369 

© Published by College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad. This 

is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License 

 

 
The Continuous Classical Optimal Control Problems for Quaternary Elliptic 

Partial Differential Equations 

 

Haider H. Diwan1 , Jamil A. Ali Al-Hawasy2* ,  and Waffa F. keidan3    
1,3Department of Mathematics, College of Science, Diyala University, Baquba, Iraq. 

2Department of Mathematics, College of Science, Mustansiriyah University, Baghdad, Iraq. 

*Corresponding Author.  

 

 Received:15 March 2023                        Accepted:22 May 2023                         Published:20 July 2024  

 doi.org/10.30526/37.3.3339 

 
 Abstract 

     In this paper, the Quaternary Continuous Classical Optimal Control Problem (QCCOCP) for 

the Quaternary Linear Elliptic Partial Differential Equations (QLEPDEqs) is studied. The 

mathematical model for the proposed problem is formulated, and it consists of the QLEPDEqs, the 

Objective Function (OF), and the set of state controls. The method of Galerkin (MG) is used to 

prove the existence theorem of a unique state vector solution (QSVS) of the Weak Form (WF) for 

the QLEPDEqs when the Quaternary Classical Continuous Control Vector (QCCCV) is fixed. 

Furthermore, the existence of a Quaternary Classical Continuous Optimal Control Vector 

(QCCOCV) ruled by the QLEPDEqs is stated and proved. The Quaternary Adjoint Equations 

(QAJEqs) associated with the QLEPDEqs are formulated and then studied. The Fréchet Derivative 

(FD) for the OF is derived. Finally, the necessary condition theorem (NCTH) for the optimality of 

the QCCOCP is proved.  

Keywords: Quaternary Continuous Classical Optimal Control Vector, Quaternary Partial 

Differential Equations, Objective Function, and Adjoint Equations.  

 

 1. Introduction 

     Optimal control problems have an essential role in important areas of applied mathematics that 

relate to many important aspects of life. One of the important applications of life is in medicine 

[1,2], aircraft [3,4], economics [5,6], robotics [7,8], weather conditions [9,10], biology [11,12], 

Aerospace [13-14], Electrical Machines [15-16],  and many other important applications[17-21]. 

Because of this importance, many researchers have been interested in studying optimal control 

problems related to Nonlinear Ordinary Differential Equations (NLODEs) [22], or those related to 

different types of NLPDEs like parabolic, hyperbolic, and elliptic [23–25], or those related to 

NLOPDEs of a couple of these three kinds [26]. In the current work, the study of the optimal 

control problem is motivated to deal with the study of the QCCOCP related to the QLEPDEqs. 

The mathematical model for the proposed problem is formulated; the MG is used to study and 

prove the existence theorem for a unique QSVS for the Wf of the QLEPDEqs for fixed CCOCV. 

The existence theorem for a CCOCV associated with the QLEPDEqs is stated and proved. The 

https://creativecommons.org/licenses/by/4.0/
https://orcid.org/0009-0004-8480-1952
mailto:Scimathms2211@uodiyala.edu.iq
http://orcid.org/0000-0002-7225-8030
mailto:Jhawassy17@uomustansiriyah.edu.iq
https://orcid.org/0009-0002-5749-5289
mailto:wafaafa5@yahoo.com


IHJPAS. 2024, 37( 3 ) 

370 
 

QAEqs related to the QLEPDEqs are formulated and then studied. The FD of the OF is derived; 

finally, the NCTH is stated and proved. 

2.  Description of the Problem  

     Let Ω be an abounded and open connected subset in 𝑅2 with a Lipschitz (LIP) boundary 𝜕Ω in 

the QCCOCP, including the QLEPDEqs:  

−∆𝑦1 + 𝑦1 + 𝑦2 + 𝑦3 − 𝑦4 =  𝑏1 (𝑥) + 𝑢1                                                                                    (1)        

−∆𝑦2 − 𝑦1 + 𝑦2 + 𝑦3 − 𝑦4 =  𝑏2 (𝑥) + 𝑢2                                                                                   (2)                                     

−∆𝑦3 − 𝑦1 − 𝑦2 + 𝑦3 − 𝑦4 =  𝑏3 (𝑥) + 𝑢3                                                                                   (3)                                       

−∆𝑦4 + 𝑦1 + 𝑦2 + 𝑦3 + 𝑦4 =  𝑏4 (𝑥) + 𝑢4                                                                                   (4)                                      

With a Drichlet Boundary  Condition(DBC) 

 𝑦𝑖(𝑥) = 0  ,           ∀ 𝑖 = 1,2,3,4              𝑖𝑛  𝜕Ω                                                                               (5)      

 �⃗� =  (𝑦1, 𝑦2, 𝑦3, 𝑦4)   ∈ (𝐻0
2(Ω))

4
 is the  QSVS , �⃗⃗� = (𝑢1, 𝑢2, 𝑢3, 𝑢4)  ∈ (𝐿2(Ω))

4
 is the QCCCV 

and 𝑏𝑖(𝑥) ∈ (𝐿2(Ω))4 ∀ 𝑖 = 1,2,3,4, is give, ∀𝑥 = (𝑥1, 𝑥2) ∈ Ω. 

The set of  ACV   is �⃗⃗⃗� ⊂ (𝐿2(Ω))4, s.t.  

�⃗⃗⃗� = {(�⃗⃗� ∈ (𝐿2(Ω))4 ∣ (𝑢1, 𝑢2, 𝑢3, 𝑢4) ∈ �⃗⃗⃗� ⊂ 𝑅4 𝑎. 𝑒 𝑖𝑛 Ω} . 

Where 𝑈 ⃗⃗⃗⃗ = 𝑈1 × 𝑈2 × 𝑈3 × 𝑈4 is a convex set  

The OF is  

𝐽0(�⃗⃗�) =
1

2
∑ ‖𝑦𝑖 − 𝑦𝑖𝑑‖0

2𝑛
𝑖=1 +

𝛼

2
∑ ‖𝑢𝑖‖0

2 𝑛
𝑖=1    ∀�⃗⃗� ∈ 𝑈.⃗⃗⃗⃗                                                                    (6)             

Where 𝛼 is a positive real number, �⃗� is the QSVS of (1-5) corresponding to the QCCCV �⃗⃗� and 

(𝑦1𝑑 , 𝑦2𝑑 , 𝑦3𝑑 , 𝑦4𝑑) is the desired data. 

The  QCCOCP is:    𝐽0 (�⃗⃗̅�) = 𝑀𝑖𝑛
�⃗⃗⃗�∈�⃗⃗⃗�

  𝐽0 (�⃗⃗�) .  

