IHJPAS. 36(2)2023 

331 
This work is licensed under a Creative Commons Attribution 4.0 International License 

 

   

 

 

 

 

 

 

 

 
 

 

 

Abstract  

This paper is concerned with the quaternary nonlinear hyperbolic boundary value problem 

(QNLHBVP) studding constraints quaternary optimal classical continuous control vector 

(CQOCCCV), the cost function (CF), and the equality and inequality quaternary state and control 

constraints vector (EIQSCCV). The existence of a CQOCCCV dominating by the QNLHBVP is 

stated and demonstrated using the Aubin compactness theorem (ACTH) under appropriate 

hypotheses (HYPs). Furthermore, mathematical formulation of the quaternary adjoint equations 

(QAEs) related to the quaternary state equations (QSE) are discovere  so as its weak form (WF) . 

The directional derivative (DD) of the Hamiltonian (Ham) is calculated. The necessary and 

sufficient conditions for optimality (NCSO) theorems for the proposed problem are stated and 

proved. 

 

Keywords: Necessary and Sufficient Conditions fro optimality, Nonlinear Hyperbolic System, 

Quaternary Optimal Classical Continuous Control vector. 

 

1. Introduction 

Optimal control problems (OCPs) are important in a wide range of practical applications, 

including robotics robotics[1], economics[2], weather conditions[3], community health[4], and a 

variety of other scientific fields. Nonlinear ODEs [5]or nonlinear PDEs (NLPDEs) [6] usually 

dominate OCPs. This significance pushed many researchers to be concerned about OCPs in 

general and optimal classical continuous control problems (OCCCPs) in particular. During the last 

decade much emphasis has been place on studying the OCPs for system dominating by nonlinear 

PDEs (NLPDEs) of the three types in general; hyperbolic, elliptic and parabolic [7-9]. Later the 

study of this subject, in particular  for hyperbolic type  of PDEs was generalized to deal with 

CCOCPs dominated by coupled NLPDEs of it [10], and then to CCOCPs dominating by triple 

NLPDEs of it [11]. the problem in each type of these OCCCPs was typically comprised of an 

initial and boundary value problem, the CF and the constraints on the state and the control vectors 

(CSCV). The study of each one of these problems had been included of; the existence theorem of 

constraints OCCC vector satisfying the SCCV had been stated and demonstrated under appropriate 

HYPs, the mathematical formulation for the QAEs related to the given QSEs had been obtained, 

doi.org/10.30526/36.2.2992 

Article history: Received 5 September 2022, Accepted 9 October 2022, Published in April 2023. 

 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

Constraints Optimal Classical Continuous Control Vector Problem for 

Quaternary Nonlinear Hyperbolic System  

 

 
Mayeada Abd Alsatar Hassan 

Department of Mathematics, College of 

Science, Mustansiriyah University, 

Baghdad, Iraq. 

mayeadabd1989@uomustanriyah.edu.iq 

 

 

 

Jamil A. Ali Al-Hawasy 

Department of Mathematics, College of 

Science, Mustansiriyah University, 

Baghdad, Iraq. 

jhawassy17@uomustansiriyah.edu.iq 

 

 

 
 

https://creativecommons.org/licenses/by/4.0/
mailto:mayeadabd1989@uomustanriyah.edu.iq
mailto:jhawassy17@uomustansiriyah.edu.iq
mailto:jhawassy17@uomustansiriyah.edu.iq


IHJPAS. 36(2)2023 

332 
 

and the DD for the Ham had been derived. The theorems of necessity and sufficient conditions for 

optimality had been stated and demonstrated.   

  All of these concerns motivated us to consider extending the study of the CCOCP 

dominating by triple NLPDEs of hyperbolic type to a CCOCP dominating by QNLHBVP. As a 

result of this expansion, there was a need to generalize the mathematical model and then to 

generalize all the proofs related to this generalization, and accordingly. The authors created new 

Theorems, Lemma and then proved them in this paper. The existence theorem (ETH) of a 

CQOCCCV dominating by the QNLHBVPs with EIQSCCV was stated and demonstrated in this 

work using the ACTH under appropriate HYPs. Moreover mathematical formulation of the QAEs 

related to QSEs was discovered as was the WF of the QAEs. The derivative of DD was obtained. 

Lastly, both the theorems for the NCSO of the proposed problems were stated and demonstrated. 

2.  Problem Description: 

Let  𝐼 = [0, 𝑇], T < ∞, Ω ⊂ ℝ2, be an open bounded regular region with boundary Γ = 𝜕Ω, 𝑄 =
Ω × 𝐼, Σ = Γ × 𝐼. The CQOCCCV including of the QSEs are given by the following QNLHBVP: 

𝑦1𝑡𝑡 − ∆𝑦1 + 𝑦1 − 𝑦2 + 𝑦3 + 𝑦4 = 𝑓1(𝑥, 𝑡, 𝑦1, 𝑢1), in 𝑄,                                                          (1)    

𝑦2𝑡𝑡 − ∆𝑦2 + 𝑦1 + 𝑦2 − 𝑦3 − 𝑦4 = 𝑓2(𝑥, 𝑡, 𝑦2, 𝑢2), in 𝑄,                                                          (2) 

𝑦3𝑡𝑡 − ∆𝑦3 − 𝑦1 + 𝑦2 + 𝑦3 + 𝑦4 = 𝑓3(𝑥, 𝑡, 𝑦3, 𝑢3), in 𝑄,                                                          (3) 

𝑦4𝑡𝑡 − ∆𝑦4 − 𝑦1 + 𝑦2 − 𝑦3 + 𝑦4 = 𝑓4(𝑥, 𝑡, 𝑦4, 𝑢4), in 𝑄,                                                           (4) 
with the following boundary conditions (BCs) and the initial conditions (ICs) 

𝑦𝑖 (𝑥, 𝑡) = 0, on Σ, for 𝑖 = 1,2,3,4 .                                                                                             (5) 

𝑦1(𝑥, 0) = 𝑦𝑖
0(𝑥),and 𝑦𝑖𝑡 (𝑥, 0) = 𝑦𝑖

1(𝑥), in Ω for 𝑖 = 1,2,3,4.                                                  (6) 

where �⃗� = (𝑦1, 𝑦2, 𝑦3, 𝑦4) ∈ 𝑯
𝟏(𝛀) = (𝐻1(Ω))4is the quaternary solution vectors (QSVs), 

corresponding to the quaternary classical continuous control vector (QCCCV) �⃗⃗� =
(𝑢1, 𝑢2, 𝑢3, 𝑢4) ∈ 𝑳

𝟐(𝐐) = (𝐿2(𝑄))4 and (𝑓1, 𝑓2, 𝑓3, 𝑓4) ∈ 𝑳
𝟐(𝐐) is a vector of a given function on 

(𝑄 × ℝ × 𝑈1) × (𝑄 × ℝ × 𝑈2) × (𝑄 × ℝ × 𝑈3) × (𝑄 × ℝ × 𝑈4), with 𝑈𝑖 ⊂ ℝ, ∀𝑖 = 1,2,3,4 .   

The QSCCs are �⃗⃗� ∈ 𝑊 ⃗⃗⃗⃗⃗, 𝑊 ⃗⃗⃗⃗⃗ ⊂ 𝑳
𝟐(𝐐) where �⃗⃗⃗⃗� = {�⃗⃗⃗� ∈ 𝑈 ⃗⃗⃗⃗ ⊂ ℝ

4, 𝑎. 𝑒 𝑖𝑛 𝑄}, with is a convex 
(CO). 

The CF is given and The EINEQSCC on the QSCCs are resp. 

 𝐺0(�⃗⃗�) = Σ
𝑖=1

4

∫
𝑄

𝑔0𝑖  (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 )𝑑𝑥𝑑𝑡 ,                                                                                         (7) 

𝐺1(�⃗⃗�) = Σ
𝑖=1

4

∫
𝑄

𝑔1𝑖  (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 )𝑑𝑥𝑑𝑡 = 0,                                                                                    (8) 

𝐺2(�⃗⃗�) = Σ
𝑖=1

4

∫
𝑄

𝑔2𝑖  (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 )𝑑𝑥𝑑𝑡 ≤ 0,                                                                                     (9) 

The set of admissible quaternary control (AQC) is: 

  �⃗⃗⃗⃗�𝐴 = {�⃗⃗� ∈ 𝑊 ⃗⃗⃗⃗⃗ ∣ 𝐺1(�⃗⃗�) = 0, 𝐺2(�⃗⃗�) ≤ 0}. 

