Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 60 The Necessary Condition for Optimal Boundary Control Problems for Triple Elliptic Partial Differential Equations Department of Mathematics, College of Science , Mustansiriyah University, Baghdad, Iraq. hawasy20@yahoo.com jhawassy17@mustansiriyah.edu.iq Abstract In this work, we prove that the triple linear partial differential equations (PDEs) of the elliptic type (TLEPDEs) with a given classical continuous boundary control vector (CCBCVr) has a unique "state" solution vector (SSV) by utilizing the Galerkin's method (GME). Also, we prove the existence of a classical continuous boundary optimal control vector (CCBOCVr) ruled by the TLEPDEs. We study the existence solution for the triple adjoint equations (TAJEs) related with the triple state equations (TSEs). The Fréchet derivative (FDe) for the objective function is derived. At the end we prove the necessary "conditions" theorem (NCTh) for optimality for the problem. Keywords: boundary optimal control, triple linear partial differential equations of elliptic type, Fréchet derivative, necessary conditions. 1. Introduction In many scopes, the optimal control problem (OCPr) has a significant base of life problems, different examples for applications of such problems are studied in medicine [1], in aircraft [2], in electric power [3], in economic growth [4], and many other fields. This role push many investigators to study the OCPr for nonlinear ordinary differential equations (NONODEs) as [5], or for different types of linear PDEs (LPDEs) hyperbolic, parabolic and elliptic as in [6,7] and [8] respectively. However, many others interested to study the OCPr for couple nonlinear PDEs (CNONLPDEs) of these three types [9,10] and [10], whilst [11,12] and [13] studied these three types of the CNONLPDEs but involved a Neumann boundary control (NBC). On the other hand, [14,15], and [16] in 2019 studied OCPr for triple PDEs (TPDEs) of the three types, while [17] studied OCPr involving NBC Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/34.1.2557 Article history: Received 