 

3. The WF of the QLEPDEqs:  

     To obtain the WF of problem (1-5) consider:  

𝑾 = 𝑊1 × 𝑊2 × 𝑊3 × 𝑊4 = 𝐻0
1(Ω) × 𝐻0

1(Ω) × 𝐻0
1(Ω) × 𝐻0

1(Ω ) = (𝐻0
1(Ω))

4
    

     = {�⃗⃗⃗�: �⃗⃗⃗� = (𝑤1, 𝑤2, 𝑤3, 𝑤4) ∈ (𝐻0
1(Ω))

4
 𝑤𝑖𝑡ℎ 𝑤𝑖 = 0  𝑜𝑛 𝜕Ω, ∀𝑖 = 1,2,3,4}. 

MBS of (1-4) by 𝑤𝑖 ∈ 𝑊𝑖    ( 𝑖 = 1,2,3,4) resp., then, integrating w.r.t. 𝑥 . And finally using the 

generalized Greens theorem for the first term in the L.H.S of the four obtained equations, to get: 

(∇𝑦1, ∇𝑤1) + (𝑦1, 𝑤1) + (𝑦2, 𝑤1) + (𝑦3, 𝑤1) − (𝑦4, 𝑤1) = 

(𝑏1, 𝑤1) + (𝑢1, 𝑤1)                                                                                                                         (7) 

(∇𝑦2, ∇𝑤2) − (𝑦1, 𝑤2) + (𝑦2, 𝑤2) + (𝑦3, 𝑤2) − (𝑦4, 𝑤2) = 

 (𝑏2 , 𝑤2) + (𝑢2, 𝑤2)                                                                                                                      (8)   

(∇𝑦3, ∇𝑤3) − (𝑦1, 𝑤3) − (𝑦2, 𝑤3) + (𝑦3, 𝑤3) − (𝑦4, 𝑤3) =  

 (𝑏3, 𝑤3) +   (𝑢3, 𝑤3)                                           (9) 

(∇𝑦4, ∇𝑤4) + (𝑦1, 𝑤4) + (𝑦2, 𝑤4) + (𝑦3, 𝑤4) + (𝑦4, 𝑤4) =                   

   (𝑏4, 𝑤4) + (𝑢4, 𝑤4)                                                                                                                    (10) 

By blending to gather equations (7-10), one gets: 

 𝐵(�⃗�, �⃗⃗⃗�) = Ǎ(�⃗⃗⃗�)                                                                                                                          (11) 

Where 𝐵(. , . ): �⃗⃗⃗⃗� × �⃗⃗⃗⃗� → 𝑅  is a bilinear form and  Ǎ(.):�⃗⃗⃗⃗�→R is a linear from, such that (s.t.) 



IHJPAS. 2024, 37( 3 ) 

371 
 

 𝐵(�⃗�, �⃗⃗⃗�) = (∇𝑦1, ∇𝑤1) + (𝑦1, 𝑤1) + (𝑦2, 𝑤1) + (𝑦3, 𝑤1) − (𝑦4, 𝑤1) + (∇𝑦2, ∇𝑤2) − (𝑦1, 𝑤2) +

(𝑦2, 𝑤2) + (𝑦3, 𝑤2) − (𝑦4, 𝑤2) + (∇𝑦3, ∇𝑤3) − (𝑦1, 𝑤3) − (𝑦2, 𝑤3) + (𝑦3, 𝑤3) − (𝑦4, 𝑤4) +

(∇𝑦4, ∇𝑤4) + (𝑦1, 𝑤4) + (𝑦2, 𝑤4) + (𝑦3, 𝑤4) + (𝑦4, 𝑤4 ) , and  

Ǎ(�⃗⃗⃗�) = (𝑏1 + 𝑢1, 𝑤1) + (𝑏2 + 𝑢2, 𝑤2) + (𝑏3 + 𝑢3, 𝑤3) + (𝑏4 + 𝑢4, 𝑤4). 

The following hypotheses (HYPs) are required in the study of the existence of a unique QSVS of 

the WF (11).  

3.1 HYPs  

1)  𝐵(. , . ) satisfies the following:  

a) 𝐵(�⃗�, �⃗⃗⃗�) is coercive, i.e.  

   
𝐵(�⃗⃗�,�⃗⃗�)

‖�⃗⃗�‖1
= ‖�⃗�‖1  > 0,    �⃗� ∈ �⃗⃗⃗⃗�   ⇒  𝐵(�⃗�, �⃗�) = ‖�⃗�‖1

2 → ∞       𝑎𝑠 ‖�⃗�‖1 → ∞. 

b) 𝐵(�⃗�, �⃗⃗⃗�) is continuous, i.e. ∃ 𝜖1 > 0, s.t. |𝐵(�⃗�, �⃗⃗⃗�)|  ≤ 𝜖1‖�⃗�‖1‖�⃗⃗⃗�‖1, ∀�⃗�, �⃗⃗⃗� ∈ �⃗⃗⃗⃗�. 

2) Ǎ(�⃗⃗⃗�) is a bounded on �⃗⃗⃗⃗� , where  �⃗⃗� ∈ (𝐿2(Ω))4 is bounded, i.e. ∃  𝜖2 > 0  𝑠. 𝑡. 

|Ǎ(�⃗⃗⃗�)|  ≤ 𝜖2‖𝑤‖1, ∀ �⃗⃗⃗� ∈ �⃗⃗⃗⃗�. 

The  MG  is used to find the approximation solution (app. sl.) of the WF (11) which is found 

through choosing a finite subspace �⃗⃗⃗⃗�𝑛 ⊂ �⃗⃗⃗⃗� (where 𝑊𝑛 be the set piece wise affine functions 

(PWAFs)  in Ω), therefore (11), will reduced to the following  app. problem (app. pro.)   

𝐵(�⃗�𝑛 , �⃗⃗⃗�) = Ǎ(�⃗⃗⃗�), ∀ �⃗�𝑛 , �⃗⃗⃗�  ∈ �⃗⃗⃗⃗�𝑛                                                                                  (12) 

Theorem 3.1 [27] For each �⃗⃗⃗� ∈ 𝑊,⃗⃗⃗⃗⃗⃗  there is a sequence { �⃗⃗�𝑛 }, with �⃗⃗�𝑛 ∈ �⃗⃗⃗⃗�𝑛 for each n, s.t  �⃗⃗�𝑛 →

�⃗⃗⃗� strongly (ST) in �⃗⃗⃗⃗� .  

3.2  Existence and Uniqueness Solution of the WF Theorem 3.2  

For every fixed QCCCV �⃗⃗� ∈ (𝐿2(Ω))4, the WF (12) has a unique app. sl. �⃗�𝑛 ∈ �⃗⃗⃗⃗�𝑛.  