The CQOCCCV is to find �⃗⃗� ∈ �⃗⃗⃗⃗�𝐴, s.t.  𝐺0(�⃗⃗�) = 𝑚𝑖𝑛
�⃗⃗⃗�∈�⃗⃗⃗⃗�𝐴 

𝐺0 (�⃗⃗⃗�) .       

Let�⃗⃗� = {�⃗� = (𝑣1, 𝑣2, 𝑣3, 𝑣4) ∈ 𝑯
𝟏(𝛀), 𝑣1 = 𝑣2 = 𝑣3 = 𝑣4 = 0 𝑜𝑛 𝜕Ω},V⃗⃗⃗ = 𝑯𝟎

𝟏(𝛀) = (𝐻0
1(Ω))4             

, 𝑳𝟐(𝑰, 𝑽) = (𝐿2(𝐼, 𝑉))4 and 𝑉 = 𝐻0
1(Ω), the inner product (IP) and the norm(Nr) in 𝑳𝟐(𝐐) are 

denoted by (�⃗�, �⃗�) and ∥ �⃗� ∥𝑳𝟐(𝐐)= Σ
𝑖=1

4

∥ 𝑣1 ∥𝐿2(Q)
2

 resp.,  the Nr in 𝑳𝟐(𝑰, 𝑽) by ∥ �⃗� ∥𝑳𝟐(𝑰,𝑽)=

Σ
𝑖=1

4

∥ 𝑣1 ∥𝐿2(𝐼,𝑉)
2

  , and 𝑳𝟐(𝑰, 𝑽∗) is the dual of 𝑳𝟐(𝑰, 𝑽).   

The WF of ((1)-(6)) with �⃗� ∈ 𝑯𝟎
𝟏(𝛀) is given (a.e. on I and ∀𝑣𝑖 , 𝑦𝑖 (0, 𝑡) ∈ 𝑉,∀𝑖 = 1,2,3,4 ) by : 

(𝑦1𝑡𝑡 , 𝑣1) + (∇𝑦1, ∇𝑣1) + (𝑦1, 𝑣1) − (𝑦2, 𝑣1) + (𝑦3, 𝑣1) + (𝑦4, 𝑣1) = (𝑓1, 𝑣1),                   (10) 



IHJPAS. 36(2)2023 

333 
 

(𝑦1
0, 𝑣1) = (𝑦1(0), 𝑣1), and (𝑦1𝑡

1 , 𝑣1) = (𝑦1𝑡 (0), 𝑣1),                                                           (11) 

(𝑦2𝑡𝑡 , 𝑣2) + (∆𝑦2, ∇𝑣2) + (𝑦1, 𝑣2) + (𝑦2, 𝑣2) − (𝑦3, 𝑣2) − (𝑦4, 𝑣2) = (𝑓2, 𝑣2),                  (12) 

(𝑦2
0, 𝑣2) = (𝑦2(0), 𝑣2), and (𝑦2𝑡

1 , 𝑣2) = (𝑦2𝑡 (0), 𝑣2) ,                                                          (13) 

(𝑦3𝑡𝑡 , 𝑣3) + (∇𝑦3, ∇𝑣3) − (𝑦1, 𝑣3) + (𝑦2, 𝑣3) + (𝑦3, 𝑣3) + (𝑦4, 𝑣3) = (𝑓3, 𝑣3),                   (14) 

(𝑦3
0, 𝑣3) = (𝑦3(0), 𝑣3), and (𝑦3𝑡

1 , 𝑣3) = (𝑦3𝑡 (0), 𝑣3),                                                           (15) 

(𝑦4𝑡𝑡 , 𝑣4) + (∇𝑦4, ∇𝑣4) − (𝑦1, 𝑣4) + (𝑦2, 𝑣4) − (𝑦3, 𝑣4) + (𝑦4, 𝑣4) = (𝑓4, 𝑣4),                    (16) 

(𝑦4
0, 𝑣4) = (𝑦4(0), 𝑣4), and (𝑦4𝑡

1 , 𝑣4) = (𝑦4𝑡 (0), 𝑣4),                                                            (17) 

Assums (A): Suppose that  𝑓𝑖  is of Carathéodory type (CaraT) on  𝑄 × (ℝ × 𝑈𝑖 ) satisfies 

(w.r.t. 𝑦𝑖 &𝑢𝑖  ) the following 

 (i)|𝑓𝑖 (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 )| ≤ 𝐹𝑖 (𝑥, 𝑡)+∣ 𝑢𝑖 | + 𝛽𝑖 |𝑦𝑖 |, where 𝑦𝑖 , 𝑢𝑖 ∈ ℝ, 𝛽𝑖 > 0 and 𝐹𝑖 ∈ 𝐿
2(Q). 

 (ii) 𝑓𝑖  is satisfied Lipschitz condition (LPC) w.r.t. 𝑦𝑖, i.e.  

       |𝑓𝑖 (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 ) − 𝑓𝑖 (𝑥, 𝑡, �̅�𝑖 , 𝑢𝑖 )| ≤ 𝐿𝑖 |𝑦𝑖 − �̅�𝑖 | , 𝑦𝑖 , �̅�𝑖 , 𝑢𝑖 ∈ ℝ, 𝐿𝑖 > 0, for (𝑥, 𝑡) ∈ 𝑄.  

Proposition 2.1[12]: Let 𝐷 ⊂ ℝ2 be measurable, 𝑓: 𝐷 × ℝ𝑛 ⟶ ℝ𝑚 is of CaraT satisfies: 

‖𝑓(𝑣, 𝑥)‖ ≤ (𝑣) + (𝑣)‖𝑥‖𝛼, 

where 𝑥 ∈ 𝐿𝑝(𝐷 × ℝ𝑛 ), ∈ 𝐿1(𝐷 × ℝ ), ∈ 𝐿
𝑝

𝑝−𝛼(𝐷 × ℝ ), 𝛼 ∈ [0, ∞).  

Then the functional (funl) 𝐹(𝑥) = ∫  
 

𝐷
𝑓(𝑣, 𝑥(𝑣))𝑑𝑣 is continuous (cont.).  

 

Theorem2.1 (ETH of a Unique QSVs)[13]: If Assums (A) hold, then for each given �⃗⃗� ∈ 𝑳𝟐(𝐐), 

the WF (10- 17) has a unique QSVs �⃗� = (𝑦1, 𝑦2, 𝑦3, 𝑦4) ∈ 𝑳
𝟐(𝑰, 𝑽) with �⃗�𝑡 = (𝑦1𝑡 , 𝑦2𝑡 , 𝑦3𝑡 , 𝑦4𝑡 ) ∈

𝑳𝟐(𝐐), �⃗�𝑡𝑡 = (𝑦1𝑡𝑡 , 𝑦2𝑡𝑡 , 𝑦3𝑡𝑡 , 𝑦4𝑡𝑡 ) ∈ 𝑳
𝟐(𝑰, 𝑽∗). 

Assums (B): Consider 𝑔𝑙𝑖 (for 𝑖 = 1,2,3,4 & 𝑙 = 0,1,2) is of the CaraT on  𝑄 × (ℝ × 𝑈𝑖 ) and 

satisfies:|𝑔𝑙𝑖 (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 )| ≤ 𝐺𝑙𝑖 (𝑥, 𝑡) + 𝐶𝑙𝑖 (𝑦𝑖 )
2 + 𝐶𝑙𝑖 (𝑢𝑖 )

2, where 𝐺𝑙𝑖 ∈ 𝐿
1(𝑄), 𝑦𝑖 ∈ ℝ &𝑢𝑖 ∈ 𝑈𝑖. 

Lemma 2.1: With Assums (B), the funl �⃗⃗� → 𝐺𝑙 (�⃗⃗�), ∀𝑙 = 0,1,2 is cont. on  𝑳
𝟐(𝐐). 

Proof: The proof is obtained from the Assums (B) and Proposition 1.  

Lemma 2.2[12]: Let 𝑔: 𝑄 × ℝ → ℝ is of CaraT on 𝑄 × (ℝ × ℝ) and satisfies 

|𝑔 (𝑥, 𝑡, 𝑦, 𝑢)| ≤ 𝐺 (𝑥, 𝑡) + 𝑐 𝑦 
2 + �́�𝑢 

2, where  𝐺(𝑥, 𝑡) ∈ 𝐿1(Q),𝑢 ∈ 𝑈, 𝑐, 𝑐́ ≥ 0, 𝑈 ⊂ ℝ, is 

compact(COM). Then ∫
𝑄

𝑔 (𝑥, 𝑦, 𝑢 )𝑑𝑥 is cont. on 𝐿
2(𝑄) w.r.t. 𝑦. 

