
𝜕𝑡
= −Δ𝜑(𝑥, 𝑡) + 𝑄∗
(𝑡)𝜑(𝑥, 𝑡) + ℎ(𝑡)𝑢ℎ(𝑥, 𝑡) Π
𝜑(𝑥, 0) = 0 Ω (11)

1961
consequently
||𝜑|| 𝐿∞([0,𝑇];𝐻0
1(Ω)) ≤ 𝑘4||ℎ ∥ 𝐿2([0,𝑇])
where 𝑘𝑖, {𝑖 = 1,2,3,4}, and 𝑘 are positive constants. Let the map 𝜙 = 𝜑 − 𝜓 which is solution of
{
𝜕𝜙(𝑥,𝑡)
𝜕𝑡
= −Δ𝜙(𝑥, 𝑡) + 𝑄∗
(𝑡)𝜙(𝑥, 𝑡) + ℎ(𝑡)𝜑(𝑥, 𝑡) Π
𝜙(𝑥, 0) = 0 Ω (12)
𝜙 ∈ 𝐶([0, 𝑇]; 𝐻0
1
(Ω)), and we have
||𝜙|| 𝐶([0,𝑇];𝐿0
1(Ω)) ≤ 𝑘||ℎ|| 𝐿2([0,𝑇])
2
furthermore
||𝑢(𝑄∗
+ ℎ) − 𝑢(𝑄∗
) − 𝑑(𝑢(𝑄∗
))ℎ|| 𝐶(0,𝑇;𝐻0
1(Ω)) ≤ 𝑘||ℎ|| 𝐿2([0,𝑇])
2
.
Next, we consider the family of optimality systems
{
𝜕𝑝 𝑖(𝑥,𝑡)
𝜕𝑡
= Δ𝑝𝑖(𝑥, 𝑡) − 𝑄𝜀
∗
(𝑡)𝑝𝑖(𝑥, 𝑡) 𝑄
𝑝𝑖(𝑥, 𝑇) = (
𝜕𝑢(𝑇)
𝜕𝑥 𝑖
− 𝜒̃ 𝜔
∗
𝑦𝑖
𝑑
) Ω (13)
where 𝜒̃ 𝜔
∗
is the adjoint of 𝜒̃ 𝜔 defined from 𝐿2
(𝜔) ⟶ 𝐿2
(Ω) by
𝜒̃ 𝜔
∗
𝑢(𝑥) = {
𝑢(𝑥) 𝑥 ∈ 𝜔
0 𝑥 ∈ Ω𝜔
the following lemma gives the differential of 𝐽𝜀(𝑄), respecting to 𝑄.
4.2. Lemma
If Qε ∈ L2
(0, T) the optimal control solution of (4), ψ is the solution of (10) and pi is
the solution of (13), then
lim
β⟶0
Jε(Qε+βh)−Jε(Qε)
β
= ∑ ‍n
i=1 ∫ ‍ω
χ̃ω
∗
χ̃ω [∫ ‍
T
0
∂pi
∂t
∂ψ(x,t)
∂xi
dt + ∫ ‍
T
0
pi
∂
∂xi
(
∂ψ
∂t
)dt] dx
+ ∫ ‍
T
0
2εhQεdt.
(14)
proof. The quadratic cost Jε(Qε) defined by (5), can be write in the following form
Jε(Qε) =
1
2
∑ ‍n
i=1 ∫ ‍ω
(χ̃ ω
∂u
∂xi
− yi
d
)2
dx + ε ∫ ‍
T
0
Qε
2
(t)dt (15)
let uβ = u(Qε + βh) and u = u(Qε), using (15) we have
lim
β⟶0
β
= lim
β⟶0
∑ ‍n
i=1
1
2
∫ ‍ω
(χ̃ω
∂uβ
∂xi
−yi
d
)2−(χ̃ω
∂u
∂xi
−yi
d
)2
β
dx
+ lim
β⟶0
ε ∫ ‍
T
0
(Qε+βh)2−Qε
2
β
(t)dt.