Proof: Let {�⃗⃗�1, �⃗⃗�2 … … . . �⃗⃗�𝑛  } be a basis of �⃗⃗⃗⃗�𝑛 for each n, with    𝑛 = 4𝑁 and let  

�⃗�𝑛 = �⃗�𝑛(𝑥1, 𝑥2) = ∑ 𝐶𝑗�⃗⃗�𝑗(𝑥1, 𝑥2)𝑛
𝑗=1   

                           = (∑ 𝐶𝑗 �⃗⃗�1𝑗
𝑛
𝑗=1 , ∑ 𝐶𝑗 �⃗⃗�2𝑗

𝑛
𝑗=1 , ∑ 𝐶𝑗�⃗⃗�3𝑗

𝑛
𝑗=1 , ∑ 𝐶𝑗�⃗⃗�4𝑗

𝑛
𝑗=1 )                                          (13) 

Where    �⃗⃗�𝑗  = (𝑎1𝜓𝑘  , 𝑎2𝜓𝑘  , 𝑎3𝜓𝑘, 𝑎4𝜓𝑘  )  . 

Where 𝑎1 = (1 −
𝐿+(2𝐿𝑚𝑜𝑑3)

3
), 𝑎2 = (

1

2
(𝑃𝑚𝑜𝑑3)(𝐿𝑚𝑜𝑑3)), 𝑎3 = (

1+(𝐿𝑚𝑜𝑑3)−(𝑃𝑚𝑜𝑑3)

3
), 𝑎4 =

(
(2𝑃𝑚𝑜𝑑3)+𝑃

3
−  1), for 𝐿 = 0,1,2,3,  𝑃 = 𝐿 + 1 = 1,2,3,4, 𝐾 = 1,2, … … 𝑁   

Ĵ = 𝐾 + 𝑁[((𝑃 − 1)𝐿)𝑚𝑜𝑑4] + 𝑁[
𝐿(𝐿−1)

2
] , and 𝐶𝑗 is an unknown constant for each 𝑗 = 1,2, … , 𝑛  

By substituting   �⃗�𝑛 from (13), with  �⃗⃗⃗� = �⃗⃗�𝑖 in (11), to get  

𝐵(∑ 𝐶𝑗 �⃗⃗�𝑗
𝑛
𝑗=1 , �⃗⃗�𝑖) = Ǎ(�⃗⃗�𝑖)         ∀ 𝑖 = 1,2, … 𝑛                                                                             (14)   

Equation (14) can be rewritten as the following linear system: 

𝐴𝐶 = 𝑏                                                                                                                                         (15) 

Where 𝐴 = (𝑎𝑖𝑗)𝑛×𝑛, 𝑎𝑖𝑗 = 𝐵(𝜓𝑗 , 𝜓𝑖), ∀𝑖, 𝑗 = 1,2 … , 𝑁,𝑏 = (𝑏1, 𝑏2, … , 𝑏𝑛)𝑇  ,   𝑏𝑖 = Ǎ(�⃗⃗�𝑖) 

, ∀𝑖 = 1,2, … , 𝑁 , and  𝐶 = (𝑐1, 𝑐2, … , 𝑐𝑛)𝑇. 

Now, let  

𝐴𝐶 = 0 = 0 ⇒ ∑ 𝐶𝑗𝑎𝑖𝑗 = 0𝑛
𝑗=1 , ∀𝑖 = 1,2, … 𝑛,             

                     ⇒ 𝐵(∑ 𝐶𝑗 �⃗⃗�𝑗 , �⃗⃗�𝑖) = 0𝑁
𝑗=1  ,   ∀𝑖 = 1,2, … 𝑛                                                                    (16) 



IHJPAS. 2024, 37( 3 ) 

372 
 

From HYP 3.1(1-a), once get that: 

‖∑ 𝐶𝑗 �⃗⃗�𝑗
𝑛
𝑗=1 ‖

1

2
= 𝐵(∑ 𝐶𝑗�⃗⃗�𝑗

𝑛
𝑗=1 , ∑ 𝐶𝑗 �⃗⃗�𝑗

𝑛
𝑗=1 ) = 𝐵(∑ 𝐶𝑗 �⃗⃗�𝑗

𝑛
𝑗=1 , ∑ 𝐶𝑖�⃗⃗�𝑖

𝑛
𝑗=1 )  

                        = ∑ 𝐶𝑖𝐵𝑛
𝑗=1 (∑ 𝐶𝑗�⃗⃗�𝑗 , �⃗⃗�𝑖

𝑛
𝑗=1 ) = 0  , by (16)  

⇒  ∑ 𝐶𝑗 �⃗⃗�𝑗
𝑛
𝑗=1 = 0.  

But { �⃗⃗�1, �⃗⃗�2, … … … �⃗⃗�𝑛 } are linearly independent, thus there exists 𝐶𝑗 = 0,     ∀ 𝑗 = 1,2, … 𝑛, 

which means equation (15) has a unique solution.                                   

Now, from the WF (12) and theorem (3.2), one gets that there exists a sequence of the WF,  

𝐵(�⃗�𝑛 , �⃗⃗�𝑛) = Ǎ(�⃗⃗�𝑛),            ∀ �⃗�𝑛 , �⃗⃗�𝑛 ∈ �⃗⃗⃗⃗�𝑛 , ∀ 𝑛                                                                         (17) 

Which has a sequence of solutions  {�⃗�𝑛}𝑛=1
∞   .                                                                          

Theorem 3.3 :( Existence and Uniqueness Solution of the WF)  

The sequence of solutions {�⃗�𝑛}𝑛=1
∞  (of the sequence of WF (17)) converges strongly to �⃗� (the 

unique solution of (11)).  

Proof: Since�⃗�𝑛 is a solution of (17), then from hypo.3.1 (1-a and 2), one gets:               

‖�⃗�𝑛‖1
2  = 𝐵(�⃗�𝑛 , �⃗�𝑛) ≤  |Ǎ(�⃗�𝑛)|  ≤  𝜖2‖�⃗�𝑛‖1. 