Theorem 2.2 (LP Cont. Theorem)[13]: In addition to Assums (A), if �⃗� and �⃗� + 𝛿�⃗� are the QSVs 

corresponding to the bounded QCCCVs �⃗⃗� and �⃗⃗� + 𝛿�⃗⃗� resp. in 𝐿2(𝑄), then for 𝛿 ∈ ℝ+ 

∥ 𝛿�⃗� ∥𝐿∞(𝐼,𝑳𝟐(𝛀))≤ 𝛿 ∥ 𝛿�⃗⃗� ∥𝑳𝟐(𝑸),∥ 𝛿�⃗�∈ ∥𝑳𝟐(𝑰,𝑽)≤ 𝛿 ∥ 𝛿�⃗⃗� ∥𝑳𝟐(𝑸)and∥ 𝛿�⃗� ∥𝐿2𝑄)≤ 𝛿 ∥ 𝛿�⃗⃗� ∥𝑳𝟐(𝑸). 

Assums (C): Assume that for each (𝑙 = 0,1,2 &𝑖 = 1,2,3,4), the functions 𝑓𝑖 , 𝑓𝑖𝑦𝑖  , 𝑓𝑖𝑢𝑖  , 𝑔𝑙𝑖𝑦𝑖 , 𝑔𝑙𝑖𝑢𝑖  

are of CaraT on 𝑄 × (ℝ × 𝑈′), where (𝑈′ is an open set containing 𝑈), s.t.( for(𝑥, 𝑡) ∈ 𝑄) : 

|𝑓𝑖𝑦𝑖 (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 )| ≤ 𝐿𝑖 , |𝑓𝑖𝑦𝑖 (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 )| ≤ 𝐿′𝑖, |𝑔𝑖𝑦𝑖 (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 )| ≤ 𝐺𝑙𝑖5(𝑥, 𝑡) + 𝐺𝑙𝑖5(𝑥, 𝑡) ∣ 𝑦𝑖 ∣, 

|𝑔𝑖𝑢𝑖 (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 )| ≤ 𝐺𝑙𝑖6(𝑥, 𝑡) + 𝐺𝑙𝑖6(𝑥, 𝑡) ∣ 𝑦𝑖 ∣,where𝑦𝑖 , 𝑢𝑖 ∈ ℝ, 𝐺𝑙𝑖5, 𝐺𝑙𝑖6 ∈ 𝐿
2(𝑄),𝐺𝑙𝑖5, 𝐺𝑙𝑖6 ≥ 0. 

 

Main Results  

3.Existence of the CQOCCCV 

Theorem 3.1: In addition to Assums ((A) & (B)), if the set �⃗⃗⃗� is CO and com., �⃗⃗⃗⃗�𝐴 ≠ 𝜙, the function 

𝑓𝑖  (∀𝑖 = 1,2,3,4) has the form: 𝑓𝑖 (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 ) = 𝑓𝑖1(𝑥, 𝑡, 𝑦𝑖 ) + 𝑓𝑖2(𝑥, 𝑡)𝑢𝑖 ,  

with  |𝑓𝑖1(𝑥, 𝑡, 𝑦𝑖 )| ≤ 𝑖 (𝑥, 𝑡) + 𝑐𝑖 ∣ 𝑦𝑖 ∣,|𝑓𝑖2(𝑥, 𝑡)| ≤ 𝐾𝑖, 𝑖 ∈ 𝐿
2(Q), 𝑐𝑖 ≥ 0.  



IHJPAS. 36(2)2023 

334 
 

𝑔1𝑖 is independent of 𝑢𝑖 , 𝑔0𝑖 and 𝑔2𝑖are CO w.r.t. 𝑢𝑖  for fixed (𝑥, 𝑡, 𝑦𝑖 ), ∀ 𝑖 = 1,2,3,4. Then there 

is a CQOCCCV. 

Proof: From the Assum on �⃗⃗⃗� ⊂ ℝ , �⃗⃗⃗⃗� is weakly compact (WCOM), since �⃗⃗⃗⃗�𝐴 ≠ 𝜙, then there is 

a minimum sequence(Seq.) {�⃗⃗�𝑘 } = {(𝑢1𝑘 , 𝑢2𝑘 , 𝑢3𝑘 , 𝑢4𝑘 )} ∈ �⃗⃗⃗⃗�𝐴, ∀𝑘 s.t.  

𝑙𝑖𝑚
𝑘→∞

𝐺0(�⃗⃗�𝑘 ) = 𝑖𝑛𝑓
�⃗⃗⃗�𝑘∈�⃗⃗⃗⃗�𝐴

𝐺0(�⃗⃗̅�).    

Since �⃗⃗�𝑘 ∈ �⃗⃗⃗⃗�𝐴, ∀𝑘 and �⃗⃗⃗⃗� is WCOM, there exists a subsequence of {�⃗⃗�𝑘 } say again {�⃗⃗�𝑘 } s.t. . �⃗⃗�𝑘 →

�⃗⃗�  weakly (WK) in 𝑳𝟐(𝐐) and ∥ �⃗⃗�𝑘 ∥𝑳𝟐(𝐐)≤ 𝑑,∀𝑘. From Theorem 1, corresponding to the Seq. 

QCV {�⃗⃗�𝑘 } the WF of the QSEs has “a unique” solution {�⃗�𝑘 = �⃗�𝑢𝑘 } and ∥ �⃗�𝑘 ∥𝑳𝟐(𝑰,𝑽), ∥ �⃗�𝑘𝑡 ∥𝑳𝟐(𝑸) 

are bounded, then by Alaoglu’s theorem (ATH), there exists a Subsequence of {�⃗�𝑘 } and {�⃗�𝑘𝑡 }, say 

again {�⃗�𝑘 } and {�⃗�𝑘𝑡 }, s.t. �⃗�𝑘 → �⃗�  WK in 𝑳
𝟐(𝑰, 𝑽), �⃗�𝑘𝑡 → �⃗�𝑡   WK in (𝐿

2(𝑄))4. Now for each 𝑘. 

and by applying the ACTH[14] , there is a Subsequence of{�⃗�𝑘 } say a gain {�⃗�𝑘 } s.t. �⃗�𝑘 → �⃗� strongly 

(ST) in 𝑳𝟐(𝐐).  

Now, for each 𝑘, substituting the QSVs �⃗�𝑘 in the WF ((10), (12), (14), (16)), multiplying both 

sides (MBSs) of each one by 𝜙𝑖 (𝑡), ∀𝑖 = 1,2,3,4 (with 𝜙𝑖 ∈ 𝐶
2[0, 𝑇], s.t. 𝜙𝑖 (𝑇) = 𝜙𝑖

′(𝑇) =

0, 𝜙𝑖 (0) ≠ 0, 𝜙𝑖
′(0) ≠ 0), rewriting the 1st terms in the LHS of each one, then integrating both 

sides (IBS) on [0, 𝑇], and then integrating by parts (IBPs) for the 1st terms, yield to 

∫  
0

𝑇
𝑑

𝑑𝑡
(𝑦1𝑘𝑡 , 𝑣1)𝜙1

 𝑑𝑡 + ∫  
0

𝑇

[(∇𝑦1𝑘 , ∇𝑣1) + (𝑦1𝑘 , 𝑣1) − (𝑦2𝑘 , 𝑣1) + (𝑦3𝑘 , 𝑣1) + (𝑦4𝑘 , 𝑣1)]𝜙1
 𝑑𝑡 

=∫  
0

𝑇

(𝑓11(𝑥, 𝑡, 𝑦1𝑘 ), 𝑣1)𝜙1
 (𝑡)𝑑𝑡 + ∫  

0

𝑇

(𝑓12(𝑥, 𝑡)𝑢1𝑘 , 𝑣1)𝜙1
 (𝑡)𝑑𝑡 ,                                       (18) 

∫  
0

𝑇
𝑑

𝑑𝑡
(𝑦2𝑘𝑡 , 𝑣2)𝜙2

 𝑑𝑡 + ∫  
0

𝑇

[(∇𝑦2𝑘 , ∇𝑣2) + (𝑦1𝑘 , 𝑣2) + (𝑦2𝑘 , 𝑣2) − (𝑦3𝑘 , 𝑣2) − (𝑦4𝑘 , 𝑣2)]𝜙2
 𝑑𝑡  

= ∫  
0

𝑇

(𝑓21(𝑥, 𝑡, 𝑦2𝑘 ), 𝑣2)𝜙2
 (𝑡)𝑑𝑡 + ∫  

0

𝑇

(𝑓22(𝑥, 𝑡)𝑢2𝑘 , 𝑣2)𝜙2
 (𝑡)𝑑𝑡,                                         (19) 