3,February,2020, Accepted15,March,2020, Published in January 2021 Jamil A. Ali Al-Hawasy Nabeel A. Thyab Al-Ajeeli file:///C:/Users/المجلة/Desktop/math%20عدد%20خاص/hawasy20@yahoo.com file:///C:/Users/المجلة/Desktop/math%20عدد%20خاص/hawasy20@yahoo.com mailto:jhawassy17@mustansiriyah.edu.iq 61 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 "OCPrNBC" for TPDEs of parabolic type (TPDEsP). All these investigations push us to seek the OCPrNBC governed by the TLEPDEs. In this paper and at first, we prove that the TLEPDEs with a given CCBCVr has a unique SSV utilizing the GME. Second we prove the existence theorem of CCBOCVr ruled by the TLEPDEs. We study the existence for the solution of the TAJE related with the TSEs. The FDe of the objective function is derived. At the end, the NCTh of optimality of is demonstrated. 2. Problem Description Let Ω be a bounded and open connected subset in R2 with "Lipshitz boundary" ∂Ω , the OCPr is considered by the "state vector equation" which consists of the TLEPDEs with the NBC. A1y1 + y1 − y2 − y3 = f1(x), in Ω (1) A2y2 + y1 + y2 + y3 = f2(x), in Ω (2) A3y3 + y1 − y2 + y3 = f3(x), in Ω (3) ∑ a1ij ∂y1 ∂n1 = u1 , 2 i,j=1 on ∂Ω (4) ∑ a2ij ∂y2 ∂n2 = u2 , 2 i,j=1 on ∂Ω (5) ∑ a3ij ∂y3 ∂n3 = u3 , 2 i,j=1 on ∂Ω (6) where Aryr = − ∑ ∂ ∂xi (arij(x) ∂yr ∂xj ) , r = 1,2,3, arij = arij(xij) ∈ L ∞(Ω) , and2i,j=1 (u1, u2, u3) = (u1(x), u2(x), u3(x)) ∈ (L2(∂Ω)) 3 is the NBC vector (NBCV), (y1, y2, y3) = (y1(x), y2(x), y3(x)) ∈ (H 1(Ω)) 3 is the SSV corresponding to NBCV, (f1, f2, f3) = (f1(x), f2 (x), f3(x)) ∈ (L2(Ω)) 3 is given functions, for all X ∈ Ω , and 𝑛𝑙, ∀ 𝑙 = 1,2,3 , is a unit vector normal on Σ. The controls are defined in the set W⃗⃗⃗ ⊂ (L2(∂Ω)) 3 , with W⃗⃗⃗ = {(u1, u2 , u3) ∈ (L2(∂Ω)) 3 |(u1, u2 , u3) ∈ U⃗⃗ ⊂ R 3 a. e in ∂Ω} Where U⃗⃗ is a convex set. The objective functional is defined Min u⃗⃗ ∈W⃗⃗⃗⃗ Go(u⃗ ) = 1 2 ‖y1 − y1d‖L2(Ω) 2 + 1 2 ‖y2 − y2d‖L2(Ω) 2 + 1 2 ‖y3 − y3d‖L2(Ω) 2 + α 2 ‖u1‖L2(∂Ω) 2 + α 2 ‖u2‖L2(∂Ω) 2 + α 2 ‖u3‖L2(∂Ω) 2 (7) Let V⃗⃗ = (V)3 = (H1(Ω)) 3 . The symbols (v , v)L2(Ω) , and ‖ V ‖L2(Ω) (‖ V ‖L2(∂Ω)) are the inner product (IP) and the norm in L2(Ω) (L2(∂Ω)), by (v , v) H1(Ω) , ‖V‖H1(Ω) the IN and the norm in H1(Ω) , By (v⃗ , v⃗ )L2(Ω) = ∑ (vi , vi) 2 i=1 and ‖ v⃗ ‖(L2(Ω)) 3 = ∑ ‖ vi‖L2(Ω) 3 i=1 the IP and the norm in (L2(Ω)) 3 , by (v⃗ , v⃗ )L2(Ω) = ∑ (vi , vi) 3 i=1 and ‖ v⃗ ‖ (H1(Ω)) 3 = ∑ ‖ vi‖H1(Ω) 3 i=1 the IP and the norm in V⃗⃗ and V⃗⃗ ∗ is the dual of V⃗⃗ . 