∴ ‖�⃗�𝑛‖1  ≤ 𝜖3 𝑤ℎ𝑒𝑟𝑒 𝜖3 =
𝜖2

𝜖
 > 0    ∀𝑛  

i.e. {�⃗�𝑛} is bounded in �⃗⃗⃗⃗� , ∀ n  then by the Alaglou theorem,  there exists a subsequence of {�⃗�𝑛} 

(for simplicity say again {�⃗�𝑛}), such that   �⃗�𝑛 → �⃗�  weakly (WK)  in �⃗⃗⃗⃗�) 

Now, we have the following two steps: 

First, since   �⃗�𝑛 → �⃗�   WK  in  �⃗⃗⃗⃗� and    �⃗⃗�𝑛 → �⃗⃗⃗�     ST  in �⃗⃗⃗⃗� then  

|𝐵(�⃗�𝑛 , �⃗⃗�𝑛) − 𝐵(�⃗�, �⃗⃗⃗�)| ≤ |𝐵(�⃗�𝑛 , �⃗⃗�𝑛 − �⃗⃗⃗�| + |𝐵(�⃗�𝑛 − �⃗�, �⃗⃗⃗�)|  

≤ 𝜖1‖�⃗�𝑛‖1 ‖�⃗⃗�𝑛 − �⃗⃗⃗�‖
1

+ |𝐵(�⃗�𝑛 − �⃗�, �⃗⃗⃗�)|  → 0  

⇒ 𝐵(�⃗�𝑛 , �⃗⃗�𝑛) → 𝐵(�⃗�, �⃗⃗⃗�). 

Second, since   �⃗⃗�𝑛 → �⃗⃗⃗�   WK  in �⃗⃗⃗⃗� , then   Ǎ(�⃗⃗�𝑛) → Ǎ(�⃗⃗⃗�).  

From the above two steps, we conclude that    𝐵(�⃗�, �⃗⃗⃗�) = Ǎ(�⃗⃗⃗�)   ∀�⃗⃗⃗� ∈ �⃗⃗⃗⃗�. 

Thus 𝑦⃗⃗⃗ ⃗ is solution of (11) . 

To prove   �⃗�𝑛 → �⃗�  ST  in �⃗⃗⃗⃗� , from HYP 3.1 (1-a), one has  

‖�⃗� − �⃗�𝑛‖1
2 = 𝐵(�⃗� − �⃗�𝑛 , �⃗� − �⃗�𝑛) = 𝐵(�⃗� − �⃗�𝑛 , �⃗�) − 𝐵(�⃗�, �⃗�𝑛) + 𝐵(�⃗�𝑛 , �⃗�𝑛)  

                  = 𝐵(�⃗� − �⃗�𝑛 , �⃗�) + Ǎ(�⃗�) − Ǎ(�⃗�𝑛)  → 0 

i.e. ‖�⃗� − �⃗�𝑛‖ = 0. Thus  �⃗�𝑛 → �⃗�.  

The Uniqueness of the Solution:  

Let �⃗� , �⃗�𝑛 be two solutions of (11), i.e.  

𝐵(�⃗�1, �⃗⃗⃗�) = Ǎ(�⃗⃗⃗�) ,          ∀�⃗⃗⃗� ∈ �⃗⃗⃗⃗�  

𝐵(�⃗�2, �⃗⃗⃗�) = Ǎ(�⃗⃗⃗�),            ∀�⃗⃗⃗� ∈ �⃗⃗⃗⃗�  

Subtract the second above equation from the first one, and then setting �⃗⃗⃗� = �⃗�1 − �⃗�2 , one gets  

𝐵(�⃗�1 − �⃗�2, �⃗�1 − �⃗�2) = 0 ,     ∀ �⃗⃗⃗� ∈ �⃗⃗⃗⃗� , i.e. 

From HYP 3.1 (1-a), one obtains: �⃗�1 = �⃗�2 .                                                                       

 

4. Existence of a QCCOCV   

Lemma 4.1: The operator  �⃗⃗� → �⃗��⃗⃗⃗�  from  (𝐿2(Ω))4 is   LIP continuous, i.e.  

‖𝛿𝑦⃗⃗⃗⃗⃗‖
0

≤ ĉ ‖𝛿𝑢⃗⃗⃗⃗⃗‖
0
 , for ĉ > 0. 



IHJPAS. 2024, 37( 3 ) 

373 
 

Proof: Let ú⃗⃗ = (ú1, ú2, ú3, ú4) be a given QCCCV of the WF (7-10) and  

ý⃗⃗ = (ý1, ý2, ý3, ý4) its corresponding QSVS, i.e. 

(∇ý1, ∇𝑤1) + (ý1, 𝑤1) + (ý2, 𝑤1) + (ý3, 𝑤1) − (ý4, 𝑤1) = 

(𝑏1, 𝑤1) + (ú1, 𝑤1)                                                                                                                          (18) 

(∇ý2, ∇𝑤2) − (ý1, 𝑤2) + (ý2, 𝑤2) + (ý3, 𝑤2) − (ý4, 𝑤2) = 

(𝑏2, 𝑤2) + (ú2, 𝑤2)                                                                                                                       (19) 

(∇ý3, ∇𝑤3) − (ý1, 𝑤3) − (ý2, 𝑤3) + (ý3, 𝑤3) − (ý4, 𝑤3) = 

(𝑏3, 𝑤3) + (ú3, 𝑤3)                                                                                                                     (20) 

(∇ý4, ∇𝑤4) + (ý1, 𝑤4) + (ý2, 𝑤4) + (ý3, 𝑤4) + (ý4, 𝑤4) = 

(𝑏4, 𝑤4) + (ú4, 𝑤4)                                                                                                                     (21) 

By subtracting equations (7 -10) from (18-21)  resp. then substituting  

𝛿𝑦𝑖 = ý𝑖 − 𝑦𝑖  , 𝛿𝑢𝑖 = ú𝑖 − 𝑢𝑖 , ∀𝑖 = 1,2,3,4   in the obtained equations, we get: 

 (∇𝛿𝑦1, ∇𝑤1) + (𝛿𝑦1, 𝑤1) + (𝛿𝑦2, 𝑤1) + (𝛿𝑦3, 𝑤1) − (𝛿𝑦4, 𝑤1) = (𝛿𝑢1, 𝑤1)                          (22) 

(∇𝛿𝑦2, ∇𝑤2) − (𝛿𝑦1, 𝑤2) + (𝛿𝑦2, 𝑤2) + (𝛿𝑦3, 𝑤2) − (𝛿𝑦4, 𝑤2) = (𝛿𝑢2, 𝑤2)                          (23) 

(∇𝛿𝑦3, ∇𝑤3) − (𝛿𝑦1, 𝑤3) − (𝛿𝑦2, 𝑤3) + (𝛿𝑦3, 𝑤3) − (𝛿𝑦4, 𝑤3) = (𝛿𝑢3, 𝑤3)                          (24) 

(∇𝛿𝑦4, ∇𝑤4) + (𝛿𝑦1, 𝑤4) + (𝛿𝑦2, 𝑤4) + (𝛿𝑦3, 𝑤4) + (𝛿𝑦4, 𝑤4) = (𝛿𝑢4, 𝑤4)                           (25) 

Blending together these equalities, setting 𝑤𝑖 = 𝛿𝑦𝑖 , ∀𝑖 = 1,2,3,4 in (22-25) resp., applying 