∫  
0

𝑇
𝑑

𝑑𝑡
(𝑦3𝑘 , 𝑣3)𝜙3

 𝑑𝑡 + ∫  
0

𝑇

[(∇𝑦3𝑘 , ∇𝑣3) − (𝑦1𝑘 , 𝑣3) + (𝑦2𝑘 , 𝑣3) + (𝑦3𝑘 , 𝑣3) + (𝑦4𝑘 , 𝑣3)]𝜙3
 𝑑𝑡  

= ∫  
0

𝑇

(𝑓31(𝑥, 𝑡, 𝑦3𝑘 ), 𝑣3)𝜙3
 (𝑡)𝑑𝑡 + ∫  

0

𝑇

(𝑓32(𝑥, 𝑡)𝑢3𝑘 , 𝑣3)𝜙3
 (𝑡)𝑑𝑡,                                          (20) 

∫  
0

𝑇
𝑑

𝑑𝑡
(𝑦4𝑘 , 𝑣4)𝜙4

 𝑑𝑡 + ∫  
0

𝑇

[(∇𝑦4𝑘 , ∇𝑣4) − (𝑦1𝑘 , 𝑣4) + (𝑦2𝑘 , 𝑣4) − (𝑦3𝑘 , 𝑣4) + (𝑦4𝑘 , 𝑣4)]𝜙4
 𝑑𝑡   

= ∫  
0

𝑇

(𝑓41(𝑥, 𝑡, 𝑦4𝑘 ), 𝑣4)𝜙4
 (𝑡)𝑑𝑡 + ∫  

0

𝑇

(𝑓42(𝑥, 𝑡)𝑢4𝑘 , 𝑣4)𝜙4
 (𝑡)𝑑𝑡,                                  (21) 

At this point, the same steps which were utilized in the proof of Theorem 2.1, can be utilized here 

to passage the limit in the WF of ((18) – (21)), to acquire  

(𝑦1𝑡 , 𝑣1) + (∇𝑦1, ∇𝑣1) + (𝑦1, 𝑣1) − (𝑦2, 𝑣1) + (𝑦3, 𝑣1) + (𝑦4, 𝑣1) 

= (𝑓11(𝑥, 𝑡, 𝑦1) + 𝑓12(𝑥, 𝑡)𝑢1, 𝑣1), ∀𝑣1 ∈ 𝑉   a.e. on I,                                                         (22) 

(𝑦2𝑡 , 𝑣2) + (∆𝑦2, ∇𝑣2) + (𝑦1, 𝑣2) + (𝑦2, 𝑣2) − (𝑦3, 𝑣2) − (𝑦4, 𝑣2) 

= (𝑓21(𝑥, 𝑡, 𝑦2) + 𝑓22(𝑥, 𝑡)𝑢2, 𝑣2), ∀𝑣2 ∈ 𝑉  a.e. on I,                                                      (23) 

(𝑦3𝑡 , 𝑣3) + (∇𝑦3, ∇𝑣3) − (𝑦1, 𝑣3) + (𝑦2, 𝑣3) + (𝑦3, 𝑣3) + (𝑦4, 𝑣3) 

= (𝑓31(𝑥, 𝑡, 𝑦3) + 𝑓32(𝑥, 𝑡)𝑢3, 𝑣3), ∀𝑣3 ∈ 𝑉     a.e. on I,                                              (24) 

(𝑦4𝑡 , 𝑣4) + (∇𝑦4, ∇𝑣4) − (𝑦1, 𝑣4) + (𝑦2, 𝑣4) − (𝑦3, 𝑣4) + (𝑦4, 𝑣4) 

= (𝑓41(𝑥, 𝑡, 𝑦4) + 𝑓42(𝑥, 𝑡)𝑢4, 𝑣4), ∀𝑣4 ∈ 𝑉    a.e. on I,                                                         (25) 

Same manner also can be utilized to that the ICs are held. Thus �⃗� is QSVs 



IHJPAS. 36(2)2023 

335 
 

From the other side, since 

 𝐺1(�⃗⃗�) = Σ
𝑖=1

4

∫
𝑄

𝑔1𝑖  (𝑥, 𝑡, 𝑦𝑖𝑘 )𝑑𝑥𝑑𝑡, with 𝑔1𝑖 (∀𝑖 = 1,2,3,4) is cont. w.r.t. 𝑦𝑖, then by Lemma 2.1, 

∫
𝑄

𝑔1𝑖  (𝑥, 𝑡, 𝑦𝑖𝑘 )𝑑𝑥𝑑𝑡 is cont. w.r.t. 𝑦𝑖 but �⃗�𝑘 → �⃗�  ST in 𝑳
𝟐(𝑸)  , therefore 

∫  
𝑄

 

𝑔1𝑖 (𝑥, 𝑡, 𝑦𝑖𝑘 ) 𝑑𝑥𝑑𝑡 → ∫  
𝑄

 

𝑔1𝑖  (𝑥, 𝑡, 𝑦𝑖 )𝑑𝑥𝑑𝑡. Thus 𝐺1(�⃗⃗�) = 𝑙𝑖𝑚
𝑘→∞

𝐺1(�⃗⃗�𝑘 ) = 0 . 

As well, since for 𝑙 = 0,2 &𝑖 = 1,2,3,4,  𝑔𝑙𝑖 (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 ) is cont. w.r.t. (𝑦𝑖 , 𝑢𝑖 ) and 𝑈𝑖 is COM with 

𝑢𝑖 ∈ 𝑈𝑖 a.e. in 𝑄, then using Lemma 2.2 to get  

∫  
𝑄

 

𝑔𝑙𝑖 (𝑥, 𝑡, 𝑦𝑖𝑘 , 𝑢𝑖𝑘 ) 𝑑𝑥𝑑𝑡 → ∫  
𝑄

 

𝑔𝑙𝑖  (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖𝑘 )𝑑𝑥𝑑𝑡 ,                                            (26)  

But 𝑔𝑙𝑖 (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 ) is CO and cont. w.r.t. 𝑢𝑖  , then   

∫  
𝑄

 

𝑔𝑙𝑖  (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 )𝑑𝑥𝑑𝑡 is weakly lowe semi cont. (WLSC) w.r.t. 𝑢𝑖 , ∀ 𝑙 = 0,2 &𝑖 = 1,2,3,4, i.e. 

∫  
𝑄

 

𝑔𝑙𝑖  (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 )𝑑𝑥𝑑𝑡 ≤ 𝑙𝑖𝑚
𝑘→∞

𝑖𝑛𝑓 ∫  
𝑄

 

[𝑔𝑙𝑖  (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖𝑘 ) − 𝑔𝑙𝑖  (𝑥, 𝑡, 𝑦𝑖𝑘 , 𝑢𝑖𝑘 )]𝑑𝑥𝑑𝑡  

                                       + 𝑙𝑖𝑚
𝑘→∞

𝑖𝑛𝑓 ∫  
𝑄

 

𝑔𝑙𝑖  (𝑥, 𝑡, 𝑦𝑖𝑘 , 𝑢𝑖𝑘 )𝑑𝑥𝑑𝑡  ≤ 𝑙𝑖𝑚
𝑘→∞

𝑖𝑛𝑓 ∫  
𝑄

 

𝑔𝑙𝑖  (𝑥, 𝑡, 𝑦𝑖𝑘 , 𝑢𝑖𝑘 )𝑑𝑥𝑑  

⟹ Σ
𝑖=1

4

∫  
𝑄

 

𝑔𝑙𝑖  (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 )𝑑𝑥𝑑𝑡 ≤ Σ
𝑖=1

4

∫  
𝑄

 

𝑔𝑙𝑖  (𝑥, 𝑡, 𝑦𝑖𝑘 , 𝑢𝑖𝑘 )𝑑𝑥𝑑𝑡.  