3. Weak Formulation: The weak form (WFO) for (1-3) is obtained by multiplying their both sides by v1 ∈ V , v2 ∈ V and v3 ∈ V respectively, then integrating them and then using the generalized Green's theorem is applied for the terms that contain the derivatives of order two, to get: a1(y1, v1 ) − (y2 + y3 , v1 )L2(Ω) = (f1 , v1)L2(Ω) + (u1 , v1)L2(∂Ω) , ∀ v1 ∈ V (8) 62 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 a2(y2, v2 ) + (y1 + y3 , v2 )L2(Ω) = (f2 , v2)L2(Ω) + (u2 , v2)L2(∂Ω) , ∀ v2 ∈ V (9) a3(y3 , v3) + (y1 − y2 , v3 )L2(Ω) = (f3 , v3)L2(Ω) + (u3 , v3)L2(∂Ω) , ∀ v3 ∈ V (10) Adding (8) , (9) and (10) , to get : a(y⃗ , v⃗ ) = F ( v⃗ ), ∀ v⃗ ∈ V (11) where a(y⃗ , v⃗ ) = a1 (y1 , v1 ) − (y2 + y3 , v1 )L2(Ω) + a2 (y2 , v2 ) (y1 + y3 , v2 )L2(Ω) + a3 (y3 , v3 ) + (y1 − y2 , v3 )L2(Ω) (12a) ar(yr , vr ) = ∫ (∑ arij ∂yr ∂xi ∂vr ∂xj + yrvr 2 i,j=1 ) Ω dx , with ar(yr , vr ) ≥ C1r ‖yr‖H1(Ω) 2 , where C1r ≥ 0 , r = 1 , 2 , 3 |ar(yr , vr )| ≤ C2r ‖yr‖H1(Ω) 2 ‖Vr‖H1(Ω) , 2 , where C2r ≥ 0 , r = 1 , 2 , 3 a(y⃗ , y⃗ ) ≥ ∝1 ‖ y⃗ ‖ (H1(Ω)) 3 2 |a(y⃗ , v⃗ )| ≤ ∝2 ‖ y⃗ ‖ (H1(Ω)) 3 ‖ v⃗ ‖ (H1(Ω)) 3 Where ‖ y⃗ ‖ (H1(Ω)) 3 2 = ‖ y⃗ ‖ (L2(Ω)) 3 2 + ‖∇ y⃗ ‖ (L2(Ω)) 3 2 , and F( v⃗ ) = (f1 , v1)L2(Ω) + (u1 , v1)L2(∂Ω) + (f2 , v2)L2(Ω) + (u2 , v2)L2(∂Ω) + (f3 , v3)L2(Ω) + (u3 , v3)L2(∂Ω) (12b) Assumptions (A): a) a( y⃗ , v⃗ ) is coercive , i. e , a ( y⃗ , y⃗ ) ≥ C‖ y⃗ ‖ (H1(Ω)) 3 2 . b) |a( y⃗ , y⃗ )| ≤ C1‖ y⃗ ‖ (H1(Ω)) 3 2 ‖ v⃗ ‖ (H1(Ω)) 3 where c1 > 0 . c) |F( v⃗ )| ≤ C2‖ v⃗ ‖ (L2(Ω)) 3 , ∀ v⃗ ∈ V , c2 > 0 . To find the solution of (11), the GME is applied, and an approximation (APP) subspace V⃗⃗ n ⊂ V⃗⃗ ( V⃗⃗ n is the set of continuous function in Ω ) is chose, thus (11) will be in the following APP form: a( y⃗ n , v⃗ ) = F( v⃗ ), ∀ y⃗ n , v⃗ ∈ V⃗⃗ n (13) Theorem 3.1: If u⃗ ∈ (L2(∂Ω)) 