(16)
consequently

 ISSN: 1693-6930
1962
lim
𝛽⟶0
𝐽 𝜀(𝑄 𝜀+𝛽ℎ)−𝐽 𝜀(𝑄 𝜀)
𝛽
= lim
𝛽⟶0
∑ ‍𝑛
𝑖=1
1
2
∫ ‍𝜔
𝜒̃ 𝜔
(
𝜕𝑢 𝛽
𝜕𝑥 𝑖
−
𝜕𝑢
𝜕𝑥 𝑖
)
𝛽
(𝜒̃ 𝜔
𝜕𝑢 𝛽
𝜕𝑥𝑖
+ 𝜒̃ 𝜔
𝜕𝑢
𝜕𝑥𝑖
− 2𝑦𝑖
𝑑
)𝑑𝑥
+ ∫ ‍
𝑇
0
(2𝜀ℎ𝑄 𝜀 + 𝛽𝜀ℎ2
)𝑑𝑡
= ∑ ‍𝑛
𝑖=1 ∫ ‍𝜔
𝜒̃ 𝜔
𝜕𝜓(𝑥,𝑇)
𝜕𝑥𝑖
𝜒̃ 𝜔(
𝜕𝑢(𝑥,𝑇)
𝜕𝑥𝑖
− 𝜒̃ 𝜔
∗
𝑦𝑖
𝑑
)𝑑𝑥 + ∫ ‍
𝑇
0
2𝜀ℎ𝑄 𝜀 𝑑𝑡
= ∑ ‍𝑛
𝑖=1 ∫ ‍𝜔
𝜒̃ 𝜔
𝜕𝜓(𝑥,𝑇)
𝜕𝑥𝑖
𝜒̃ 𝜔 𝑝𝑖(𝑥, 𝑇)𝑑𝑥 + 2𝜀 ∫ ‍
𝑇
0
ℎ𝑄 𝜀 𝑑𝑡
(17)
from (13) and (17), we deduce that
lim
𝛽⟶0
𝐽 𝜀(𝑄 𝜀+𝛽ℎ)−𝐽 𝜀(𝑄 𝜀)
𝛽
= ∑ ‍𝑛
𝑖=1 ∫ ‍𝜔
𝜒̃ 𝜔
∗
𝜒̃ 𝜔 [∫ ‍
𝑇
0
𝜕𝑝 𝑖
𝜕𝑡
𝜕𝜓(𝑥,𝑡)
𝜕𝑥 𝑖
𝑑𝑡 + ∫ ‍
𝑇
0
𝑝𝑖
𝜕
𝜕𝑥 𝑖
(
𝜕𝜓
𝜕𝑡
)𝑑𝑡] 𝑑𝑥
+ ∫ ‍
𝑇
0
2𝜀ℎ𝑄𝜀 𝑑𝑡.
(18)
which finishes the proof of this Lemma. Now, we are ready to characterize the optimal control
solution of (5), using our defined family of optimality systems.
4.3. Theorem
If Qε ∈ L2
(0, T) is an optimal control, and uε = u(Qε) its associate state solution of the
system (1), then
Qε(t) =
−1
2ε
∑ ‍n
i=1 〈χ̃ω
∂u(x,t)
∂xi
; χ̃ωpi(t)〉L2(ω) (19)
is solution of the problem (??), where pi ∈ C([0, T]; H0
1
(Ω)) is the unique solution of the adjoint
system (13). Proof. Let h ∈ L∞
(0, T) such that Qε + βh ∈ L2
(0, T) for β > 0 . The minimum of Jε is
achieved at Qε, then
0 ≤ lim
β⟶0
β
. (20)
using Lemma (4.2) replacing
∂ψ
∂t
in the system (10), we have
(21)
and from the system (13) we have
0 ≤ ∑ ‍n
i=1 ∫ ‍ω
χ̃ω
∗
χ̃ω [∫ ‍
T
0
∂ψ
∂xi
(
∂pi
∂t
+ Δpi + Q(t)pi) + h(t)
∂u
∂xi
pidt] dx + ∫ ‍
T
0
2εhQεdt
= ∫ ‍
T
0
2εhQεdt + ∑ ‍n
i=1 ∫ ‍
T
0
h(t)〈χ̃ω
∂u
∂xi
; χ̃ωpi〉L2(ω)
= ∫ ‍
T
0
h(t) [2εQε(t) + ∑ ‍n
i=1 〈χ̃ω
∂u
∂xi
; χ̃ωpi(t)〉L2(ω)] dt.