Thus 𝐺𝑙 (�⃗⃗�) ≤ 𝑙𝑖𝑚
𝑘→∞

𝑖𝑛𝑓
�⃗⃗⃗�𝑘∈�⃗⃗⃗⃗�𝐴

𝐺𝑙 (�⃗⃗�𝑘 ), then 𝐺2(�⃗⃗�) ≤ 0 , since �⃗⃗�𝑘 ∈ �⃗⃗⃗⃗�𝐴, ∀𝑘, and  

𝐺0(�⃗⃗�) ≤ 𝑙𝑖𝑚
𝑘→∞

𝑖𝑛𝑓
�⃗⃗⃗�𝑘∈�⃗⃗⃗⃗�𝐴

𝐺0(�⃗⃗�𝑘 ) = 𝑙𝑖𝑚
𝑘→∞

𝐺0(�⃗⃗�𝑘 ) = 𝑖𝑛𝑓
�⃗⃗⃗�𝑘∈�⃗⃗⃗⃗�𝐴

𝐺0(�⃗⃗̅�) ⟹ 𝐺0(�⃗⃗�) = 𝑚𝑖𝑛
�⃗⃗⃗�𝑘∈�⃗⃗⃗⃗�𝐴

𝐺0(�⃗⃗̅�), then �⃗⃗� is a 

QOCCCV. 

Theorem 3.2: Neglecting the index 𝑙 from 𝐺𝑙  and 𝑔𝑙𝑖. The QAEs �⃗� = (𝑍1, 𝑍2, 𝑍3, 𝑍4) of the QSEs 

in ((1)-(6)) can be formulated as  

𝑍1𝑡𝑡 − ∆𝑍1 + 𝑍1 + 𝑍2 − 𝑍3 − 𝑍4 = 𝑍1𝑓1𝑦1 (𝑥, 𝑡, 𝑦1, 𝑢1) + 𝑔1𝑦1 (𝑥, 𝑡, 𝑦1, 𝑢1), in 𝑄,                (27)         

𝑍1 = 0  on Σ,   𝑍1(𝑥, 𝑇) =  𝑍1𝑡 (𝑥, 𝑇) = 0 on Ω ,                                                      (28) 
𝑍2𝑡𝑡 − ∆𝑍2 − 𝑍1 + 𝑍2 + 𝑍3 + 𝑍4 = 𝑍2𝑓2𝑦2 (𝑥, 𝑡, 𝑦2, 𝑢2) + 𝑔2𝑦2 (𝑥, 𝑡, 𝑦2, 𝑢2), in 𝑄 ,              (29) 

𝑍2 = 0 on Σ ,  𝑍2(𝑥, 𝑇) =  𝑍2𝑡 (𝑥, 𝑇) = 0 on Ω,                                                            (30) 
 𝑍3𝑡𝑡 − ∆𝑍3 + 𝑍1 − 𝑍2 + 𝑍3 − 𝑍4 = 𝑍3𝑓3𝑦3 (𝑥, 𝑡, 𝑦3, 𝑢3) + 𝑔3𝑦3 (𝑥, 𝑡, 𝑦3, 𝑢3) ,  in 𝑄,            (31)  

𝑍3 = 0 on Σ,  𝑍3(𝑥, 𝑇) =  𝑍3𝑡 (𝑥, 𝑇) = 0 on Ω ,                                                                      (32) 
𝑍4𝑡𝑡 − ∆𝑍4 + 𝑍1 − 𝑍2 + 𝑍3 + 𝑍4 = 𝑍4𝑓4𝑦4 (𝑥, 𝑡, 𝑦4, 𝑢4) + 𝑔4𝑦4 (𝑥, 𝑡, 𝑦4, 𝑢4) ,   in 𝑄,             (33) 

𝑍4 = 0 on Σ,  𝑍4(𝑥, 𝑇) =  𝑍4𝑡 (𝑥, 𝑇) = 0 on Ω,                                                                        (34) 

And the Ham is defined as: 𝐻(𝑥, 𝑡, �⃗�, �⃗⃗�, �⃗�) = ∑  
𝑖=1

4

(𝑍𝑖 𝑓𝑖 (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 ) + 𝑔𝑖 (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 )) , 

Where 𝐺 (�⃗⃗�) = Σ
𝑖=1

4

∫
𝑄

𝑔 𝑖 (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 )𝑑𝑥𝑑𝑡. 

Then the DD of G is  

𝐷𝐺(�⃗⃗�, �⃗⃗̅� − �⃗⃗�) = lim  
→0

𝐺(�⃗⃗⃗�+ 𝛿�⃗⃗⃗�)−𝐺(�⃗⃗⃗�)
= ∫  

𝑄
𝐻�⃗⃗⃗�(𝑥, 𝑡, �⃗�, �⃗⃗�, �⃗�)(�⃗⃗̅� − �⃗⃗�)𝑑𝑥𝑑𝑡 . 

Proof: The WF of the QAEs ∀𝑣𝑖 ∈ 𝑉 and 𝑖 = 1,2,3,4 is  

(𝑍1𝑡𝑡 , 𝑣1) + (∇𝑍1, ∇𝑣1) + (𝑍1, 𝑣1) + (𝑍2, 𝑣1) − (𝑍3, 𝑣1) − (𝑍4, 𝑣1) 
= (𝑍1𝑓1𝑦1 , 𝑣1) + (𝑔1𝑦1 , 𝑣1) , ∀𝑣1 ∈ 𝑉  a.e. on 𝐼,                                                               (35) 

( 𝑍1(𝑇), 𝑣1) = (𝑍1𝑡 (𝑇), 𝑣1) = 0,                                                                                       (36) 
(𝑍2𝑡𝑡 , 𝑣2) + (∇𝑍2, ∇𝑣2) − (𝑍1, 𝑣2) + (𝑍2, 𝑣2) + (𝑍3, 𝑣2) + (𝑍4, 𝑣2) 
= (𝑍2𝑓2𝑦2 , 𝑣2) + (𝑔2𝑦2 , 𝑣2) , ∀𝑣2 ∈ 𝑉  a.e. on  ,                                                                (37) 



IHJPAS. 36(2)2023 

336 
 

( 𝑍2(𝑇), 𝑣2) = ( 𝑍2𝑡 (𝑇), 𝑣2) = 0,                                                                                       (38) 
(𝑍3𝑡𝑡  , 𝑣3) + (∇𝑍3 , ∇𝑣3) + (𝑍1 , 𝑣3) − (𝑍2 , 𝑣3) + (𝑍3 , 𝑣3) − (𝑍4 , 𝑣3) 
= (𝑍3𝑓3𝑦3 , 𝑣3) + (𝑔3𝑦3 , 𝑣3) , ∀𝑣3 ∈ 𝑉  a.e. on 𝐼,                                                                (39) 

( 𝑍3(𝑇) , 𝑣3) = (𝑍3𝑡 (𝑇) , 𝑣3) = 0 ,                                                                                      (40) 
(𝑍4𝑡𝑡  , 𝑣4) + (∇𝑍4 , ∇𝑣4) + (𝑍1 , 𝑣4) − (𝑍2 , 𝑣4) + (𝑍3 , 𝑣4) + (𝑍4 , 𝑣4) 
= (𝑍4𝑓4𝑦4 , 𝑣4) + (𝑔4𝑦4 , 𝑣4) , ∀𝑣4 ∈ 𝑉  a.e. on,                                                                   (41) 

(𝑍4(𝑥, 𝑇) , 𝑣4) = (𝑍4𝑡 (𝑇) , 𝑣4) = 0,                                                                                    (42) 

The WF  ((35-(42)) has a unique solution �⃗� = (𝑍1, 𝑍2, 𝑍3, 𝑍4) ∈ (𝐿
2(𝑄))4 (this it can proved so as 

the proof of existence a unique QSVs for the WF ((11)-(15)). 

Now, replacing 𝑣𝑖 = 𝛿𝑦𝑖  in (35), (37), (39) and (41), for 𝑖 = 1,2,3,4 resp. 