3 , is a given NBCV, then problem (13) has a unique APP solution(APPS) y⃗ n ∈ V⃗⃗ n Proof: let {φ⃗⃗ 1 , φ⃗⃗ 2 , … … … , φ⃗⃗ n} span V⃗⃗ n , then the APPS of (13) is written by: y⃗ n = ∑ dj φ⃗⃗ j (x1 , x2) n j=1 (14) where ∅⃗⃗ j = ((4 ℓ mod (4 − ℓ))φk , (4 mod (ℓ + 1)) φk , ((4 + ℓ2)mod (ℓ)) φk ) , ℓ = 1, 2,3 j = k + n(ℓ − 1) and dj = dℓk is unknown constant , ∀ j = 1 , 2 , … … , n , with n = 3N By substituting (14) in (13) , with v⃗ = φ⃗⃗ i , we get: ∑ dj a(φ⃗⃗ j , φ⃗⃗ i) = F(φ⃗⃗ i ) , ∀ i = 1 , 2 , … , n n j=1 (15) It is clear that (15) is equivalent to the algebraic system. An×n Dn ×1 = bn ×1 (16) where An×n = (aij )n×n , aij = a (φ⃗⃗ j , φ⃗⃗ i) , bn ×1 = (b1, b2, … . bn) T , bi = F (φ⃗⃗ i ) 63 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 and Dn ×1 = (d1, d2, … . dn) T, i, j = 1 , 2 , … , n. Since An×n Dn ×1 = 0 ⟹ 𝑎(∑ 𝑑𝑗 𝑛 𝑗=1 �⃗� 𝑗, �⃗� 𝑖) = 0, then from using( A – a ) 𝑐‖∑ 𝑑�⃗� 𝑗 𝑛 𝑗=1 ‖(𝐻¹(Ω)) 2 2 ≤ ∑ 𝑑𝑖𝑎 𝑛 𝑖=1 (∑ 𝑑𝑗�⃗� 𝑗 𝑛 𝑗=1 , �⃗� 𝑖) = 0 The uniqueness (16) is obtained from its corresponding homogeneous system. Proposition 3.1 [6]: For any v⃗ in V⃗⃗ , V⃗⃗ n has a sequence {φ⃗⃗ n} with φ⃗⃗ n ∈ V⃗⃗ n , ∀ n for which φ⃗⃗ n → V⃗⃗ strongly in V⃗⃗ . Now, by Theorem 3.1, the following sequence of the WFO has a sequence for the solutions { y⃗ n }n=1 ∞ a( y⃗ n , φ⃗⃗ n ) = F( φ⃗⃗ n ), ∀ y⃗ n , φ⃗⃗ n ∈ V⃗⃗ n , ∀ n (17) Theorem 3.2: The sequence { y⃗ n }n=1 ∞ converges to y⃗ strongly in (H1(Ω)) 3 . Proof: we have y⃗ n is a solution of (17), then by ( A – a & c ) : ‖ y⃗ n ‖ (H1(Ω)) 3 ≤ C̅1, where C̅1 > 0, ∀n From the Alaoglu's theorem (Agth)[8], {y⃗ n} has a subsequence. It is not loss of generality to say again {y⃗ n} for which y⃗ n → y⃗ , weakly in V⃗⃗ . Now, let v⃗ ∈ V⃗⃗ be fixed, then L V⃗⃗ (w⃗⃗⃗ ) = a(w⃗⃗⃗ , v⃗ ) is a bounded linear functional i.e, L V⃗⃗ ∈ V⃗⃗ . To prove the sequence of the solutions {y⃗ n }n=1 ∞ of the WFO (17) converges to the solution of the WFO(11). Step 1: since y⃗ n → y⃗ weakly in V⃗⃗ and by Proposition3.1, φ⃗⃗ n → V⃗⃗ strongly in V⃗⃗ , then |a( y⃗ n , φ⃗⃗ n ) − a ( y⃗ , v⃗ )| ≤ |a( y⃗ n , φ⃗⃗ n − v⃗ ) + a( y⃗ n − y⃗ , v⃗ )| ≤ C1‖ y⃗ n ‖ (H1(Ω)) 3 ‖ φ⃗⃗ n − v⃗ ‖ (H1(Ω)) 3 + C2‖ y⃗ n − y⃗ ‖ (H1(Ω)) 3 ‖ v⃗ ‖ (H1(Ω)) 3 → 0 (18) Hence a( y⃗ n , φ⃗⃗ n ) → a( y⃗ , v⃗ ) (19) Step 2: since φ⃗⃗ n → v⃗ strongly in V⃗⃗ ⟹ φ⃗⃗ n ⟶ v⃗ weakly in V⃗⃗ , then F( φ⃗⃗ n ) ⟶ F( v⃗ ) The above two steps give the following a( y⃗ , v⃗ ) = F( v⃗ ) , ∀ v⃗ ∈ V⃗⃗ which means y⃗ is a solution in (11). Now, to prove y⃗ n → y⃗ strongly in V⃗⃗ , by using ( A − a), it follows that : C ‖ y⃗ − y⃗ n‖ (H1(Ω)) 3 ≤ a( y⃗ − y⃗ n , y⃗ ) − a( y⃗ , y⃗ n ) + a( y⃗ n , y⃗ n) = a( y⃗ − y⃗ n , y⃗ ) = Ly⃗⃗ ( y⃗ − y⃗ n ) → 0 Thus {y⃗ n} converges to y⃗ strongly in (H 1(Ω)) 3 . 4. Existence of a CCBOCVr: Lemma 4.1: The operator u⃗ − y⃗ u⃗⃗ is Lipschitz continuous from (L2(∂Ω)) 3 into (L2(Ω)) 3 and is satisfied ‖ ∆y⃗⃗⃗⃗ ‖ (L2(Ω)) 3 ≤ C3 ‖∆u⃗⃗⃗⃗ ‖(L2(∂Ω)) 3 , with C3 > 0 . 64 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 Proof: Let u1́ , u2́ , u3́ be controls of the WFO (11) y1́, y2́ and y3́ be their corresponding SSV, subtracting the obtaining WFO from (11), by letting ∆y1 = y1́ − y1 and ∆u = u1́ − u with v⃗ = ∆y⃗⃗⃗⃗ , to get a( ∆y⃗⃗⃗⃗ , ∆y⃗⃗⃗⃗ ) = ( ∆u1 , ∆y1 )(L2(∂Ω)) + ( ∆u2 , ∆y2 )(L2(∂Ω)) + ( ∆u3 , ∆y3 )(L2(∂Ω)) (20) which gives after using(A-a), the Cauchy inequality(CSIn) and then the trace operator to obtain C‖ ∆y⃗⃗⃗⃗ ‖ (H1(Ω)) 3 2 ≤ |a ( ∆y⃗⃗⃗⃗ , ∆y⃗⃗⃗⃗ ) | ≤ C1‖ ∆u⃗⃗⃗⃗ ‖(L2(∂Ω)) 3 ‖ ∆y⃗⃗⃗⃗ ‖ (H1(Ω)) 3 Then, ‖ ∆y⃗⃗⃗⃗ ‖ (H1(Ω)) 3 ≤ C2‖ ∆u⃗⃗⃗⃗ ‖(L2(∂Ω)) 3 where C2 = C1 C Since ‖ ∆y⃗⃗⃗⃗ ‖ (L2(Ω)) 3 ≤ C ‖ ∆y⃗⃗⃗⃗ ‖ (H1(Ω)) 3 , then the above inequality becomes ‖ ∆y⃗⃗⃗⃗ ‖ (L2(Ω)) 3 ≤ C3‖ ∆u⃗⃗⃗⃗ ‖(L2(∂Ω)) 3 , where C3 = C ∙ C2 (21) Lemma 4.2 [3]:The norm ‖ ‖L2(Ω) (or the norm ‖ ‖L2(∂Ω) ) is weakly lower semi continuous (WELSC) Lemma 4.3: The objective function (7) is WELSC. Proof: The norm ‖ ‖L2(∂Ω) is WELSC (by lemma 4.2 ), but when u⃗ n → u⃗ weakly in (L2(Ω)) 3 , then by using lemma 4.1 gives y⃗ n → y⃗ = y⃗ u⃗⃗ weakly in (L2(Ω)) 3 , then by using lemma 4.2 , ‖y⃗ − y⃗ n‖L2(Ω) 2 is WELSC, i.e , G0( u⃗ ) is WELSC. Lemma 4.4 [3]:The norm ‖ ‖L2(Ω) (‖ ‖L2(∂Ω) ) is strictly convex (SC) . Remark 4.1: by applying lemma 4.4, G0( u⃗ ) is (SC). Theorem 4.1: If Ui , ∀ i = 1 , 2 , 3 is bounded, then