(22)
note that h = h(t) is an arbitrary function with Qε + βh ∈ L2
(0, T) for all small β, by a standard
control argument involving the sign of the variation h depending on the size of Qε, we obtain the
desired characterization of Qε, namely,

1963
Qε(t) =
−1
2ε
∑ ‍n
i=1 〈χ̃ω
∂u(x,t)
∂xi
; χ̃ωpi(t)〉L2(ω) (23)
the existence of a solution to the adjoint (13) is similar to existence of solution to the state
equation since
(
∂u(T)
∂xi
− χ̃ω
∗
yi
d
)‍‍‍in‍‍‍C([0, T], H0
1
(Ω))‍‍‍‍‍‍
4.4. Remarks
In the case of Neumann boundary conditions, all contributions can be easily
generalized. The map 𝑢 ⟶ 𝑄(𝑡)𝑢 is not use as a special case. The same results hold with other
types of damping.
5. Simulations
For simulations, we Choose the one dimensional bi-linear equation
{
𝜕𝑢
𝜕𝑡
+ 𝛼
𝜕2 𝑢
𝜕𝑥2 = 𝛽𝑄(𝑡)𝑢 [0,1]
𝑢(𝑥, 0) = 𝑢0(𝑥) = 𝑥2
, [0,1]
𝑢 = 0 ‍‍‍𝑎𝑡‍‍‍𝑥 = 0,1
(24)
the operator −𝛼
𝜕2
𝜕𝑥2 admits a set of eigenfunctions 𝜙 𝑛(. ) associated to the eigenvalues 𝜆 𝑛
given by:
𝜙 𝑛(𝑥) = √2sin(𝑛𝜋𝑥); 𝜆 𝑛 = 𝛼𝑛2
𝜋2
, 𝑛 ≥ 1.
while the operator of the system (24) and the perturbation 𝑄(𝑡)𝑢 commute, the solution of (24)
can be write as
𝑢 𝑚(𝑡) = ∑ ‍𝑛=𝑀
𝑛=1 𝑒 𝛼𝑛2 𝜋2 𝑡
< 𝑒 𝛽 ∫ ‍
𝑡
0 𝑄 𝑚(𝑠)𝑑𝑠
𝑥2
, √2sin(𝑛𝜋𝑥) > √2sin(𝑛𝜋𝑥). (25)
and its gradient
𝜕𝑢 𝑚(𝑡)
𝜕𝑥
= ∑ ‍𝑛=𝑀
𝑛=1 𝑒 𝛼𝑛2 𝜋2 𝑡
< 𝑒 𝛽 ∫ ‍
𝑡
0 𝑄 𝑚(𝑠)𝑑𝑠
𝑥2
, √2sin(𝑛𝜋𝑥) > √2𝑛𝜋cos(𝑛𝜋𝑥). (26)
where the optimal control 𝑄 𝑚 is calculated by choosing 𝜀 =
1
𝑚
and
{
𝑄 𝑚+1(𝑡) =
−𝑚
2
〈𝜒̃ 𝜔
𝜕𝑢 𝑚(𝑥,𝑡)
𝜕𝑥
; 𝜒̃ 𝜔 𝑝 𝑚(𝑡)〉 𝐿2(𝜔)
𝑄0 = 0
(27)
and 𝑝 is the solution of
{
𝜕𝑝 𝑚(𝑥,𝑡)
𝜕𝑡
= 𝛼
𝜕2 𝑝 𝑚(𝑥,𝑡)
𝜕𝑥2 − 𝛽𝑄 𝑚(𝑡)𝑝 𝑚(𝑥, 𝑡) [0,1]
𝑝 𝑚(𝑥, 𝑇) = (
𝜕𝑢 𝑄(𝑇)
𝜕𝑥
− 𝜒̃ 𝜔
∗
𝑔 𝑑
(𝑥)) [0,1]
(28)
the formula (25) is the mild solution of the system (24) calculate using the semi group
associated to the operator −𝛼
𝜕2
𝜕𝑥2 and the formula (26) its derivative, see [21]. The solution of
the (28) with final condition is
(29)

 ISSN: 1693-6930
1964
the formula (27) is the minimizing bounded sequence of optimal control deduced from the
theorem 4.3. It admits a subsequence convergent, which allow us to consider the following
convergent algorithm for numerical implementation of the above results. The Algorithim show
in Table 1.