∫  
0

𝑇

(𝛿𝑦1 , 𝑍1𝑡𝑡 )𝑑𝑡 + ∫  
0

𝑇

[(∇𝑍1, ∇𝛿𝑦1 ) + (𝑍1, 𝛿𝑦1 ) + (𝑍2, 𝛿𝑦1 ) − (𝑍3, 𝛿𝑦1 ) − (𝑍4, 𝛿𝑦1 )]𝑑𝑡  

= ∫  
0

𝑇

(𝑍1𝑓1𝑦1 , 𝛿𝑦1 ) + (𝑔1𝑦1 , 𝛿𝑦1 )𝑑𝑡,                                                                                  (43) 

∫  
0

𝑇

(𝛿𝑦2 , 𝑍2𝑡𝑡 )𝑑𝑡 + ∫  
0

𝑇

[(∇𝑍2, ∇𝛿𝑦2 ) − (𝑍1, 𝛿𝑦2 ) + (𝑍2, 𝛿𝑦2 ) + (𝑍3, 𝛿𝑦3 ) + (𝑍4, 𝛿𝑦2 )]𝑑𝑡  

= ∫  
0

𝑇

(𝑍2𝑓2𝑦2 , 𝛿𝑦2 ) + (𝑔2𝑦2 , 𝛿𝑦2 )𝑑𝑡,                                                                                  (44) 

∫  
0

𝑇

(𝛿𝑦3 , 𝑍3𝑡𝑡 )𝑑𝑡 + ∫  
0

𝑇

[(∇𝑍3, ∇𝛿𝑦3 ) + (𝑍1, 𝛿𝑦3 ) − (𝑍2, 𝛿𝑦3 ) + (𝑍3, 𝛿𝑦3 ) − (𝑍4, 𝛿𝑦3 )]𝑑𝑡  

= ∫  
0

𝑇

(𝑍3𝑓3𝑦3 , 𝛿𝑦3 ) + (𝑔3𝑦3 , 𝛿𝑦3 )𝑑𝑡 ,                                                                                (45) 

∫  
0

𝑇

(𝛿𝑦4 , 𝑍4𝑡𝑡 )𝑑𝑡 + ∫  
0

𝑇

[(∇𝑍4, ∇𝛿𝑦4 ) + (𝑍1, 𝛿𝑦4 ) − (𝑍2, 𝛿𝑦4 ) + (𝑍3, 𝛿𝑦4 ) + (𝑍4, 𝛿𝑦4 )]𝑑𝑡  

= ∫  
0

𝑇

(𝑍4𝑓4𝑦4 , 𝛿𝑦4 ) + (𝑔4𝑦4 , 𝛿𝑦4 )𝑑𝑡,                                                                                      (46) 

Now, take �⃗⃗̅�, �⃗⃗� ∈ 𝑳𝟐(𝐐),set 𝛿𝑢⃗⃗ ⃗⃗ ⃗ = �⃗⃗̅� − �⃗⃗�, �⃗⃗� = �⃗⃗� + 𝛿𝑢⃗⃗ ⃗⃗ ⃗ ∈ 𝑳𝟐(𝐐) for > 0, then by Theorem 1, 

�⃗� = �⃗��⃗⃗⃗� & �⃗� = �⃗��⃗⃗⃗�𝜀  are their corresponding QSVs. Setting 𝛿𝑦
⃗⃗ ⃗⃗ ⃗ = (𝛿𝑦1 , 𝛿𝑦2 , 𝛿𝑦3 , 𝛿𝑦4 ) = �⃗� −

�⃗�, substituting 𝑣𝑖 = 𝑍𝑖  for 𝑖 = 1,2,3,4 in ((10)- (17)), IBSs on [0, 𝑇], then integrating by parts 

twice (IBPs2) the 1st in the LHS of each obtained equation, finding the FrD of 𝑓𝑖  (∀𝑖 = 1,2,3,4) in 

the RHS of each one equation (which is exists from the Assums C), then from the result of Theorem 

2.2 and the Minkowiski inequality (MIN), once get  

∫  
0

𝑇

(𝛿𝑦1 , 𝑍1𝑡𝑡 )𝑑𝑡 + ∫  
0

𝑇

[(∇𝛿𝑦1 , ∇𝑍1) + (𝛿𝑦1 , 𝑍1) + (𝛿𝑦2 , 𝑍1) + (𝛿𝑦3 , 𝑍1) + (𝛿𝑦4 , 𝑍1)]𝑑𝑡  

= ∫  
0

𝑇

(𝑓1𝑦1 𝛿𝑦1 + 𝑓1𝑢1 𝛿𝑢1, 𝑍1)𝑑𝑡 + 𝑂11( ) ,                                                                  (47) 

∫  
0

𝑇

(𝛿𝑦2 , 𝑍2𝑡𝑡 )𝑑𝑡 + ∫  
0

𝑇

[(∇𝛿𝑦2 , ∇𝑍2) + (𝛿𝑦1 , 𝑍2) + (𝛿𝑦2 , 𝑍2) + (𝛿𝑦3 , 𝑍2) + (𝛿𝑦4 , 𝑍2)]𝑑𝑡  

= ∫  
0

𝑇

(𝑓2𝑦2 𝛿𝑦2 + 𝑓2𝑢2 𝛿𝑢2, 𝑍2)𝑑𝑡 + 𝑂12( ),                                                                    (48) 

∫  
0

𝑇

(𝛿𝑦3 , 𝑍3𝑡𝑡 )𝑑𝑡 + ∫  
0

𝑇

[(∇𝛿𝑦3 , ∇𝑍3) + (𝛿𝑦1 , 𝑍3) − (𝛿𝑦2 , 𝑍3) + (𝛿𝑦3 , 𝑍3) + (𝛿𝑦4 , 𝑍3)]𝑑𝑡  

= ∫  
0

𝑇

(𝑓3𝑦3 𝛿𝑦3 + 𝑓3𝑢3 𝛿𝑢3, 𝑍3)𝑑𝑡 + 𝑂13( ),                                                                    (49) 

∫  
0

𝑇

(𝛿𝑦4 , 𝑍4𝑡𝑡 )𝑑𝑡 + ∫  
0

𝑇

[(∇𝛿𝑦4 , ∇𝑍4) − (𝛿𝑦1 , 𝑍4) − (𝛿𝑦2 , 𝑍4) + (𝛿𝑦3 , 𝑍4) + (𝛿𝑦4 , 𝑍4)]𝑑𝑡  



IHJPAS. 36(2)2023 

337 
 

= ∫  
0

𝑇

(𝑓4𝑦4 𝛿𝑦4 + 𝑓4𝑢4 𝛿𝑢4, 𝑍4)𝑑𝑡 + 𝑂14( ),                                                                  (50) 

where 𝑂1𝑖 ( ) =∥ 𝛿𝑦𝑖 ∥𝑄
2 + ∥ 𝛿𝑢𝑖  ∥𝑄

2 → 0 , as → 0, ∀𝑖 = 1,2,3,4. 

Subtracting ((47) – (50)) from ((43)- (46)) resp., collecting the obtain equations, to acquire  

∫  
0

𝑇

∑  
𝑖=1

4

(𝑓𝑖𝑢𝑖 𝛿𝑢𝑖 , 𝑍𝑖 )𝑑𝑡 + 𝑂1( ) = ∫  
0

𝑇

∑  
𝑖=1

4

(𝑔𝑖𝑦𝑖 , 𝛿𝑦𝑖 )𝑑𝑡  , ∀𝑖 = 1,2,3,4 ,                              (51) 

Where 𝑂1( ) = ∑  
𝑖=1

4

𝑂1𝑖 ( ) → 0 as → 0. 

From the other side, by employing the Assums (C), the definition of the FrD the result of Theorem 

2.2, and using the MIN, one has 

𝐺(�⃗⃗� ) − 𝐺(�⃗⃗�) = ∑  
𝑖=1

4

∫  
𝑄

(𝑔𝑖𝑦𝑖  𝛿𝑦𝑖 + 𝑔𝑖𝑢𝑖  𝛿𝑢𝑖 )𝑑𝑥𝑑𝑡 + 𝑂2( ),                               (52) 

Where 𝑂2( ) =∥ 𝛿𝑦⃗⃗ ⃗⃗ ⃗⃗ ⃗ ∥𝑳𝟐(𝐐)
2 + ∥ 𝛿𝑢  

⃗⃗⃗⃗⃗⃗⃗⃗ ∥
𝑳𝟐(𝐐)
2 → 0 as → 0, ∀𝑖 = 1,2,3,4. 

Now, by using  (51) in (52), to obtain 

𝐺(�⃗⃗� ) − 𝐺(�⃗⃗�) = ∫  
0

𝑇

∑  
𝑖=1

4

(𝑍𝑖 𝑓𝑖𝑢𝑖 + 𝑔𝑖𝑢𝑖 )𝛿𝑢𝑖 𝑑𝑥𝑑𝑡 + 𝑂3( ) , 

Where 𝑂3( ) = 𝑂1( ) + 𝑂2( ). 

Lastly, dividing both sides by , then taking the limit → 0, yields to 

𝐷𝐺(�⃗⃗�, �⃗⃗̅� − �⃗⃗�) = ∫  
𝑄

𝐻�⃗⃗⃗�(𝑥, 𝑡, �⃗�, �⃗⃗�, �⃗�)(�⃗⃗̅� − �⃗⃗�)𝑑𝑥𝑑𝑡 . 

