Microsoft Word - 101-112


   

101 
  

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 

 

  

 

 

The Continuous Classical Optimal Control Problems for Triple Nonlinear 
Elliptic Boundary Value Problem  

 
 

 

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

 

 
 

 
 

A b s t r a c t  
     In this research, our aim is to study the optimal control problem (OCP) for triple nonlinear 
elliptic boundary value problem (TNLEBVP). The Mint-Browder theorem is used to prove 
the existence and uniqueness theorem of the solution of the state vector for fixed control 
vector. The existence theorem for the triple continuous classical optimal control vector 
(TCCOCV) related to the TNLEBVP is also proved. After studying the existence of a unique 
solution for the triple adjoint equations (TAEqs) related to the triple of the state equations, we 
derive The Fréchet derivative (FD) of the cost function using Hamiltonian function. Then the 
theorems of necessity conditions and the sufficient condition for optimality of the constraints 
problem are proved. 
 
K e y w o r d s :  Triple nonlinear elliptic value problem, continuous classical optimal control 
vector, Mint-Browder theorem, triple adjoint equations, Fréchet derivative necessity and 
sufficient theorems. 
   
1 .  I n t r o d u c t i o n  
     The OCP is one of the most important subject not only in mathematics, but in all branches 
of science, for instance, in engineering such  as robotics [1]. And aeronautics [2]. In the 
medicine and mathematical biology, such as modeling and optimal controlling the infectious 
diseases [3]. In the life sciences, such as sustainable forest management [4].  
In the past few decades, there were many studies and papers published in OCPs for systems 
that related to nonlinear ordinary differential equations [5]. or systems related to nonlinear 
partial differential equation (NLPDEqs) either of: a hyperbolic type [6]. Or of a parabolic type 
[7]. Or of an elliptic type [8]. 

Ibn Al Haitham Journal for Pure and Applied Science 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

Doi: 10.30526/33.3.2477 

Article history: Received 4  September 2019, Accepted 4 November 2019, Published 
in July 2020.

Jamil A. Al-Hawasy Doaa Kateb Jasim 

Jhawassy17@uomustansiriyah.edu.iq  hawasy20@yahoo.com



   

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 

 or OCP are related to couple of NLPDEqs of: a hyperbolic [9]. Or of hyperbolic but include a 
boundary control [10]. Or of a parabolic type [11].Or of a parabolic type but includes a 
boundary control [12]. Or of an elliptic type [13]. Of an elliptic type that includes a Numann 
boundary control [14]. While other papers deals with the optimal control problems that are 
related to triple linear partial differential equation of : an elliptic type [15]. Or of an parabolic 
type [16]. 
In this work, the Minty-Browder theorem is used to prove the existence theorem for a unique 
solution (continuous state vector) for the TNLEBVP for fixed TCCOCV, and to state and 
prove the theorem for the existence TCCOCV related to the TNLEBVP, so as the theorem of 
the existence of a unique solution of the TAEqs related to the TNLEBVP. The FD of the cost 
function is derived. At the end the theorem of necessity conditions is stated and proved so as 
is the sufficient condition theorem for optimality of the constrained problem. 

2 .  T h e  Pr o b l e m  D e s c r i p t i o n  
     Let Λ be an open (bounded) connected subset in ℝ ℝ with Lipschitz boundary ∂Λ. 
Consider the TCCOC of the TNLEBVP 

𝐵  𝜉 𝜉 𝜉 𝜉 𝑎 𝑥, 𝜉 , 𝑣 𝑎 𝑥, 𝑣 , 𝑖𝑛 𝛬 ( 1 )  
𝐵  𝜉 𝜉 𝜉 𝜉 𝑝 𝑥, 𝜉 , 𝑣 𝑝 𝑥, 𝑣 , 𝑖𝑛 𝛬 ( 2 )  
𝐵  𝜉 𝜉 𝜉 𝜉 𝑘 𝑥, 𝜉 , 𝑣 𝑘 𝑥, 𝑣 , 𝑖𝑛 𝛬 ( 3 )  

with the Dirchlet boundary condition   
𝜉 𝜉 𝜉  0 ,  i n  ∂Λ  ( 4 )  

W h e r e  B ξ ∑ b, ,  𝑟 1,2,3  ,  b b x ∈ L Λ , ∀i, j 1,2,  x x , x   

ξ⃗ ξ x , ξ x , ξ x ∈ H Λ  is the classical solution of the system (1)-(4), v⃗

v x , v x , v x ∈ L Λ  is the CCV,  the functions a x, ξ , v , p x, ξ , v  a n d  
 k x, ξ , v  a r e  d e f i n e d  o n  Λ ℝ V ,  Λ ℝ V  and   Λ ℝ V   respectively, and 
the functions a x, v , p x, v  and k x, v   are defined on Λ V , Λ V   and  Λ V  
respectively with V , V  , V ⊂ ℝ. 

T h e  c o n t r o l  c o n s t r a i n t  i s  ( v , v , v ∈ U U U U⃗,   U⃗⊂  L Λ , where U⃗ is 
the control  set has the form    

�⃗� 𝑢 ∈ 𝐿 𝛬 |𝑢 𝑢 , 𝑢 , 𝑢 ∈ 𝑉 𝑉 𝑉 �⃗� 𝑎. 𝑒. 𝑖𝑛 𝛬  

With V⃗ ⊂ ℝ  that is convex and compact set. 
T h e  c o s t  f u n c t i o n  i s   

𝑌 �⃗� 𝑦
 

𝑥, 𝜉 , 𝑣 𝑑𝑥 𝑦
 

𝑥, 𝜉 , 𝑣 𝑑𝑥 𝑦
 

𝑥, 𝜉 , 𝑣 𝑑𝑥  ( 5 )   

T h e  s t a t e  – c o n t r o l  c o n s t r a i n t s  a r e   
𝑌 �⃗� 𝑦

 
𝑥, 𝜉 , 𝑣 𝑑𝑥 𝑦

 
𝑥, 𝜉 , 𝑣 𝑑𝑥 𝑦

 
𝑥, 𝜉 , 𝑣 𝑑𝑥 0 ( 6 )   

Y v⃗ y
 

x, ξ , v dx y
 

x, ξ , v dx y
 

x, ξ , v dx 0 ( 7 )   

T h e  s e t  o f  t h e  a d m i s s i b l e  c o n t r o l s  is �⃗� �⃗� ∈ �⃗�|𝑌 �⃗� 0, 𝑌 �⃗� 0  
The TCCOC problem is to minimize the cost function (5) subject to the state constraints of 

(6) and (7), i.e. to find v⃗ such that v⃗ ∈ U⃗  and Y v⃗ min
⃗∈ ⃗

 Y u⃗ . 