HYP3.1(1-a), then using the Cauchy Schwarz inequality (C-S-I) to the R.H.S. to obtain: 

‖𝛿𝑦⃗⃗⃗⃗⃗‖
1

2
≤ ‖𝛿𝑢1‖0‖𝛿𝑦1‖0 + ‖𝛿𝑢2‖0‖𝛿𝑦2‖0 + ‖𝛿𝑢3‖0‖𝛿𝑦3‖0 + ‖𝛿𝑢4‖0‖𝛿𝑦4‖0                     (26)  

Since ‖𝛿𝑦𝑖‖0  ≤  ‖𝛿𝑦⃗⃗⃗⃗⃗‖
0

 ≤ 𝑐‖𝛿𝑦⃗⃗⃗⃗⃗‖
1
 𝑎𝑛𝑑 ‖𝛿𝑢𝑖‖0  ≤ ‖𝛿𝑢⃗⃗⃗⃗⃗‖

0
  ∀𝑖 = 1,2,3,4 , then (26) gives, 

‖𝛿𝑦⃗⃗⃗⃗⃗‖
1

 ≤  č‖𝛿𝑢⃗⃗⃗⃗⃗‖
0
, with  ĉ =

3𝑐

𝜖
. 

So the operator �⃗⃗� → �⃗��⃗⃗⃗�  is (LICS) on (𝐿2(Ω))
4
 . 

Lemma 4.2[28]: The norm ‖. ‖0 is W L Sc.  

Lemma 4.3: The OF in (6) is W L Sc.  

Proof: since   �⃗⃗�𝑛 → �⃗⃗� WK in (𝐿2(Ω)) then (by lemma 4.1),   �⃗�𝑛 → �⃗�  WK   in   (𝐿2(Ω))   which 

gives by lemma 4.2,   ‖�⃗� − �⃗�𝑛‖0
2 is W L Sc.      

i .e  𝐽0(�⃗⃗�)   is   W L Sc.                                                                                                           

Lemma 4.4[11]: The norm ‖. ‖0
2   is strictly convex.  

Remark 4.1: From Lemma 4.4, one can conclude that 𝐽0(�⃗⃗�)  is strictly convex.   

Theorem 4.1: If 𝐽0(�⃗⃗�) is coercive, then there exists a QCCOCV for the CCOCVP.   

Proof: From the convexity of  �⃗⃗⃗� , and the coercivity of  𝐽0(�⃗⃗�), with 𝐽0(�⃗⃗�) ≥ 0       there exist a 

minimizing sequence {�⃗⃗�𝑛}  ∈ �⃗⃗⃗� ∀𝑛 s.t.:  lim
𝑛→∞

𝐽0(�⃗⃗�𝑛) = 𝑖𝑛𝑓
�⃗⃗⃗�∈�⃗⃗⃗�

  𝐽0(�⃗⃗�) .                                                                            

Hence, there exists a constant 𝜖 > 0, s.t. 

 ‖�⃗⃗�𝑛‖ ≤ 𝜖   ∀𝑛   .                                                                                                                           (27) 

Then by ALTH, there exists a subsequence of {�⃗⃗�𝑛} s.t  �⃗⃗�𝑛 → �⃗⃗� WK in  (𝐿2(Ω))
4
.  

But for each QCCCV �⃗⃗�𝑛 (∀𝑛) the SVEs has a unique QSVS �⃗�𝑛.  

Now, using (27), HYPs 3. (1-a and 2) and the C-S-I, it yields: 

‖�⃗�𝑛‖1
2 = 𝐵(�⃗�𝑛 , �⃗�𝑛) = Ǎ(�⃗�𝑛)  

             ≤ ‖𝑏1‖0‖𝑦1𝑛‖0 + ‖𝑢1𝑛‖0‖𝑦1𝑛‖0 + ‖𝑏2‖0‖𝑦2𝑛‖0 +  ‖𝑢2𝑛‖0‖𝑦2𝑛‖0 + 

                    ‖𝑏3‖0‖𝑦3𝑛‖0 + ‖𝑢3𝑛‖0‖𝑦3𝑛‖0 + ‖𝑏4‖0‖𝑦4𝑛‖0 + ‖𝑢4𝑛‖0‖𝑦4𝑛‖0 

≤ ℎ1‖𝑦1𝑛‖0 + 𝜖1‖𝑦1𝑛‖0 + ℎ2‖𝑦2𝑛‖0 + 𝜖2‖𝑦2𝑛‖0 + ℎ3‖𝑦3𝑛‖0 + 



IHJPAS. 2024, 37( 3 ) 

374 
 

                   𝜖3‖𝑦3𝑛‖0 + ℎ4‖𝑦4𝑛‖0 + 𝜖4‖𝑦4𝑛‖0  

              ≤ 2(𝛾1 + 𝛾2 + 𝛾3 + 𝛾4)‖�⃗�𝑛‖1 = 𝜛‖�⃗�𝑛‖1. 

Where  𝛾1 = 𝑚𝑎𝑥(ℎ1, 𝜖1) ,   𝛾2 = 𝑚𝑎𝑥(ℎ2, 𝜖2) ,   𝛾3 = 𝑚𝑎𝑥(ℎ3, 𝜖3) ,   𝛾4 = 𝑚𝑎𝑥(ℎ4, 𝜖4) and     

𝜛 = 𝑚𝑎𝑥(𝛾1 + 𝛾2+ 𝛾3 + 𝛾4). 

Then ‖�⃗�𝑛‖1 ≤ 𝛾, for each n, with 𝛾 =
𝜛

𝜖
> 0 . 

By ALTH there exists a subsequence of {�⃗�𝑛}  s.t    �⃗�𝑛 → �⃗�   WK   in �⃗⃗⃗⃗�  

Since for each n,   �⃗�𝑛    satisfies the WF (11), then ∀�⃗⃗⃗� = (𝑤1, 𝑤2, 𝑤3, 𝑤4) ∈ �⃗⃗⃗⃗�,∀𝑛 

(∇𝑦1𝑛 , ∇𝑤1) + (𝑦1𝑛 , 𝑤1) + (𝑦2𝑛 , 𝑤1) + (𝑦3𝑛 , 𝑤1) − (𝑦4𝑛 , 𝑤1) + (∇𝑦2𝑛 , ∇𝑤2) −  