4.The NCSO and SCSO  

4.1Theorem:  

(a) with Assums (A), (B) &( C), if �⃗⃗⃗⃗� is CO., the �⃗⃗� ∈ 𝑊𝐴⃗⃗ ⃗⃗ ⃗⃗  is CQOCCCV, then there exist 𝜆𝑙 ∈ ℝ, 

𝑙 = 0,1,2 with 𝜆0 ≥ 0, 𝜆2 ≥ 0 , ∑  
𝑙=0

2

∣ 𝜆𝑙 ∣= 1, s.t. the following Kuhn-Tucher Lagrange (KTL) 

conditions are held: 

∑  
𝑙=0

2

𝜆𝑙  𝐷𝐺𝑙 (�⃗⃗�, �⃗⃗̅� − �⃗⃗�) ≥ 0, ∀ �⃗⃗̅� ∈ �⃗⃗⃗⃗�,                                                             (53) 

𝜆2𝐺2(�⃗⃗�) = 0,                                                                                                            (54) 

(b) Inequality (53) is equivalent to: 

𝐻�⃗⃗⃗�(𝑥, 𝑡, �⃗�, �⃗⃗�, �⃗�)�⃗⃗�(𝑡) = min
�⃗⃗⃗�∈�⃗⃗⃗�

𝐻�⃗⃗⃗�(𝑥, 𝑡, �⃗�, �⃗⃗�, �⃗�)�⃗⃗̅�(𝑡) , 𝑎. 𝑒. 𝑜𝑛 𝑄,                (55) 

Where  𝐻�⃗⃗⃗� (𝑥, 𝑡, �⃗�, �⃗⃗�, �⃗�) = ∑  
𝑖=1

4

(𝑍𝑖 𝑓𝑖𝑢𝑖 (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 ) + 𝑔𝑖𝑢𝑖 (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 )). 

Proof: From Lemma 2.1, the funl. 𝐺𝑙 (�⃗⃗�) (for 𝑙 = 0,1,2) is cont. w.r.t. �⃗⃗̅� − �⃗⃗� and linear in �⃗⃗̅� − �⃗⃗�, 

the 𝐷𝐺𝑙 (�⃗⃗�) is M-differential for any M, then applying the KTL theorem[15], there exist 𝜆𝑙 ∈

ℝ , 𝑙 = 0,1,2 with 𝜆0, 𝜆2 ≥ 0  , ∑  
𝑙=0

2

∣ 𝜆𝑙 ∣= 1 s.t.  ((53)-(54)) are satisfied, then by utilizing 

Theorem 3.2, (53) becomes 

∫  
𝑄

(𝑍1𝑓1𝑢1 , 𝑍2𝑓2𝑢2 , 𝑍3𝑓3𝑢3 , 𝑍4𝑓4𝑢4 ). (�⃗⃗̅� − �⃗⃗�)𝑑𝑥𝑑𝑡 ≥ 0, ∀ �⃗⃗̅� ∈ �⃗⃗⃗⃗�,                           (56) 

where  𝑔𝑖 = ∑  
𝑙=0

2

𝜆𝑙 𝑔𝑙𝑖 and 𝑍𝑖 = ∑  
𝑙=0

2

𝜆𝑙 𝑍𝑙𝑖 , (∀𝑖 = 1,2,3,4). 

(b) Let {�̅�𝑘⃗⃗⃗⃗ ⃗}  be dense Seq (DSeq) in �⃗⃗⃗⃗�, 𝜇 is Lebesgue measure (LM) on 𝑄 and let 𝑆 ⊂ 𝑄 be a 

measurable set (MS) s.t. 



IHJPAS. 36(2)2023 

338 
 

�⃗⃗̅�(𝑥, 𝑡) = {
�̅�𝑘⃗⃗⃗⃗ ⃗(𝑥, 𝑡), 𝑖𝑓 (𝑥, 𝑡) ∈ 𝑆

�⃗⃗�(𝑥, 𝑡), 𝑖𝑓 (𝑥, 𝑡) ∉ 𝑆
. 

Which makes (56), gives 

∫  
𝑆

(𝑍1𝑓1𝑢1 + 𝑔1𝑢1 , 𝑍2𝑓2𝑢2 + 𝑔2𝑢2 , 𝑍3𝑓3𝑢3 + 𝑔3𝑢3 , 𝑍4𝑓4𝑢4 + 𝑔4𝑢4 ). (�⃗⃗̅�𝑘 − �⃗⃗�)𝑑𝑥𝑑𝑡 ≥ 0 ,  

or 

(𝑍1𝑓1𝑢1 + 𝑔1𝑢1 , 𝑍2𝑓2𝑢2 + 𝑔2𝑢2 , 𝑍3𝑓3𝑢3 + 𝑔3𝑢3 , 𝑍4𝑓4𝑢4 + 𝑔4𝑢4 ). (�⃗⃗̅�𝑘 − �⃗⃗�) ≥ 0, 𝑎. 𝑒. 𝑜𝑛 𝑄, 

i.e. this inequality holds on 𝑄\𝑄𝑘   with 𝜇(𝑄𝑘) = 0, ∀𝑘, where 𝜇 is a LM, i.e. it is satisfies on 

𝑄\∪𝑘 𝑄𝑘, with 𝜇(∪𝑘 𝑄𝑘 ) = 0, but {�̅�𝑘⃗⃗⃗⃗ ⃗}  is a DSeq in �⃗⃗⃗⃗�, then there is �⃗⃗̅� ∈ �⃗⃗⃗⃗� , s.t. 

(𝑍1𝑓1𝑢1 + 𝑔1𝑢1 , 𝑍2𝑓2𝑢2 + 𝑔2𝑢2 , 𝑍3𝑓3𝑢3 + 𝑔3𝑢3 , 𝑍4𝑓4𝑢4 + 𝑔4𝑢4 ). (�⃗⃗̅� − �⃗⃗�) ≥ 0, 𝑎. 𝑒. 𝑜𝑛 𝑄, ∀�⃗⃗̅� ∈ �⃗⃗⃗⃗�. 

i.e. (53) gives (56). The converse is clear. 

 

4.2Theorem: (The SCSO) 

In addition to the assums (A), (B) &( C). Suppose �⃗⃗⃗⃗� is CO., 𝑓𝑖 , 𝑔𝑖 are affine w.r.t. (𝑦𝑖 , 𝑢𝑖 ) for 

each (𝑥, 𝑡), 𝑔0𝑖, 𝑔2𝑖  are CO. w.r.t. (𝑦𝑖 , 𝑢𝑖 ), ∀(𝑥, 𝑡), 𝑖 = 1,2,3,4. Then the NCSO of Theorem 4.1, 

with 𝜆0 > 0 are also sufficient. 

Proof: Assume �⃗⃗� ∈ �⃗⃗⃗⃗�𝐴, is satisfied the KTL condition ((53)- (54)). Let 𝐺(�⃗⃗�) = ∑  
𝑙=0

2

𝜆𝑙 𝐺𝑙 (�⃗⃗�), 

then using Theorem 3.2, to get 

𝐷𝐺(�⃗⃗�, �⃗⃗̅� − �⃗⃗�) = ∑  
𝑙=0

2

𝜆𝑙 ∫  
𝑄

∑  
𝑖=1

4

𝑍𝑙𝑖 𝑓𝑙𝑖𝑢𝑖 + 𝑔𝑙𝑖𝑢𝑖 𝛿𝑢𝑖 𝑑𝑥𝑑𝑡 ≥ 0,  

Since 

𝑓𝑖 (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 ) = 𝑓𝑖1(𝑥, 𝑡)𝑦𝑖 + 𝑓𝑖2(𝑥, 𝑡)𝑢𝑖 + 𝑓𝑖3(𝑥, 𝑡). 