Let 𝑊 𝑊 𝑊 𝑊 𝐻 𝛬 𝐻 𝛬 𝐻 𝛬 , y  ‖w‖  and  ‖𝑤‖ are denoted by the 

norm in H Λ  and ( H Λ respectively, y ‖𝑤‖  ( ‖𝑤‖  are denoted the norm in 𝐿 Λ  



   

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 

and in 𝐿 Λ  respectively and the inner product in 𝑊 is denoted by 𝑤, 𝑤 , with ‖𝑤‖

‖𝑤 ‖ ‖𝑤 ‖ ‖𝑤 ‖ , W∗⃗ is dual of W⃗. 

3 .  W e a k  F o r m u l a t i o n  o f  t h e  T N L E B V P   
The weak form (WF) of (1)-(4) is obtained through multiplying both sides of Equations (1)-
(3) byw ∈ W , w ∈ W  and w ∈ W  respectively, then integrating the obtained equations. 
Finally, using the generalize Green's theorem for the 1st term in left hand side (L.H.S) of the 
three obtained equations, once get ∀𝑤 , 𝑤 , 𝑤 ∈ 𝑊  
𝑏 𝜉 , 𝑤 𝜉 , 𝑤 𝜉 , 𝑤 𝜉 , 𝑤 𝑎 𝜉 , 𝑣 , 𝑤  𝑎 𝑣 , 𝑤  (8) 
𝑏 𝜉 , 𝑤 𝜉 , 𝑤 𝜉 , 𝑤 𝜉 , 𝑤 𝑝 𝜉 , 𝑣 , 𝑤 𝑝 𝑣 , 𝑤    (9) 
𝑏 𝜉 , 𝑤 𝜉 , 𝑤 𝜉 , 𝑤 𝜉 , 𝑤 𝑘 𝜉 , 𝑣 , 𝑤 𝑘 𝑣 , 𝑤  (10)  

where     b ξ , w ∑ b, .
 

 dx,  ξ , w ξ
 

w dx,  Θ, w Θ w dx
 

, 

with 𝛩 𝑎  𝑜𝑟 𝑝  𝑜𝑟 𝑘  , 𝑟, 𝑝 1,2,3 , 𝚤 1,2. 
By blending to gather equations (8), (9) and (10), once get 

B ξ⃗, w⃗ a ξ , v , w p ξ , v , w k ξ , v , w   a v , w
p v , w k v , w  (11) 

where 𝐵 𝜉, 𝑤 𝑏 𝜉 , 𝑤 𝜉 , 𝑤 𝜉 , 𝑤 𝜉 , 𝑤 𝑏 𝜉 , 𝑤 𝜉 , 𝑤
                                   𝜉 , 𝑤  𝜉 , 𝑤 𝑏 𝜉 , 𝑤 𝜉 , 𝑤 𝜉 , 𝑤 𝜉 , 𝑤  
H y p o t h e s e s  A :  

a ) B ξ⃗, w⃗  i s  c o e r c i v e ,  i . e .   
⃗, ⃗

⃗
ϵ ξ⃗ 0 , ξ⃗ ∈ W⃗ 

b ) B ξ⃗, w⃗ ϵ ξ⃗ ‖w⃗‖ ,  ϵ 0 

c )  the functions a x, ξ , v  , p x, ξ , v  and k x, ξ , v  are of Carathéodory type on 
Λ× ℝ ×V ,  Λ× ℝ ×V   and   Λ ℝ V  respectively and satisfy the following sublinearity 
conditions with respect to (w.r.t.) ξ , v , ξ , v  and ξ , v  respectively. 

|a x, ξ , v |≤ϑ x 𝒸 |ξ | �̅� |v |  ,  |p x, ξ , v |≤ϑ x 𝔠 |ξ | �̅� |v | ,  
|k x, ξ , v |≤ϑ x 𝔠 |ξ | �̅� |v |  

∀(x,ξ , v ∈ Λ ℝ U    with ϑ ∈ L Λ ,     𝔠 , �̅� 0 , i 1,2,3.  
d )  a x, ξ , v ,  p x, ξ , v  and k x, ξ , v  are monotone w.r.t. ξ , ξ , ξ        
    respectively for  each x ∈ Λ  , v ∈ V ,  v ∈ V , v ∈ V . 
e) a x, 0, v 0, ∀x ∈ Λ, v ∈ V ,   p x, 0, v 0, ∀x ∈ Λ, v ∈ V , 
      k x, 0, v 0, ∀x ∈ Λ, v ∈ V . 
f) the functions a x, v  , p x, v  and k x, v  are of Carathéodory type on     
    Λ×V , Λ×V  and Λ V  respectively and satisfy the following conditions 
 |a x, v |≤ϑ x 𝔠 |v | ,    |p x, v |≤ϑ x 𝔠 |v | ,   |k x, v |≤ϑ x 𝔠 |v | 
  ∀(x,v ∈ Λ U  , 𝑖 1,2,3  with ϑ ∈ L Λ ,  𝔠 0 , 𝑟 4,5,6. 
T h e o r e m  3 . 1  ( T h e  M i n t y - B r o w d e r  t h e o r e m ) [ 1 7 ] .   let W be a reflexive Banach 
space and  D: W → W∗ be a nonlinear continuous map such that  

𝐷𝑤 𝐷𝑤 , 𝑤 𝑤 0, ∀𝑤 , 𝑤 ∈ 𝑊, 𝑤 𝑤       𝑎𝑛𝑑           𝑙𝑖𝑚
‖ ‖→

,

‖ ‖
∞ 

Then the equation  Dξ a has a unique (solution) ξ ∈ W for every a ∈ W∗.  



   

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P r o p o s i t i o n  3 . 1  [ 1 8 ] .  Let 𝑎: Λ ℝ ⟶ ℝ  is of Carathéodory type, and the functional 𝐴 
is defined by 𝐴 𝜉 𝑎 𝑥, 𝜉 𝑥

 
𝑑𝑥, where Λ is a measurable subset of ℝ , and suppose 

that  
‖𝑎 𝑥, 𝜉 ‖ 𝜗 𝑥 𝜂 𝑥 ‖𝜉‖ ,∀ 𝑥, 𝜉 ∈ Λ ℝ , 𝜉 ∈ 𝐿 Λ ℝ  

where 𝜗 ∈ 𝐿 Λ ℝ , 𝜂 ∈ 𝐿 Λ ℝ  , and 𝛼 ∈ 0, 𝑃 , if 𝑃 ∈ 1, ∞  , and 𝜂 ≡ 0, if 𝑃 ∞. 
Then 𝐴 is continuous on 𝐿 Λ ℝ . 
T h e o r e m  3 . 2 :  In addition to the hypo.(A-a&d), If at least one of the functions a , p  or k  

in hypo.(A-d) is strictly monotone. Then for any fixed controlv⃗ ∈ U⃗  , the WF (11) has a 

unique solution ξ⃗ ∈ W⃗. 