(𝑦1𝑛 , 𝑤2) + (𝑦2𝑛 , 𝑤2) + (𝑦3𝑛 , 𝑤2) − (𝑦4𝑛 , 𝑤2) + (∇y3𝑛 , ∇𝑤3) − (𝑦1𝑛 , 𝑤3) −              

(𝑦2𝑛 , 𝑤3) + (𝑦3𝑛 , 𝑤3) − (𝑦4𝑛 , 𝑤3) + (∇𝑦4𝑛 , ∇𝑤4) + (𝑦1𝑛 , 𝑤4) + (𝑦2𝑛 , 𝑤4) +       

(𝑦3𝑛 , 𝑤4) + (𝑦4𝑛 , 𝑤4) = (𝑏1𝑛 , 𝑤1) + (𝑢1𝑛 , 𝑤1) + (𝑏2𝑛 , 𝑤2) + (𝑢2𝑛 , 𝑤2)  

                                         +(𝑏3𝑛 , 𝑤3) + (𝑢3𝑛 , 𝑤3) + (𝑏4𝑛 , 𝑤4) + (𝑢4𝑛 , 𝑤4)                                 (28)  

To show (28) converges to the following equations: 

(∇𝑦1, ∇𝑤1) + (𝑦1, 𝑤1) + (𝑦2, 𝑤1) + (𝑦3, 𝑤1) − (𝑦4, 𝑤1) + (∇𝑦2, ∇𝑤2) − (𝑦1, 𝑤2)  

+(𝑦2, 𝑤2) + (𝑦3, 𝑤2) − (𝑦4, 𝑤2) + (∇y3, ∇𝑤3) − (𝑦1, 𝑤3) − (𝑦2, 𝑤3) + (𝑦3, 𝑤3)  

−(𝑦4, 𝑤3) + (∇𝑦4, ∇𝑤4) + (𝑦1, 𝑤4) + (𝑦2, 𝑤4) + (𝑦3, 𝑤4) + (𝑦4, 𝑤4)  

= (𝑏1, 𝑤1) + (𝑢1, 𝑤1) + (𝑏2, 𝑤2) + (𝑢2, 𝑤2) + (𝑏3, 𝑤3) + (𝑢3, 𝑤3) + (𝑏4, 𝑤4)  

          +(𝑢4, 𝑤4),  �⃗⃗⃗� ∈ �⃗⃗⃗⃗�,∀𝑛                                                                                                         (29) 

First  

Since 𝑦𝑖𝑛 → 𝑦𝑖 WK  in �⃗⃗⃗⃗� then from theorem 3.2, 𝑦𝑖𝑛 → 𝑦𝑖 ST  in �⃗⃗⃗⃗�, which gives  

𝑦𝑖𝑛  → 𝑦𝑖 , and 
𝜕𝑦𝑖𝑛

𝜕𝑥𝑖
→

𝜕𝑦𝑖

𝜕𝑥𝑖
  ST in 𝐿2(Ω), and by using the C-S-I and HYP  3.1 (1-b)  

|(∇𝑦1𝑛 , ∇𝑤1) + (𝑦1𝑛 , 𝑤1) + (𝑦2𝑛 , 𝑤1) + (𝑦3𝑛 , 𝑤1) − (𝑦4𝑛 , 𝑤1) + (∇𝑦2𝑛 , ∇𝑤2) − 

(𝑦1𝑛 , 𝑤2) + (𝑦2𝑛 , 𝑤2) + (𝑦3𝑛 , 𝑤2) − (𝑦4𝑛 , 𝑤2) + (∇y3𝑛 , ∇𝑤3) − (𝑦1𝑛 , 𝑤3) −  

(𝑦2𝑛 , 𝑤3) + (𝑦3𝑛 , 𝑤3) − (𝑦4𝑛 , 𝑤3) + (∇𝑦4𝑛 , ∇𝑤4) + (𝑦1𝑛 , 𝑤4) + (𝑦2𝑛 , 𝑤4) +  

(𝑦3𝑛 , 𝑤4) + (𝑦4𝑛 , 𝑤4) − (∇𝑦1, ∇𝑤1) − (𝑦1, 𝑤1) − (𝑦2, 𝑤1) − (𝑦3, 𝑤1) + (𝑦4, 𝑤1)  

−(∇𝑦2, ∇𝑤2) + (𝑦1, 𝑤2) − (𝑦2, 𝑤2) − (𝑦3, 𝑤2) + (𝑦4, 𝑤2) − (∇y3, ∇𝑤3) +   

(𝑦1, 𝑤3) + (𝑦2, 𝑤3) − (𝑦3, 𝑤3) + (𝑦4, 𝑤3) − (∇𝑦4, ∇𝑤4) − (𝑦1, 𝑤4) −  

(𝑦2, 𝑤4) − (𝑦3, 𝑤4) − (𝑦4, 𝑤4)|  

≤ ‖∇𝑦1𝑛 − ∇𝑦1‖0‖𝑤1‖0 + ‖𝑦1𝑛 − 𝑦1‖0‖𝑤1‖0 + ‖𝑦2𝑛 − 𝑦2‖0‖𝑤1‖0         

    +‖𝑦3𝑛 − 𝑦3‖0‖𝑤1‖0 + ‖𝑦4𝑛 − 𝑦4‖0‖𝑤1‖0 + ‖∇𝑦2𝑛 − ∇𝑦2‖0‖𝑤2‖0  

    +‖𝑦1𝑛 − 𝑦1‖0‖𝑤2‖0 + ‖𝑦2𝑛 − 𝑦2‖0‖𝑤2‖0 + ‖𝑦3𝑛 − 𝑦3‖0‖𝑤2‖0  

    +‖𝑦4𝑛 − 𝑦4‖0‖𝑤2‖0 + ‖∇𝑦3𝑛 − ∇𝑦3‖0‖𝑤3‖0 + ‖𝑦1𝑛 − 𝑦1‖0‖𝑤3‖0  

    +‖𝑦2𝑛 − 𝑦2‖0‖𝑤3‖0 + ‖𝑦3𝑛 − 𝑦3‖0‖𝑤3‖0 + ‖𝑦4𝑛 − 𝑦4‖0‖𝑤3‖0  

    +‖∇𝑦4𝑛 − ∇𝑦4‖0‖𝑤4‖0 + ‖𝑦1𝑛 − 𝑦1‖0‖𝑤4‖0 + ‖𝑦2𝑛 − 𝑦2‖0‖𝑤4‖0  

     +‖𝑦3𝑛 − 𝑦3‖0‖𝑤4‖0 + ‖𝑦4𝑛 − 𝑦4‖0‖𝑤4‖0   ⟶ 0 .        

Second, the convergence for the R.H.S of (28) to the L.H.S of (29) is obtained through 𝑢𝑖𝑛 →

𝑢𝑖       ∀𝑖 = 1,2,3,4    WK   in 𝐿2(Ω).  

Then from these two steps of convergences, (28) converges to (29). 