Let �⃗⃗�&�⃗⃗̅� are given QCVs, then �⃗�& �⃗̅� are their corresponding QSVs. Substituting the pair (�⃗⃗�, �⃗�)in 

((1)-(6)) and MBS by 𝛼 ∈ [0,1] once, and then substituting the pair (�⃗⃗̅�, �⃗̅�) in ((1)-(6)) and MBS 

by (1 − 𝛼) once again, finally collecting each pair from the corresponding equations together 

one gets 

(𝛼𝑦1 + (1 − 𝛼)�̅�1)𝑡𝑡 − ∆(𝛼𝑦1 + (1 − 𝛼)�̅�1) + (𝛼𝑦1 + (1 − 𝛼)�̅�1) − (𝛼𝑦2 + (1 − 𝛼)�̅�2) 

+(𝛼𝑦3 + (1 − 𝛼)�̅�3) + (𝛼𝑦1 + (1 − 𝛼)�̅�1) 

= 𝑓11(𝑥, 𝑡)(𝛼𝑦1 + (1 − 𝛼)�̅�1) + 𝑓12(𝑥, 𝑡)(𝛼𝑢1 + (1 − 𝛼)�̅�1) + 𝑓13(𝑥, 𝑡),                        (57) 

𝛼𝑦1(𝑥, 𝑡) + (1 − 𝛼)�̅�1(𝑥, 0) = 0,                                                                                         (58) 

𝛼𝑦1(𝑥, 0) + (1 − 𝛼)�̅�1(𝑥, 0) = 𝑦1
0(𝑥), 𝛼𝑦1𝑡 (𝑥, 0) + (1 − 𝛼)�̅�1𝑡 (𝑥, 0) = 𝑦1

1(𝑥),               (59) 

(𝛼𝑦2 + (1 − 𝛼)�̅�2)𝑡𝑡 − ∆(𝛼𝑦2 + (1 − 𝛼)�̅�2) + (𝛼𝑦1 + (1 − 𝛼)�̅�1) + (𝛼𝑦2 + (1 − 𝛼)�̅�2) 

−(𝛼𝑦3 + (1 − 𝛼)�̅�3) − (𝛼𝑦4 + (1 − 𝛼)�̅�4) 

 = 𝑓21(𝑥, 𝑡)(𝛼𝑦2 + (1 − 𝛼)�̅�2) + 𝑓22(𝑥, 𝑡)(𝛼𝑢2 + (1 − 𝛼)�̅�2) + 𝑓23(𝑥, 𝑡),                       (60) 

𝛼𝑦2(𝑥, 𝑡) + (1 − 𝛼)�̅�2(𝑥, 0) = 0,                                                                                         (61) 

𝛼𝑦2(𝑥, 0) + (1 − 𝛼)�̅�2(𝑥, 0) = 𝑦2
0(𝑥), 𝑦2𝑡 (𝑥, 0) + (1 − 𝛼)�̅�2𝑡 (𝑥, 0) = 𝑦2

1(𝑥) ,                (62) 

(𝛼𝑦3 + (1 − 𝛼)�̅�3)𝑡𝑡 − ∆(𝛼𝑦3 + (1 − 𝛼)�̅�3) − (𝛼𝑦1 + (1 − 𝛼)�̅�1) + (𝛼𝑦2 + (1 − 𝛼)�̅�2) 

+(𝛼𝑦3 + (1 − 𝛼)�̅�3) + (𝛼𝑦4 + (1 − 𝛼)�̅�4) 

= 𝑓31(𝑥, 𝑡)(𝛼𝑦3 + (1 − 𝛼)�̅�3) + 𝑓32(𝑥, 𝑡)(𝛼𝑢3 + (1 − 𝛼)�̅�3) + 𝑓33(𝑥, 𝑡),                         (63) 

𝛼𝑦3(𝑥, 𝑡) + (1 − 𝛼)�̅�3(𝑥, 0) = 0,                                                                        (64) 

𝛼𝑦3(𝑥, 0) + (1 − 𝛼)�̅�3(𝑥, 0) = 𝑦3
0(𝑥), 𝛼𝑦3𝑡 (𝑥, 0) + (1 − 𝛼)�̅�3𝑡 (𝑥, 0) = 𝑦3

1(𝑥),                (65) 

(𝛼𝑦4 + (1 − 𝛼)�̅�4)𝑡𝑡 − ∆(𝛼𝑦4 + (1 − 𝛼)�̅�4) − (𝛼𝑦1 + (1 − 𝛼)�̅�1) + (𝛼𝑦2 + (1 − 𝛼)�̅�2) 

−(𝛼𝑦3 + (1 − 𝛼)�̅�3) + (𝛼𝑦4 + (1 − 𝛼)�̅�4) 



IHJPAS. 36(2)2023 

339 
 

= 𝑓41(𝑥, 𝑡)(𝛼𝑦4 + (1 − 𝛼)�̅�4) + 𝑓42(𝑥, 𝑡)(𝛼𝑢4 + (1 − 𝛼)�̅�4) + 𝑓43(𝑥, 𝑡),                 (66) 

𝛼𝑦4(𝑥, 𝑡) + (1 − 𝛼)�̅�4(𝑥, 0) = 0,                                                                            (67) 

𝛼𝑦4(𝑥, 0) + (1 − 𝛼)�̅�4(𝑥, 0) = 𝑦4
0(𝑥), 𝛼𝑦4𝑡 (𝑥, 0) + (1 − 𝛼)�̅�4𝑡 (𝑥, 0) = 𝑦4

1(𝑥),          (68) 

Equalities ((57)- (68)), show that if the QCV is �⃗⃗̅�( with (�⃗⃗̅� = 𝛼�⃗⃗� + (1 − 𝛼)�⃗⃗̅�)) has corresponding 

QSVs �⃗̅� with (�̅�𝑖 = 𝑦𝑖𝑢𝑖 = 𝑦𝑖(𝛼𝑢𝑖+(1− 𝛼)𝑢𝑖)). 

This means the operator �⃗⃗� → �⃗��⃗⃗⃗� is CO-linear (COL) w.r.t. (�⃗⃗�, �⃗�) in 𝑄. Now, since 

𝑔1𝑖 (𝑥, 𝑡, 𝑦𝑖 , 𝑢𝑖 ) is affine w.r.t. (𝑦𝑖 , 𝑢𝑖 ), in 𝑄, then 𝐺1(�⃗⃗�) is COL w.r.t. (�⃗⃗�, �⃗�), also,  since 𝑔0𝑖 &𝑔2𝑖  

are CO w.r.t.(𝑦𝑖 , 𝑢𝑖 ) , in 𝑄, ∀𝑖 = 1,2,3,4 , then the funl. 𝐺0(�⃗⃗�), 𝐺2(�⃗⃗�)  are CO. w.r.t. (�⃗�, �⃗⃗�) in 𝑄 

(from the assum. on the funl 𝑔𝑙𝑖  (∀𝑙 = 0,1,2, &𝑖 = 1,2,3,4) and from the sum of two integral of 

CO function is also CO), i.e. 𝐺(�⃗⃗�) is CO w.r.t. (�⃗�, �⃗⃗�) , in 𝑄 in the CO set �⃗⃗⃗⃗�, and has a cont. DD 

satisfies 

𝐷𝐺(�⃗⃗�, �⃗⃗̅� − �⃗⃗�) ≥ 0, which means 𝐺(�⃗⃗�) has a minimum at �⃗⃗�, i.e.  

𝐺(�⃗⃗�) ≤ 𝐺(�⃗⃗̅�),∀ �̅� ⃗⃗⃗ ⃗ ∈ �⃗⃗⃗⃗�, i.e. 

𝜆0𝐺0(�⃗⃗�) + 𝜆1𝐺0(�⃗⃗�) + 𝜆2𝐺2(�⃗⃗�) ≤ 𝜆0𝐺0(�⃗⃗̅�) + 𝜆1𝐺1(�⃗⃗̅�) + 𝜆2𝐺2(�⃗⃗̅�), ∀�̅� ⃗⃗⃗ ⃗ ∈ �⃗⃗⃗⃗� 

Let �̅� ⃗⃗⃗ ⃗ ∈ �⃗⃗⃗⃗�𝐴, 𝜆2 ≥ 0 and from (54), the above inequality becomes 

𝜆0𝐺0(�⃗⃗�) ≤ 𝜆0𝐺0(�̅� ⃗⃗⃗ ⃗), ∀�̅� ⃗⃗⃗ ⃗ ∈ �⃗⃗⃗⃗�, or 𝐺0(�⃗⃗�) ≤ 𝐺0(�̅� ⃗⃗⃗ ⃗), ∀�̅� ⃗⃗⃗ ⃗ ∈ �⃗⃗⃗⃗�, thus �⃗⃗� ia a CQOCCCV. 

 

5.Conclusions and Discussions:  

In this work, the CQOCCCVP dominating by a QNLHBVP is studied. The existence of a 

CQOCCCV dominating by a QNLHBVP with EINQSCC is stated and demonstrated under 

appropriate HYP with using the ACTH. Moreover mathematical formulation of the QAEs related 

to QSEs is found so as its WF. The derivation of the DD for the Ham is attained. Lastly, both the 

NCSO and the SCSO “theorems” optimality of the proposed problem are stated and demonstrated. 

The study of the proposed problem is considered very interesting in the field of applied 

mathematics since the proposed model represents a generalization for a wave equation; from a 

side, and from the other, these results are very important because they give the green light about 

the ability for solving such problems numerically.  

 

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