 P r o o f :  l e t  D: W⃗ → W⃗∗, then the WF (11) is rewriting as          

 𝐷 𝜉 , 𝑤 a v , w p v , w k v , w          (12) 

where 𝐷 𝜉 , 𝑤 𝐵 𝜉, 𝑤 𝑎 𝜉 , 𝑣  , 𝑤 𝑝 𝜉 , 𝑣 , 𝑤   𝑘 𝜉 , 𝑣 , 𝑤   (13) 

Then D satisfies the following:  
i) D is coercive from hypo. (A-a&e&d) 

ii) f r o m  h y p o t h e s e s  ( A- a & c )  and using proposition3.1 the maping ξ⃗ ⟼ D ξ⃗ , w⃗  is 

continuous w.r.t. ξ⃗. 

iii)  from hypotheses (A-a&b) and (i) D is strictly monotone w.r.t. ξ⃗. 

Hence by Theorem3.1, there exists a unique weak solution ξ⃗ ∈ W⃗ of (11).  
4 .  E x i s t e n c e  o f  t h e  T C C O C  
L e m m a  4 . 1 :   If the functions a & a , p & p  and (k &k  are Lipschitz w.r.t. v , v  

and v  respectively, moreover the hypothesis (A). Then the transformation v⃗ ⟼ ξ⃗ ⃗  from U⃗ to 

L² Ω is Lipschitz continuous. 

P r o o f :  let V⃗ v , v , v ∈ U⃗ be a given control of WF(8)-(10) with its corresponding state 
solution ξ , ξ , ξ , then by subtracting (8)-(10) from the equations which are obtained from 

substituting δξ ξ ξ  , δv v v  𝑖 1,2,3  in (8)-(10)  respectively, setting w
δξ  , w δξ  and w δξ  and blending together the obtained equation, to give 
𝑏 𝛿𝜉 , 𝛿𝜉 𝛿𝜉 , 𝛿𝜉 𝑏 𝛿𝜉 , 𝛿𝜉 𝛿𝜉 , 𝛿𝜉 𝑏 𝛿𝜉 , 𝛿𝜉 𝛿𝜉 , 𝛿𝜉  

𝑎 𝜉 𝛿𝜉 , 𝑣 𝛿𝑣 𝑎 𝜉 , 𝑣 𝛿𝑣 , 𝛿𝜉  
𝑝 𝜉 𝛿𝜉 , 𝑣 𝛿𝑣 𝑝 𝜉 , 𝑣 𝛿𝑣 , 𝛿𝜉  
𝑘 𝜉 𝛿𝜉 , 𝑣 𝛿𝑣 𝑘 𝜉 , 𝑣 𝛿𝑣 , 𝛿𝜉  

𝑎 𝜉 , 𝑣 𝛿𝑣 𝑎 𝜉 , 𝑣 , 𝛿𝜉 𝑝 𝜉 , 𝑣 𝛿𝑣 𝑝 𝜉 , 𝑣 , 𝛿𝜉  
𝑘 𝜉 , 𝑣 𝛿𝑣 𝑘 𝜉 , 𝑣 , 𝛿𝜉  𝑎 𝑣 𝛿𝑣 , 𝛿𝜉 𝑎 𝑣 , 𝛿𝜉  
𝑝 𝑣 𝛿𝑣 , 𝛿𝜉 𝑝 𝑣 , 𝛿𝜉 𝑘 𝑣 𝛿𝑣 , 𝛿𝜉 𝑘 𝑣 , 𝛿𝜉   ( 1 4 )  

By hypotheses (A-a&d), one has: 

𝜖 ∥ 𝛿�⃗� ∥   𝑎 𝑥, 𝜉 , 𝑣 𝛿𝑣 𝑎 𝑥, 𝜉 , 𝑣 𝛿𝜉
  

𝑑𝑥   𝑝 𝑥, 𝜉 , 𝑣 𝛿𝑣
 

𝑝 𝑥, 𝜉 , 𝑣 𝛿𝜉 𝑑𝑥   𝑘 𝑥, 𝜉 , 𝑣 𝛿𝑣 𝑘 𝑥, 𝜉 , 𝑣 𝛿𝜉 𝑑𝑥
 

𝑎 𝑥, 𝑣
 

𝛿𝑣 𝑎 𝑥, 𝑣  𝛿𝜉 𝑑𝑥 𝑝 𝑥, 𝑣 𝛿𝑣 𝑝 𝑥, 𝑣 𝛿𝜉 𝑑𝑥
 

𝑘 𝑥, 𝑣
 

𝛿𝑣 𝑘 𝑥, 𝑣 𝛿𝜉 𝑑𝑥  
By using Lipchitz condition on 𝑎 & 𝑎 , 𝑝 & 𝑝  and (𝑘 &𝑘  w.r.t. 𝑣 , 𝑣  ,𝑣  respectively 
and  Cauch-Schwarz Inequality (C-S-I) of the obtained inequality, to get:  



   

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 

𝛿�⃗� 𝐿 ‖𝛿𝑣 ‖ ‖𝛿𝜉 ‖ 𝐿 ‖𝛿𝑣 ‖ ‖𝛿𝜉 ‖ 𝐿 ‖𝛿𝑣 ‖ ‖𝛿𝜉 ‖   
        

 

‖ 𝛿𝜉 ⃗ 𝐿 𝛿�⃗� ,  w i t h   L 𝑚𝑎x , , 𝐿 𝑚𝑎𝑥 , , 𝐿 𝑚𝑎𝑥 ,   ( 1 5 )  

H y p o t h e s e s  B :  
Suppose that yℓ  ∀ ℓ 0,1,2 & 𝑖 1,2,3  is of Carathéodory type on Λ ℝ V , satisfies 
the following condition w.r.t. ξ , v , i.e.  
|𝑦ℓ 𝑥, 𝜉 , 𝑣 | 𝜗ℓ 𝑥 𝔠ℓ 𝜉 �̆�ℓ 𝑣 , where ξ , v ∈ ℝ v ,ϑℓ ∈ L Λ  and 𝔠ℓ , �̆�ℓ 0. 
L e m m a  4 . 2 :  With hypotheses (B), the functional �⃗� ⟼ 𝑌ℓ �⃗� ,(∀ℓ 0,1,2,) defines on 

𝐿² Λ is continuous.  

P r o o f :  hypotheses (B) and proposition 3.1, gives that 𝑦ℓ 𝑥, 𝜉 , 𝑣
 

𝑑𝑥 ∀ℓ 0,1,2, & 𝑖

1,2,3 , is continuous on 𝐿² Λ . Hence 𝑌ℓ �⃗�  is continuous on 𝐿² Λ . 
L e m m a  4 . 3 [ 1 8 ] .  Let 𝑦 ∶ Λ ℝ ⟶ ℝ  is of Carathéodory type on Λ ℝ , with  

│𝑦 𝑥, 𝜉, 𝑣 │ 𝜂 𝑥 ℂ𝑦 ℂ′𝑢 , where 𝜂 ∈ 𝐿 Λ, ℝ , ℂ, ℂ′ 0. 
Then 𝑦 𝑥, 𝜉, 𝑣 𝑑𝑥

 
 is continuous on 𝐿 Λ, ℝ

 
 , with 𝑣 ∈ 𝑉, 𝑉 ⊂ ℝ is compact. 