Since 𝐽0(�⃗⃗�) is WLSc (by Lemma 4.3) and    �⃗⃗�𝑛 → �⃗⃗�  WK   in (𝐿2(Ω))
4
, then  

𝐽0(�⃗⃗�) ≤  lim
𝑛→∞

𝑖𝑛𝑓
�⃗⃗⃗�𝑛∈�⃗⃗⃗�

𝐽0(�⃗⃗�𝑛) = lim
𝑛→∞

𝐽0 (�⃗⃗�𝑛) = 𝑖𝑛𝑓
�⃗⃗⃗�∈�⃗⃗⃗�

𝐽0(�⃗⃗�) ⇒ 𝐽0(�⃗⃗�) = 𝑚𝑖𝑛
�⃗⃗⃗�∈�⃗⃗⃗�

𝐽0(�⃗⃗�)  

⇒ 𝑢 ⃗⃗⃗⃗   is QCCOCV.  

 



IHJPAS. 2024, 37( 3 ) 

375 
 

To prove the uniqueness: 

Let  �⃗⃗�1, �⃗⃗�2  ∈ 𝑈 ⃗⃗ ⃗⃗  be two  QCCOCV  of  𝐽0(�⃗⃗�), then  
�⃗⃗⃗�1

2
+

�⃗⃗⃗�2

2
∈ �⃗⃗⃗� , and  

   𝐽0 (
�⃗⃗⃗�1

2
+

�⃗⃗⃗�2

2
) ≤

1

2
𝐽0(�⃗⃗�1) +

1

2
𝐽0(�⃗⃗�2) =  𝐽0(�⃗⃗�),  𝙲ǃ 

Then the uniqueness is obtained from lemma 4.4.                                                                  

 

5. The NCTh for Optimality  

Theorem 5.1: Consider the OF (2.6) and the QAEqs  (𝑧1, 𝑧2, 𝑧3, 𝑧4) of the QLEPDEq (2.1-2.5) 

are given by:  

 −∆𝑧1 + 𝑧1 − 𝑧2 − 𝑧3 + 𝑧4 = (𝑦1 − 𝑦1𝑑)                                                              (30)  

 −∆𝑧2 + 𝑧1 + 𝑧2 − 𝑧3 + 𝑧4 =  (𝑦2 − 𝑦2𝑑)                                                                                   (31) 

 −∆𝑧3 + 𝑧1 + 𝑧2 + 𝑧3 + 𝑧4 = (𝑦3 − 𝑦3𝑑)                                                                                   (32) 

 −∆𝑧1 − 𝑧1 − 𝑧2 − 𝑧3 + 𝑧4 = (𝑦4 − 𝑦4𝑑)                                                                                   (33)  

𝑧𝑖 = 0          ∀𝑖 = 1,2,3,4       𝑜𝑛 𝜕Ω                                                                                             (34)  

Then the FD of 𝐽0 is given by (𝐽0(�⃗⃗�), 𝛿𝑢⃗⃗⃗⃗⃗) = (𝑧 + 𝛼𝑢⃗⃗⃗⃗⃗⃗ , 𝛿𝑢⃗⃗⃗⃗⃗ ). 

Proof: Rewriting the QAEqs ((30) -(34)) by their following WF  

(∇𝑧1, ∇𝑤1) + (𝑧1, 𝑤1) − (𝑧2, 𝑤1) − (𝑧3, 𝑤1) + (𝑧4, 𝑤1) = (𝑦1 − 𝑦1𝑑 , 𝑤1)                                  (35) 

(∇𝑧2, ∇𝑤2) + (𝑧1, 𝑤2) + (𝑧2, 𝑤2) − (𝑧3, 𝑤2) + (𝑧4, 𝑤2) = (𝑦2 − 𝑦2𝑑 , 𝑤2)                                (36) 

(∇𝑧3, ∇𝑤3) + (𝑧1, 𝑤3) + (𝑧2, 𝑤3) + (𝑧3, 𝑤3) + (𝑧4, 𝑤3) = (𝑦3 − 𝑦3𝑑 , 𝑤3)                                (37) 

(∇𝑧4, ∇𝑤4) − (𝑧1, 𝑤4) − (𝑧2, 𝑤4) − (𝑧3, 𝑤4) + (𝑧4, 𝑤4) = (𝑦4 − 𝑦4𝑑 , 𝑤4)                                 (38) 

By blending (35-38) together, we get  

(∇𝑧1, ∇𝑤1) + (𝑧1, 𝑤1) − (𝑧2, 𝑤1) − (𝑧3, 𝑤1) + (𝑧4, 𝑤1) + (∇𝑧2, ∇𝑤2) + (𝑧1, 𝑤2) + (𝑧2, 𝑤2) −

(𝑧3, 𝑤2) + (𝑧4, 𝑤2) + (∇𝑧3, ∇𝑤3) + (𝑧1, 𝑤3) + (𝑧2, 𝑤3) + (𝑧3, 𝑤3) + (𝑧4, 𝑤3) +  (∇𝑧4, ∇𝑤4) −

(𝑧1, 𝑤4) − (𝑧2, 𝑤4) − (𝑧3, 𝑤4) + ( 𝑧4, 𝑤4)  

= (𝑦1 − 𝑦1𝑑 , 𝑤1) + (𝑦2 − 𝑦2𝑑 , 𝑤2) + (𝑦3 − 𝑦3𝑑 , 𝑤3) + (𝑦4 − 𝑦4𝑑 , 𝑤4)                                   (39) 

The WF (39) has a unique solution (𝑧1, 𝑧2, 𝑧3, 𝑧4) = (𝑧1𝑢1, 𝑧2𝑢2, 𝑧3𝑢3, 𝑧4𝑢4)  ∈ �⃗⃗⃗⃗�  (this can be 

proved by the same way used in the proof of theorem 3.2).  

Now, substituting  𝑤𝑖 = 𝛿𝑧𝑖 in  ((35) – (38)) ∀𝑖 = 1,2,3,4  , then subtracting each obtained 

equations from those each obtained from substituting 𝑤𝑖 = 𝑧𝑖 in ((22) – (25)), we get : 

(𝑧2, 𝛿𝑦1) + (𝑧3, 𝛿𝑦1) + (𝛿𝑦2, 𝑧1) + (𝛿𝑦3, 𝑧1) − (𝛿𝑦4, 𝑧1) − (𝑧4, 𝛿𝑦1) =       

(𝛿𝑢1, 𝑧1) − (𝑦1 − 𝑦1𝑑  
, 𝛿𝑦1)                                                                                                         (40)  

−(𝛿𝑦1, 𝑧2) − (𝑧1, 𝛿𝑦2) + (𝛿𝑦3, 𝑧2) + (𝑧3, 𝛿𝑦2) − (𝛿𝑦4, 𝑧2) − (𝑧4, 𝛿𝑦2) =                           

(𝛿𝑢2, 𝑧2) − (𝑦2 − 𝑦2𝑑 , 𝛿𝑦2)                                                                                                          (41)  