Table 1. Algorithm
Step 1 Step 2 Step 3
Defined the initial data of the problem Until ∥𝑄(𝑚+1)-𝑄 𝑚∥≤ε repeat 𝑄 𝑚 such that ∥ 𝑄 𝑚+1 − 𝑄 𝑚 ∥≤ 𝜀
The control time 𝑇
The gradient state 𝑔 𝑑
The error 𝜀
The subregion𝜔
Compute
𝜕𝑢 𝑚
𝜕𝑡
(𝑇) by (26)
Compute 𝑝 𝑚(𝑡) by the formula
(28)
Compute 𝑄 𝑚+1 by the formula
(27)
is the solution of (4)
5.1. Remarks
a. Let consider the error 𝐸 =∥
𝜕𝑢 𝑄(𝑇)
𝜕𝑡
− 𝑔 𝑑
∥ 𝐿2(𝜔)
2
. It is depending of the subregion 𝜔 and of the
amplitude of the desired state chosen as shown bellow.
b. The truncation 𝑀 defined in (26) will be such that 𝐸 ≤ 𝜀, simulations in general, show that a
big choice of M is not preferred due to number of iterations and accumulation of errors.
c. All simulations are obtained by using complex numerical program en FORTRAN 95.
d. Optimal control problems using optimization methods as in [25] are still under
considerations.
5.2. Example
We choose Ω =]0,1[, 𝑇 = 2, 𝛼 = 𝛽 = 0.01 and applying the previous algorithm, we
propose two examples of simulations. We choose the desired states is g^d (x)=x(1-x)(x+1)
chosen for numerical considerations and ω=]0.5,0.7[.Figure 1 shows how the reached position
is very close to the desired position on 𝜔, the desired state is obtained with error
𝐸 = 2.01 × 10−4
.
Figure 1. Desired (red line) and final (blue line) gradient position on 𝜔
6. Relations Subregion-Error and Amplitude-Error
Numerically, we show how the error grow with respect to the subregion Table 2 and
with respect to the amplitude of the desired state Table 3.
Table 2. Relation Subregion-Error
Subregion𝜔 Error 𝐸
]0.4, 0.6 [ 2.01 ∗ 10−4
] 0.38, 0.7 [ 3.07 ∗ 10−4
] 0.25, 0.75 [ 1.03 ∗ 10−3
] 0.2, 0.81 [ 2.11 ∗ 10−2
] 0.03, 0.88[ 5.1 ∗ 10−2
Table 3. Relation Amplitude-Error
Amplitude Error 𝐸
0.4 2.01 ∗ 10−4
0.45 3.07 ∗ 10−4
0.6 2.01 ∗ 10−3
0.7 3.4 ∗ 10−2
0.9 1.01 ∗ 10−2

1965
7. Conclusion
This paper considers for the first time the problem of regional gradient optimal control of
infinite dimensional bilinear systems. We have shown the existence of solution of such problem
and we have proposed a characterization of its solution. The results have been tested
successfully through numerical simulations.
Acknowledgements
This project was supported by Jouf University under the research project
number 398/37.
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