T h e o r e m  4 . 1 :  In addition to hypotheses (A & B), we suppose that the set of controls �⃗�, 

with �⃗� is convex and compact, 𝑈 𝜙, where 𝑎 , 𝑝  and 𝑘  are independent of 𝑣 , 𝑣  and 𝑣  
respectively, and 𝑎 , 𝑝  and 𝑘  are linear w.r.t. 𝑣 , 𝑣  and 𝑣  respectively, i.e. 
𝑎 𝑥, 𝜉 , 𝑣 𝑎 𝑥, 𝜉    ,    𝑝 𝑥, 𝜉 , 𝑣 𝑝 𝑥, 𝜉    ,    𝑘 𝑥, 𝜉 , 𝑣 𝑘 𝑥, 𝜉            
𝑎 𝑥, 𝑣 𝑎 𝑥 𝑣         ,      𝑝 𝑥, 𝑣 𝑝 𝑥 𝑣          ,    𝑘 𝑥, 𝑣 𝑘 𝑥 𝑣 , such that 
|𝑎 𝑥, 𝜉 | 𝜗 𝑥 �̂� |𝜉 |,  |𝑝 𝑥, 𝜉 | 𝜗 𝑥 �̂� |𝜉 |,    |𝑘 𝑥, 𝜉 | 𝜗 𝑥 �̂� |𝜉 | 
where 𝜗 , 𝜗 , 𝜗 ∈ 𝐿² Λ  and  �̂� , �̂� , �̂� 0 ,|𝑎 𝑥 | 𝑛  , |𝑝 𝑥 | 𝑛  ,     |𝑘 𝑥 | 𝑛  
𝑦  is independent of𝑣   and 𝑦ℓ (for 𝑙 0,2  and  𝑖 1,2,3) is convex w.r.t. 𝑣  for fixed 𝜉 ,  
then there exists TCCOCV. 

P r o o f :  Since �⃗� is convex and compact, then �⃗� is weakly compact. 

Since �⃗� ∅ then there exists 𝑢 ∈ 𝑈  and a minimum sequence �⃗� 𝑣 , 𝑣 , 𝑣 ∈

�⃗� , such that∀ �⃗� ∈ �⃗� ,∀𝑛 :  lim
→

𝑌 �⃗�  inf
⃗∈ ⃗

𝑌 𝑢 . 

Since �⃗� is weakly compact, then there exists a subsequence of �⃗� , (let it be again �⃗� ) 

which converges weakly to some  �⃗� ∈ 𝑈, i.e. �⃗� ⟶
 

 
�⃗� weakly in 𝐿 Λ and  ‖�⃗� ‖ �̃� , ∀𝑛. 

Now, by using (12), hypotheses  and C-S-I, give  

   𝜖 𝜉 𝐷 𝜉 , 𝜉 𝑎 𝑥, 𝑣 , 𝜉 𝑝 𝑥, 𝑣 , 𝜉 𝑘 𝑥, 𝑣 , 𝜉  

             | 𝑎 𝑥 𝑣 , 𝜉 | |𝑝 𝑥 𝑣 , 𝜉 | | 𝑘 𝑥 𝑣 , 𝜉 | 
             𝑛 𝔠 ‖𝜉 ‖ 𝑛 𝔠 ‖𝜉 ‖ 𝑛 𝔠 ‖𝜉 ‖                          

             𝑛 𝔠 𝑛 𝔠 𝑛 𝔠 𝜉 𝜔 𝜉 ,  w h e r e  ω max 𝑛 𝔠 , 𝑛 𝔠 , 𝑛 𝔠 0 

Then 𝜉 𝜇 , for each 𝑛 with  𝜇 0 ( i.e.  𝜉  is bounded ∀n) 

By Alaoglu theorem(Al.Th.) [19]. there exists a subsequence of 𝜉  , (let it be again 

𝜉  such that 𝜉
    
→ 𝜉 weakly in 𝑊, which mean that 𝜉 ⟶

 

  
𝜉 weakly in 𝐿 Λ , then by 

compactness theorem(Rellich–Kondrachov [20].)𝜉 ⟶
 

  
𝜉  strongly in  𝐿 Λ .Since for each 

n, 𝜉 𝜉 , 𝜉 , 𝜉  satisfies ( 1 1 ) , i . e .  



   

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B 𝜉 , 𝑤 𝑎 𝜉 , 𝑤  𝑝 𝜉 , 𝑤 𝑘 𝜉 , 𝑤   𝑎 𝑥 𝑣 , 𝑤
𝑝 𝑥 𝑣 , 𝑤  𝑘 𝑥 𝑣 , 𝑤  ( 1 6 )  

L e t  𝑤 , 𝑤 , 𝑤 ∈ 𝐶 Λ ,  t o  s h o w  t h a t  ( 1 6 )  c o n v e r g e s  t o  ( 1 7 ) , s u c h  t h a t  

𝐵 𝜉, 𝑤 𝑎 𝜉 , 𝑣 , 𝑤 𝑝 𝜉 , 𝑣 , 𝑤 𝑘 𝜉 , 𝑣 , 𝑤   𝑎 𝑣 , 𝑤
𝑝 𝑣 , 𝑤 𝑘 𝑣 , 𝑤  ( 1 7 )  

i .  Since 𝜉
          
⎯ 𝜉  𝑤𝑒𝑎𝑘𝑙𝑦 𝑖𝑛 𝑊

∀ , ,
𝜉

          
⎯ 𝜉 weakly in 𝐿 Λ    a n d   

          
⎯  weakly in 𝐿 Λ   

ii.  from the hypotheses on 𝑎 𝑥, 𝜉 , 𝑝 𝑥, 𝜉  𝑎𝑛𝑑 𝑘 𝑥, 𝜉  and by using the result of 
lemma4.2, give that 𝑎 𝑥, 𝜉

 
Ω

𝑤 𝑑𝑥, 𝑝 𝑥, 𝜉 𝑤  𝑑𝑥
 

  and 𝑘 𝑥, 𝜉 𝑤 𝑑𝑥
 

 are 

continuous w.r.t. 𝜉 , 𝜉   and  𝜉 respectively since 𝜉 ⟶
 

  
𝜉  s t r o n g l y  i n   𝐿 Λ ,  

then the L.H.S of (16) →L.H.S of (17). 
Also the convergence for the R.H.S of (16) to the R.H.S of (17) is obtained through (𝑣
      

𝑣  𝑤𝑒𝑎𝑘𝑙𝑦 𝑖𝑛 𝐿 Λ , (𝑖 1,2,3 . 