−(𝛿𝑦1, 𝑧3) − (𝛿𝑦2, 𝑧3) − (𝛿𝑦4, 𝑧3) − (𝑧1, 𝛿𝑦3) − (𝑧2, 𝛿𝑦3) − (𝑧4, 𝛿𝑦3) =                           

(𝛿𝑢3, 𝑧3) − (𝑦3 − 𝑦3𝑑 , 𝛿𝑦3)                                                                                                          (42) 

(𝛿𝑦1, 𝑧4) + (𝛿𝑦2, 𝑧4) + (𝛿𝑦3, 𝑧4) + (𝑧1, 𝛿𝑦4) + (𝑧2, 𝛿𝑦4) + (𝑧3, 𝛿𝑦4) =                                      

(𝛿𝑢4, 𝑧4) − (𝑦4 − 𝑦4𝑑 , 𝛿𝑦4)                                                                                                          (43) 

Blending together the above quaternary equations, we get: 

(𝛿𝑢1, 𝑧1) + (𝛿𝑢2, 𝑧2) + (𝛿𝑢3, 𝑧3) + (𝛿𝑢4, 𝑧4) =  

(𝑦1 − 𝑦1𝑑 , 𝛿𝑦1) + (𝑦2 − 𝑦2𝑑 , 𝛿𝑦2) + (𝑦3 − 𝑦3𝑑 , 𝛿𝑦3) + (𝑦4 − 𝑦4𝑑 , 𝛿𝑦4)                                  (44) 

On the other hand, the OB becomes: 

𝐽0(�⃗⃗� + 𝛿𝑢⃗⃗⃗⃗⃗) =
1

2
∬  ∑ (𝑦𝑖 + 𝛿𝑦𝑖 − 𝑦𝑖𝑑)4

𝑖=1
2 

Ω
𝑑𝑥 +

𝛼

2
∬ ∑ (𝑢𝑖 − 𝑢𝑖𝑑)24

𝑖=1
 

Ω
𝑑𝑥  

But by using  (44), we have: 



IHJPAS. 2024, 37( 3 ) 

376 
 

𝐽0(�⃗⃗� + 𝛿𝑢⃗⃗⃗⃗⃗) − 𝐽0(�⃗⃗�) = (𝑧1 + 𝛼𝑢1, 𝛿𝑢1) + (𝑧2 + 𝛼𝑢2, 𝛿𝑢2) + (𝑧3 + 𝛼𝑢3, 𝛿𝑢3)  

                                      +(𝑧4 + 𝛼𝑢4, 𝛿𝑢4) +
1

2
‖𝛿𝑦⃗⃗⃗⃗⃗‖

0

2
+

𝛼

2
‖𝛿𝑢⃗⃗⃗⃗⃗‖

0

2
                                                (45) 

Using lemma 4.1, we get:  

1

2
‖𝛿𝑦⃗⃗⃗⃗⃗‖

0

2
+  

𝛼

2
‖𝛿𝑢⃗⃗⃗⃗⃗‖

0

2
=  𝜖(𝛿𝑢⃗⃗⃗⃗⃗)‖𝛿𝑢⃗⃗⃗⃗⃗‖

0
 ,   where 𝜖(𝛿𝑢⃗⃗⃗⃗⃗)   → 0     𝑎𝑠  ‖𝛿𝑢⃗⃗⃗⃗⃗‖

0
 → 0 . 

Hence (45), becomes: 

𝐽0(�⃗⃗� + 𝛿𝑢⃗⃗⃗⃗⃗) − 𝐽0(�⃗⃗�) = (𝑧 + 𝛼�⃗⃗�, 𝛿𝑢⃗⃗⃗⃗⃗) + 𝜖(𝛿𝑢⃗⃗⃗⃗⃗)‖𝛿𝑢⃗⃗⃗⃗⃗‖
0
                                                        

where 𝜖(𝛿𝑢⃗⃗⃗⃗⃗)   → 0     𝑎𝑠  ‖𝛿𝑢⃗⃗⃗⃗⃗‖
0

 → 0 . 

From the FD for  𝐽0 , one concludes that  

(𝐽0(�⃗⃗�), 𝛿𝑢⃗⃗⃗⃗⃗) =  (𝑧 + 𝑏𝑢⃗⃗⃗⃗⃗, 𝛿𝑢⃗⃗⃗⃗⃗) .                  

Theorem5.2: If the QCCCV of (1-5)  is optimal, the   𝐽0(�⃗⃗�) = 𝑧 + 𝛼�⃗⃗� = 0  with �⃗� = �⃗�𝑢 ⃗⃗⃗⃗   and 𝑧 =

𝑧𝑢 ⃗⃗⃗⃗ . 

Proof: If  �⃗⃗� is an QCCOCV of the problem, then  

𝐽0(�⃗⃗�) =  𝑚𝑖𝑛
�⃗⃗�∈�⃗⃗⃗�

 𝐽0(𝑣) , ∀𝑣 ∈ (𝐿2(Ω))
4
,     i .e  𝐽0(�⃗⃗�) = 0  

 ⇒    𝑧 + 𝛼 �⃗⃗� = 0        ⇒     �⃗⃗� =  −
𝑧

𝑏(𝑥)
   , with 𝛿𝑢⃗⃗⃗⃗⃗ =  𝑣 −  �⃗⃗�  

Then the NCO is  (𝐽0 (�⃗⃗�) , 𝛿𝑢⃗⃗⃗⃗⃗ )  ≥ 0  , ⇒    (𝑧 + 𝛼 �⃗⃗� , 𝛿𝑢⃗⃗⃗⃗⃗)  ≥ 0 

⇒(𝑧 + 𝛼 �⃗⃗�, �⃗⃗�)  ≤  (𝑧 + 𝛼�⃗⃗�, 𝑣)    ∀𝑣  ∈ (𝐿2(Ω))
4
  .                                                      

 

6. Conclusion 

     The mathematical model for the “new” proposed problem is formulated. The existence and 

uniqueness theorem for a QSVS of the WF from the QLEPDEqs is stated and proved successfully 

by using the GM when the QCCCV is given. Furthermore, the existence of a QCCOCV ruled by 

the QLEPDEqs is stated and proven. The mathematical formulations for the QAEqs, which are 

related to the QLEPDEqs, are formulated and then studied. The FD for the OF is derived. Finally, 

the NCTH “for optimality” is proved for this problem. 

 

Acknowledgement 

I extend my thanks to reviewers and the editor who contributed to presenting and improving 

this paper. 

 

Conflict of Interest  

    The authors declare that they have no conflicts of interest. 

Funding  

     There is no funding for the article. 

 

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