But 𝐶 Λ  is dense in 𝑊,  which gives 𝜉 ⟶ 𝜉 𝜉 ⃗   is a solution of the state equations in 

𝑊.  

From lemma4.2, 𝑌ℓ �⃗�  is continuous on 𝐿 Λ , for each ℓ 0,1,2. 
From the hypotheses on 𝑦ℓ 𝑓𝑜𝑟 ℓ 0,1,2  𝑎𝑛𝑑 𝑖 1,2,3 , and 𝜉 ⟶

 

  
𝜉  strongly in 𝐿 Λ , 

then 𝑌 �⃗� lim
→

𝑌 �⃗� , hence 𝑌 �⃗� 0. 

Now, to prove 𝑌ℓ �⃗� , ℓ 0,2  is W.L.Sc. w.r.t. (𝜉 , 𝑣  , 𝑖 1,2,3 . 

From hypotheses B, 𝑣 , 𝑣 , 𝑣 ∈ �⃗� almost everywhere (a.e.) in Λ and �⃗� is compact, 
hence 𝑌ℓ �⃗�  is satisfied the hypotheses of lemma4.3, and gets that  

𝑦ℓ 𝑥, 𝜉 , 𝑣 𝑑𝑥           
⎯ 𝑦ℓ 𝑥, 𝜉 , 𝑣 𝑑𝑥

  
  

Since𝑦ℓ 𝑥, 𝜉 , 𝑣   is convex w.r.t. 𝑣 , then 𝑦ℓ 𝑥, 𝜉 , 𝑣 𝑑𝑥
 

 is W.L.S. w.r.t.𝑣 , i.e. 

𝑦ℓ 𝑥, 𝜉 , 𝑣 𝑑𝑥
 

lim
 

→

𝑦ℓ 𝑥, 𝜉 , 𝑣 𝑑𝑥
 

  

                     lim
 

→

𝑦ℓ 𝑥, 𝜉 , 𝑣 𝑦ℓ 𝑥, 𝜉 , 𝑣 𝑑𝑥 lim
 

→

𝑦ℓ 𝑥, 𝜉 , 𝑣 𝑑𝑥
  

        

                     lim
 

→

𝑦ℓ 𝑥, 𝜉 , 𝑣 𝑑𝑥
 

 

H e n c e  𝑌 �⃗� lim
 

→

𝑌 �⃗�  lim
→

𝑌 �⃗�  inf
⃗∈ ⃗

𝑌 𝑢 ⟹ �⃗�  i s  a n  o p t i m a l  c o n t r o l  

 
5 .  The Necessary and the Sufficient Conditions for Optimality Hypotheses C: 

a) The functions 𝑎 , 𝑎 , 𝑝 , 𝑝 , 𝑘 , 𝑘  are of the Carathéodory type on   

    Λ × ℝ ℝ a n d  s a t i s f y  f o r  𝑥 ∈ Λ a n d  𝑑 , 𝑗 0 ,  𝑖 1,2,3 :  
𝑎 𝑥, 𝜉 , 𝑣 𝑑 ,   𝑝 𝑥, 𝜉 , 𝑣 𝑑 ,    𝑘 𝑥, 𝜉 , 𝑣 𝑑  ,   

         𝑎 𝑥, 𝜉 , 𝑣 𝑗 ,    𝑝 𝑥, 𝜉 , 𝑣 𝑗  ,     𝑘 𝑥, 𝜉 , 𝑣 𝑗    

b) The functions 𝑎 , 𝑝 , 𝑘  are of the Carathéodory type on Λ×ℝ,with      

𝑎 𝑥, 𝑣 𝑞 ,   𝑝 𝑥, 𝑣 𝑞 ,   𝑘 𝑥, 𝑣 𝑞   

 where 𝑥 ∈ Λ and 𝑞 0 , 𝑖 1,2,3 .  
c )  T he functions 𝑦ℓ  ,  𝑦ℓ ∀ℓ 0,1,2 & 𝑖 1,2,3  are of the Carathéodory     



   

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    type    on 𝛬 ℝ ℝ and satisfy the following conditions for  𝜂ℓ , �̂�ℓ  ∈ 𝐿 Λ : 
     𝑦ℓ 𝜂ℓ Υℓ |𝜉 | Υℓ |𝑣 | And 𝑦ℓ �̂�ℓ Υℓ |𝜉 | Υℓ |𝑣 |, with Υℓ , Υℓ
0 , 
        

T h e o r e m  5 . 1 :  With hypotheses A, B and C, the Hamiltonian is: 

𝐻 𝑥, 𝜉, 𝜁, �⃗� 𝜁 𝑎 𝑥, 𝑣 𝑎 𝑥, 𝜉 , 𝑣 𝑦 𝑥, 𝜉 , 𝑣 𝜁 𝑝 𝑥, 𝑣 𝑝 𝑥, 𝜉 , 𝑣  

                    𝑦 𝑥, 𝜉 , 𝑣 𝜁 𝑘 𝑥, 𝑣 𝑘 𝑥, 𝜉 , 𝑣 𝑦 𝑥, 𝜉 , 𝑣  

The adjoint vector 𝜁 , 𝜁 , 𝜁 𝜁 , 𝜁 , 𝜁  "equations "of (3.1- 3.4) are: 

𝐵 𝜁 𝜁 𝜁 𝜁 𝜁 𝑎 𝑥, 𝜉 , 𝑣 𝑦 𝑥, 𝜉 , 𝑣   , 𝑖𝑛 𝛬  ( 1 8 )  

𝐵 𝜁 𝜁 𝜁 𝜁 𝜁 𝑝 𝑥, 𝜉 , 𝑣 𝑦 𝑥, 𝜉 , 𝑣   , 𝑖𝑛 𝛬   ( 1 9 )  

𝐵 𝜁 𝜁 𝜁 𝜁 𝜁 𝑘 𝑥, 𝜉 , 𝑣 𝑦 𝑥, 𝜉 , 𝑣   , 𝑖𝑛 𝛬  ( 2 0 )  

𝜁 𝜁 𝜁 0    𝑜𝑛  𝜕Λ   ( 2 1 )  
Then the FD of 𝑌  is given by:  

�⃗� �⃗�  𝛿�⃗� 𝐻 ⃗
 

. 𝛿�⃗� 𝑑𝑥,     𝐻 ⃗

𝐻 𝑥, 𝜉, 𝜁, �⃗�

 𝐻 𝑥, 𝜉, 𝜁, �⃗�

𝐻 𝑥, 𝜉, 𝜁, �⃗�

𝜁 𝑎 𝑎 𝑦

𝜁 𝑝 𝑝 𝑦

𝜁 𝑘 𝑘 𝑦

 

P r o o f :  Rewriting the TAEqs (18)-(20) by their WF and then blending them together: 

𝐵 𝜁, 𝑤 𝜁 𝑎 𝜉 , 𝑣 , 𝑤 𝜁 𝑝 𝜉 , 𝑣 , 𝑤 𝜁 𝑘 𝜉 , 𝑣 , 𝑤  

𝑦 𝜉 , 𝑣 , 𝑤 𝑦 𝜉 , 𝑣 , 𝑤 𝑦 𝜉 , 𝑣 , 𝑤  ( 2 2 )  

w h e r e  𝐵 𝜁, 𝑤 𝑏 𝜁 , 𝑤 𝜁 , 𝑤 𝜁 , 𝑤 𝜁 , 𝑤 𝑏 𝜁 , 𝑤 𝜁 , 𝑤
𝜁 , 𝑤 𝜁 , 𝑤 𝑏 𝜁 , 𝑤 𝜁 , 𝑤 𝜁 , 𝑤 𝜁 , 𝑤  

   The WF of the TAEqs (22) has a unique solution; this can be proved using the same way 
which is used to prove the WF of the state equation (11). 

Now by substituting 𝑤 𝛿�⃗� in (22), once has: 

𝐵 𝜁, 𝛿�⃗� 𝜁 𝑎 𝜉 , 𝑣 , 𝛿𝜁 𝜁 𝑝 𝜉 , 𝑣 , 𝛿𝜁 𝜁 𝑘 𝜉 , 𝑣 , 𝛿𝜁  

𝑦 𝜉 , 𝑣 , 𝛿𝜁 𝑦 𝜉 , 𝑣 , 𝛿𝜁 𝑦 𝜉 , 𝑣 , 𝛿𝜁                          ( 2 3 ) 

Setting the solution 𝜉 𝛿�⃗� in (8)-(10) then subtracting (8)-(10) from those equations which 

are obtained by setting 𝜉 𝛿�⃗� , then setting 𝑤 𝜁 , 𝑤 𝜁 ,  𝑤 𝜁  and then blending 
them together, to get: 

𝐵 𝛿�⃗�, 𝜁 𝑎 𝜉 𝛿𝜉 , 𝑣 𝛿𝑣 𝑎 𝜉 , 𝑣 , 𝜁 𝑝 𝜉 𝛿𝜉 , 𝑣 𝛿𝑣
𝑝 𝜉 , 𝑣 , 𝜁 𝑘 𝜉 𝛿𝜉 , 𝑣 𝛿𝑣 𝑘 𝜉 , 𝑣 , 𝜁 𝑎 𝑣 𝛿𝑣 𝑎 𝑣 , 𝜁
 𝑝 𝑣 𝛿𝑣 𝑝 𝑣 , 𝜁 𝑘 𝑣 𝛿𝑣 𝑘 𝑣 , 𝜁   ( 2 4 )  

Now, from hypo. on 𝑎 , 𝑝 , 𝑘 , 𝑎 , 𝑝  𝑎𝑛𝑑 𝑘 , using proposition 3.1 and the Mean value 
theorem,  the FD of 𝑎 , 𝑝 , 𝑘 , 𝑎 , 𝑝  𝑎𝑛𝑑 𝑘  are exist, once get that:   

 𝐵 𝛿�⃗�, 𝜁 𝑎 𝛿𝜉 𝑎 𝛿𝑣 , 𝜁 𝑝 𝛿𝜉 𝑝 𝛿𝑣 , 𝜁 𝑘 𝛿𝜉 𝑘 𝛿𝑣 , 𝜁 

 𝑎 𝛿𝑣 , 𝜁 𝑝 𝛿𝑣 , 𝜁 𝑘 𝛿𝑣 , 𝜁 𝜀̃ 𝛿Ξ⃗ 𝛿Ξ⃗    ( 2 5 a )  

w h e r e  𝜀̃ 𝛿Ξ⃗ 𝛿Ξ⃗ 𝜀̃ 𝛿�⃗�, 𝛿�⃗�)
𝛿𝜉

𝛿�⃗�

⃗
,   

From the Minkowiski inequality and lemma 4.1, once obtain that: 



   

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𝜀̃ 𝛿Ξ⃗ 𝜀̃ 𝛿�⃗�, 𝛿�⃗� 𝜀̅ 𝛿�⃗� ,     𝛿Ξ⃗
𝛿𝜉
𝛿𝑣

𝒸 𝛿�⃗�     
                 

  

𝜀̃ 𝛿Ξ⃗ 𝛿Ξ⃗ 𝜀̃ 𝛿�⃗� 𝛿�⃗�   ,   where 𝜀̃ 𝛿�⃗�
        

0 , 𝑎𝑛𝑑 𝛿�⃗�
        

0  𝑎𝑠  𝛿�⃗�
        

0 

Hence 

 𝐵 𝛿�⃗�, 𝜁 𝑎 𝛿𝜉 𝑎 𝛿𝑣 , 𝜁 𝑝 𝛿𝜉 𝑝 𝛿𝑣 , 𝜁 𝑘 𝛿𝜉 𝑘 𝛿𝑣 , 𝜁

𝑎 𝛿𝑣 , 𝜁 𝑝 𝛿𝑣 , 𝜁 𝑘 𝛿𝑣 , 𝜁  𝜀̃ 𝛿�⃗� 𝛿𝑣      ( 2 5 b ) 

Now, from definition of the FD, hypotheses on 𝑦ℓ ℓ 0,2  , 𝑖 1,2,3  and by using the 
result of lemma 4.1, once obtain that: 

𝑌 �⃗� 𝛿�⃗� 𝑌 �⃗� 𝑦 𝜉 , 𝑣 𝛿𝜉 𝑦 𝜉 , 𝑣 𝛿𝑣 𝑑𝑥 𝑦 𝜉 , 𝑣 𝛿𝜉
  

𝑦 𝜉 , 𝑣 𝛿𝑣 𝑑𝑥 𝑦 𝜉 , 𝑣 𝛿𝜉 𝑦 𝜉 , 𝑣 𝛿𝑣 𝑑𝑥
 

𝜀̃ 𝛿�⃗� 𝛿𝑣      ( 2 6 )     

where 𝜀̃ 𝛿�⃗� → 0, 𝑎𝑛𝑑 𝛿�⃗�
        

0  𝑎𝑠  𝛿�⃗�
        

0 

By subtracting (23) from (25b), and substituting the rustle in (26), once get  

𝑌 �⃗� 𝛿�⃗� 𝑌 �⃗� 𝜁 𝑎 𝑎 𝑦 𝛿𝑣
 

𝑑𝑥 𝜁 𝑝 𝑝
 

𝑦 𝛿𝑣 𝜁 𝑘 𝑘 𝑦 𝛿𝑣
 

𝑑𝑥 𝜀̃ 𝛿�⃗� 𝛿𝑣   ( 2 7 )  

 Then from FD, we have that  �⃗� �⃗�  𝛿�⃗� 𝐻 ⃗
 

. 𝛿�⃗� 𝑑𝑥.                   

Note: In the proof of the theorem 5.1, we find the FD for the functional 𝑌 , so the same 
technique is used to find the FD for 𝑌  and 𝑌 . 
Theorem 5.2: Optimality Necessary Conditions  

( a) With hypotheses  A, B and C, assume �⃗� is convex, if �⃗� ∈ �⃗�  is optimal, then there exist 

multipliers 𝜆ℓ ∈ ℝ,( ℓ 0,1,2 with 𝜆 , 𝜆 0, ∑
ℓ

|𝜆ℓ| 1), such that the following The 

Kuhn- Tucker- Lagrange's Multipliers (K.T.L)  are satisfied: 

𝐻 ⃗ ⋅ 𝛿�⃗� 𝑑𝑥
 

0, ∀𝑢 ∈ �⃗�, 𝛿�⃗� 𝑢 �⃗�   ( 2 8 a )  

where  𝑦 ∑
ℓ

𝜆ℓ𝑦ℓ  and 𝜁 ∑
ℓ

𝜆ℓ𝜁ℓ , (𝑖 1,2,3) in the definition of 𝐻, and also  

𝜆 𝑌 �⃗� 0, (Transversality condition)  (28b) 

(b) If �⃗� is of the form 

�⃗� 𝑢 ∈ 𝐿 Λ, ℝ │𝑢 𝑥 ∈ 𝑉 , a. e. on Λ , with 𝑉 ⊂ ℝ,𝑖 1,2,3. 
Then (28a) is equivalent to the minimum element wise (29), where:  

𝐻 ⃗ . �⃗� min
⃗∈ ⃗

 𝐻 ⃗ . 𝑢   a.e. on Λ   ( 2 9 )  

P r o o f  : (a) From theorem 4.2, the functional 𝑌ℓ �⃗�  has a continuous FD at each �⃗� ∈ �⃗�, since 

the control �⃗� ∈ �⃗�  is optimal, then by K.T.L theorem there exist  multipliers  𝜆ℓ ∈ ℝ , 

ℓ 0,1,2, with  𝜆 ,𝜆 0, ∑
ℓ

|𝜆ℓ| 1, such that 𝜆 �⃗� ⃗ �⃗� 𝜆
 �⃗� ⃗ �⃗� 𝜆 �⃗� ⃗ �⃗� . 𝑢

�⃗� 0, ∀𝑢 ∈ 𝑈 and  𝜆 𝑌 �⃗� 0, substituting the FD of 𝑌ℓ �⃗�  ∀ℓ 0,1,2  in the above 
inequality, to get 

𝜁 𝑎 𝑎 𝑦
 

𝛿𝑣 𝜁 𝑝 𝑝 𝑦 𝛿𝑣 𝜁 𝑘 𝑘

𝑦 𝛿𝑣 𝑑𝑥 0  



   

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where 𝜁 ∑
ℓ

𝜆ℓ𝜁 ℓ,  𝑦 ∑
ℓ

𝜆ℓ𝑦 ℓ  , for 𝑖 1,2,3, 

⟹ 𝐻 ⃗ . 𝛿�⃗� 𝑑𝑥
 

0, ∀𝑢 ∈ �⃗�  𝛿�⃗� 𝑢 �⃗�. 

( b )  Le t  �⃗� 𝑢 ∈ 𝐿 Λ, ℝ │𝑢 𝑥 ∈ 𝑉 , a. e. on Λ ,  w i t h  𝑉 ⊂ ℝ,  𝑖 1,2,3 , 𝜇 i s  a  

" Lebesgue" measure on Λ, 𝑣  be a sequence in �⃗� ⃗ and assume 𝑆 ⊂ Λ be a measurable set 

such that  𝑢 𝑥
𝑢 𝑥 , if 𝑥 ∈ 𝑆
�⃗� 𝑥 , if 𝑥 ∉ 𝑆

   .  Hence (28a), becomes 

𝐻
 ⃗

. 𝑢 �⃗� 𝑑𝑥
 

0, for each such set 𝑆
         

𝐻 ⃗ . 𝑢 �⃗� 0, a.e. on Λ  

That is it satisfies in 𝜑  with  𝜑 ⋂𝜑 , 𝑤ℎ𝑒𝑟𝑒 𝜑 Λ Λ , with 𝜇 Λ 0, but 𝜑 is 

independent of 𝑛,with 𝜇 Λ 𝜑 0 and since �⃗�  is dense in �⃗� ⃗ , then 

 𝐻 ⃗ . 𝑢 �⃗� 0, a.e. on Λ        
𝐻

 ⃗
. �⃗� min

⃗∈ ⃗
 𝐻 ⃗ . 𝑢 a.e. on Λ. 

Theorem 5.3:  Optimality Sufficient Conditions: 

In addition to the hypotheses A,B&C, with �⃗� is convex , 𝑎 &𝑦 , 𝑝 &𝑦 , , 𝑘 &𝑦  are 
affine w.r.t (𝜉 , 𝑣 ,(𝜉 , 𝑣 , 𝜉 ,𝑣 , resp  𝑎 , 𝑝 , 𝑘  are affine w.r.t 𝑣 , 𝑣 , 𝑣  resp for each 𝑥 
,and 𝑦ℓ , ℓ 0,2, 𝑖 1,2,3  is convex w.r.t. (𝜉 , 𝑣  for each 𝑥. Then the necessary 
conditions in theorem5.2, with 𝜆 0, are also sufficient. 
P r o o f :  s uppose 
  𝑎 𝑥, 𝜉 , 𝑣 𝑎 𝑥 𝜉 𝑎 𝑥 𝑣 𝑎 𝑥 ,     𝑎 𝑥, , 𝑣 𝑎 𝑥 𝑣 𝑎 𝑥 ,  
  𝑝 𝑥, 𝜉 , 𝑣 𝑝 𝑥 𝜉 𝑝 𝑥 𝑣 𝑝 𝑥  ,     𝑝 𝑥, , 𝑣 𝑝 𝑥 𝑣 𝑝 𝑥 ,  
  𝑘 𝑥, 𝜉 , 𝑣 𝑘 𝑥 𝜉 𝑘 𝑥 𝑣 𝑘 𝑥  ,    𝑘 𝑥, , 𝑣 𝑘 𝑥 𝑣 𝑘 𝑥  ,  

And that�⃗� ∈ �⃗� , �⃗� is satisfied the K.T.L. and the Transversality condition i.e.  

𝐻 ⃗ 𝑥, 𝜉, 𝜁, �⃗� ⋅ 𝛿�⃗�𝑑𝑥
 

0, ∀𝑢 ∈ �⃗�         and        𝜆 𝑦 �⃗� 0 

Let 𝑌 �⃗� ∑  
ℓ

𝜆ℓ𝑦ℓ �⃗� , then�⃗� �⃗� 𝛿�⃗� ∑  
ℓ

𝜆ℓ�⃗�ℓ �⃗� 𝛿�⃗� 

                 ∑ 𝜆ℓℓ
 

𝜁 ℓ 𝑎 𝑎 𝑦 ℓ 𝛿𝑣 𝜁 ℓ 𝑝 𝑝 𝑦 ℓ 𝛿𝑣   

                      𝜁 ℓ 𝑘 𝑘 𝑦 ℓ 𝛿𝑣 𝑑𝑥  𝐻 ⃗ 𝑥, 𝜉, 𝜁, �⃗� ⋅ 𝛿�⃗�𝑑𝑥
 

0  

Let 𝑣 , 𝑣 , 𝑣  and �̅� , �̅� , �̅�  are two given controls, then 𝜉 𝜉 ,  𝜉 𝜉 , 𝜉

𝜉  and 𝜉̅ 𝜉̅ , 𝜉̅ 𝜉̅ , 𝜉̅ 𝜉̅ ) are their  corresponding solutions, substituting 

the pair (�⃗�, 𝜉  in (1)-(4) and multiplying the obtained equation by  𝜅 ∈ 0,1  once and once 

again the pair (�⃗̅�, 𝜉̅⃗  in (1)-(4) multiplying the obtained equation by 1 𝜅 , finally then 
blending together the obtained equations from each corresponding equations once  get: 

𝐵 𝜅𝜉 1 𝜅 𝜉̅ 𝜅𝜉 1 𝜅 𝜉̅ 𝜅𝜉 1 𝜅 𝜉̅ 𝜅𝜉 1 𝜅 𝜉̅  

𝑎 𝑥 𝜅𝜉 1 𝜅 𝜉̅ 𝑎 𝑥 𝜅𝑣 1 𝜅 �̅� 𝑎 𝑥 𝑎 𝑥 𝜅𝑣
1 𝜅 �̅� 𝑎 𝑥     ( 3 0 a )  

𝜅𝜉 1 𝜅 𝜉̅ 0  ( 3 0 b )   
𝐵 𝜅𝜉 1 𝜅 𝜉̅ 𝜅𝜉 1 𝜅 𝜉̅ 𝜅𝜉 1 𝜅 𝜉̅ 𝜅𝜉 1 𝜅 𝜉̅

𝑝 𝑥 𝜅𝜉 1 𝜅 𝜉̅ 𝑝 𝑥 𝜅𝑣 1 𝜅 �̅� 𝑝 𝑥 𝑝 𝑥 𝜅𝑣
1 𝜅 �̅� 𝑝 𝑥     ( 3 1 a )   

𝜅𝜉 1 𝜅 𝜉̅ 0  ( 3 1 b )   



   

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𝐵 𝜅𝜉 1 𝜅  𝜉̅ 𝜅𝜉 1 𝜅 𝜉̅ 𝜅𝜉 1 𝜅 𝜉̅ 𝜅𝜉 1 𝜅 𝜉̅

𝑘 𝑥 𝜅𝜉 1 𝜅 𝜉̅ 𝑘 𝑥 𝜅𝑣 1 𝜅 �̅� 𝑘 𝑥 𝑘 𝑥 𝜅𝑣
1 𝜅 �̅� 𝑘 𝑥    ( 3 2 a )  

𝜅𝜉 1 𝜅 𝜉̅ 0  ( 3 2 b )  

N o w ,  i f  w e  h a v e  t h e  c o n t r o l  v e c t o r  �⃗̿� �̿� , �̿� , �̿� ,  wi t h  �̿� 𝜅𝑣 1  𝜅 �̅� ,  
  �̿� 𝜅𝑣 1  𝜅 �̅� ,  �̿� 𝜅𝑣 1  𝜅 �̅� .  T h e n  f r o m  ( 3 0  a & b ) ,  ( 3 1  a & b ) ,  ( 3 2  
a & b ) ,  o n c e  g e t  t h a t   

𝜉̿ 𝜉 𝜉  𝜅𝜉 1  𝜅 𝜉̅  ,   

𝜉̿ 𝜉 𝜉  𝜅𝜉 1  𝜅 𝜉̅  ,  

𝜉̿ 𝜉 𝜉  𝜅𝜉 1  𝜅 𝜉̅  

are their corresponding solutions, i.e. 𝜉̿ , 𝜉̿ , 𝜉̿  is satisfied (1-4). So, the operator 𝑣 ⟼ 𝜉  

is convex- linear w.r.t (𝜉 , 𝑣 (𝑖 1,2,3 ), for each 𝑥 ∈ Λ. 
Now, since 𝑦 𝑥, 𝜉 , 𝑣  is affine w.r.t. (𝜉 , 𝑣 , for each 𝑥 ∈ Λ and from the convex –linear 

property of operators 𝑣 ⟼ 𝜉 , once gets that 𝑌 �⃗�  is convex-linear w.r.t 𝜉, �⃗� , ∀𝑥 ∈ Λ. 

The convexity of 𝑌ℓ �⃗�   (for ℓ= 0,2) w.r.t. 𝜉, �⃗� , for each 𝑥 ∈ Λ  is obtained from the 
hypotheses at each of 𝑦ℓ  is convex w.r.t. 𝜉 , 𝑣 ∀𝑥 ∈ Λ, ∀ℓ 0,2 , &𝑖 1,2,3  . Hence 

𝑌 �⃗�  is convex w.r.t (𝜉, �⃗�  in the convex set �⃗� �⃗� ⃗  and it has a continuous FD satisfied 

�⃗� �⃗� 𝛿�⃗� 0        
 𝑌 �⃗�  has a minimum at �⃗� ⟹ 𝑌 �⃗� 𝑌 𝑢 , ∀𝑢 ∈ �⃗� 

       
 

𝜆 𝑌 �⃗� 𝜆 𝑌 �⃗� 𝜆 𝑌 �⃗� 𝜆 𝑌 𝑢 𝜆 𝑌 𝑢 𝜆 𝑌 𝑢        (33) 
Now, let 𝑢 be an admissible control and since �⃗� is also admissible and satisfies the 

Transversality condition, then (33) becomes𝑌 �⃗� 𝑌 𝑢 , ∀𝑢 ∈ �⃗�  i.e. �⃗� is an optimal 
control for the problem. 
6 .  Conclusion 
     The existence and uniqueness theorem for the solution (continuous state vector) of the 
TNLEBVP is stated and proved successfully using the Mint-Browder theorem when the 
TCCOCV is given. Also, the existence theorem of a TCCOCV governing by the TNLEBVP 
is proved. The existence and uniqueness solution of the TAEqs related with the TNLEBVP is 
studied. The derivation of the FD of the Hamiltonian is obtained. Finally, the theorem of 
necessary conditions so as the sufficient condition theorem for optimality of the constrained 
problem are stated and proved.  
Acknowledgement 
     I wish to express my sincere thanks to the University of Mustansiriyah for providing a 
pleasurable and stimulating environment to carry out research. Also, I would like to extend 
our sincere esteems to the College of Science- Department of Mathematics for their timely 
support. 
 
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