there is a CCBOCVr for the problem (8). Proof: Since Ui , ∀ i = 1 , 2 , 3 is bounded , then Wi ( ∀ i = 1 , 2 , 3) is a bounded and then W⃗⃗⃗ is bounded Since G0( u⃗ ) ≥ 0 , then there is a minimum sequence { u⃗ n } = {(u1n , u2n , u3n)} ∈ W⃗⃗⃗ , for each n , such that: lim n→∞ G0( u⃗ n ) = infw⃗⃗⃗ ∈W⃗⃗⃗⃗ G0( w⃗⃗⃗ ) From the coercive property of G0( u⃗ ) , and its infimum, there exists a constant C > 0 such that ‖ u⃗ n‖ (L2(∂Ω)) 3 ≤ C , ∀ n (22) Then by Agth, the sequence { u⃗ n } has a subsequence. It is not loss of generality to say again { u⃗ n } for which u⃗ n → u⃗ weakly in (L2(∂Ω)) 3 . From theorem3.1, for each control u⃗ n = (u1n , u2n , u3n) the TEPDEsE has a unique APPS y⃗ n = y⃗ un . To prove (for each n) { y⃗ n } , is bounded in V⃗⃗ , using (A – a & c) , CSIn and the trace operator , to get C‖ y⃗ n ‖ (H1(Ω)) 3 2 ≤ a ( y⃗ n , y⃗ n ) = F( y⃗ n) ≤ ℓ1‖ y1n‖L2(Ω) + c1‖ y1n‖H1(Ω) + ℓ2‖ y2n‖L2(Ω) + c2‖ y2n‖H1(Ω) + ℓ3‖ y3n‖L2(Ω) + c3‖ y3n‖H1(Ω) ≤ s‖ y⃗ n‖H1(Ω) then ‖ y⃗ n‖ (H1(Ω)) 3 ≤ a , for each n with a = s C > 0 . where r1 = max(ℓ1 , c1) , r2 = max(ℓ2 , c2) , r3 = (ℓ3 , c3) and s = max(r1 , r2 , r3). 65 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 Then by Agth { y⃗ n } has a subsequence. It is not loss of generality to say again { y⃗ n } for which y⃗ n → y⃗ weakly in V⃗⃗ , since ∀n y⃗ n satisfies the WFO (11) for each, or a( y⃗ n, v⃗ ) = Fn(v⃗ ) = ( f1 , v1)L2(Ω) + ( u1n , v1)L2(∂Ω ) + ( f2 , v2)L2(Ω) + ( u2n , v2)L2(∂Ω) +( f3 , v3)L2(Ω) + ( u3n , v3)L2(∂Ω) (23) To show (23) converges to a( y⃗ , v⃗ ) = F(v⃗ ) (24) First, since yin → yi weakly in L2(Ω) ∀ i (for yin → yi weakly in Vi ), then by CSIn, one has: │a1(y1n , v1 ) − (y2n + y3n , v1 )L2(Ω) + a2(y2n , v2 ) + (y1n + y3n , v2 )L2(Ω) + a3(y3n , v3 ) + (y1n − y2n , v3 )L2(Ω) − a1(y1 , v1 ) + (y2 + y3 , v1 )L2(Ω) − a2(y2 , v2 ) − (y1 + y3 , v2 )L2(Ω) − a3(y3 , v3 ) − (y1 − y2 , v3 )L2(Ω) ≤ (C1‖ y1n − y1‖H1(Ω) + ‖ y2n − y2‖L2(Ω) + ‖ y3n − y3‖L2(Ω) )‖ v1‖L2(Ω) +(C2‖ y2n − y2‖H1(Ω) + ‖ y1n − y1‖L2(Ω) + ‖ y3n − y3‖L2(Ω) )‖ v2‖L2(Ω) +(C3‖ y3n − y3‖H1(Ω) + ‖ y1n − y1‖L2(Ω) + ‖ y2n − y2‖L2(Ω) )‖ v3‖L2(Ω) → 0 Second, we have u⃗ n → u⃗ weakly in (L 2(∂Ω)) 3 , then the terms in the right hand side of (23) converges to the those in the right hand side of (24) . Thus (23) converges to (24) . But, we have u⃗ n → u⃗ weakly in (L2(∂Ω)) 3 and G0( u⃗ ) is WELSC, then G0( u⃗ ) ≤ lim n→∞ infu⃗⃗ n ∈W⃗⃗⃗⃗ G0( u⃗ n ) = lim n→∞ G0( u⃗ n ) = infw⃗⃗⃗ ∈W⃗⃗⃗⃗ G0( w⃗⃗⃗ ) Then G0( u⃗ ) = infw⃗⃗⃗ ∈W⃗⃗⃗⃗ G0( w⃗⃗⃗ ) , i.e, u⃗ a CCBOCVr. Applying Remark 4.1, gives us u⃗ which is unique. 5. The NCTh for Optimality: Theorem 5.1: The TAJEs(z1 , z2 , z3) = (z1u1 , z2u2 , z3u3) of the WFO of the TSEs (1-6) are given by: A1 z1 + z1 + z2 + z3 = ( y1 − y1d) , in Ω (25) A2 z1 − z1 + z2 − z3 = ( y2 − y2d) , in Ω (26) A3 z1 − z1 + z2 + z3 = ( y3 − y3d) , in Ω (27) ∂z1 ∂n1 = 0 , in ∂Ω (28) ∂z2 ∂n2 = 0 , in ∂Ω (29) ∂z3 ∂n3 = 0 , in ∂Ω (30) Then the FDe of G0 is given by : ( G0 ′ ( u⃗ ) , ∆u⃗⃗⃗⃗ ) L2(∂Ω) = (z +∝ u⃗ , ∆u⃗⃗⃗⃗ ) L2(∂Ω) Proof: Rewriting the TAJEs (25-30) by its WFO, adding them, then substituting v⃗ = ∆y⃗⃗⃗⃗ , once get the following WFO which has a unique solution z = z u⃗⃗ : a1(z1 , ∆y1 ) + (z2 + z3 , ∆y1 )L2(Ω) + a2(z2 , ∆y2 ) − (z1 + z3 , ∆y1 )L2(Ω) + a3(z3 , ∆y3 ) − (z1 − z2 , ∆y3 )L2(Ω) = (y1 − y1d , ∆y1 )L2(Ω) + (y2 − y2d , ∆y2 )L2(Ω) + 66 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 (y3 − y3d , ∆y3 )L2(Ω) (31) Utilizing (12a&b) in (11), then substituting v⃗ = z once and once again v⃗ = z and setting y⃗ + ∆y⃗⃗⃗⃗ instead of y⃗ , then subtracting the second obtained equation from the first one, to get a1(∆y1, z1 ) − (∆y2, z1 )L2(Ω) − (∆y3, z1 )L2(Ω) + a2(∆y2, z2 ) + (∆y1, z2 )L2(Ω) + (∆y3, z2 )L2(Ω) + a3(∆y3, z3 ) + (∆y1, z3 ) − (∆y2, z3 ) = (∆u1 , z1 )L2(∂Ω) + (∆u2 , z2 )L2(∂Ω) + (∆u3 , z3 )L2(∂Ω) (32) Subtracting (32) from (31), to get (∆u⃗⃗⃗⃗ , z ) L2(∂Ω) = (y⃗ − y⃗ d , ∆y⃗⃗⃗⃗ )L2(Ω) (33) Now, for the cost function, we have G0(u⃗ + ∆u⃗⃗⃗⃗ ) − G0(u⃗ ) = (y1 − y1d , ∆y1 )L2(Ω) + (y2 − y2d , ∆y2 )L2(Ω) + (y3 − y3d , ∆y3 )L2(Ω) + 1 2 ‖∆y⃗⃗⃗⃗ ‖ (L2(Ω)) 3 2 + ∝ 2 ‖∆u⃗⃗⃗⃗ ‖ (L2(∂Ω)) 3 2 (34) From (33) & (34), we get G0(u⃗ + ∆u⃗⃗⃗⃗ ) − G0(u⃗ ) = (z +∝ u⃗ , ∆u⃗⃗⃗⃗ )L2(∂Ω) + 1 2 ‖∆y⃗⃗⃗⃗ ‖ (L2(Ω)) 3 2 + ∝ 2 ‖∆u⃗⃗⃗⃗ ‖ (L2(∂Ω)) 3 2 (35) From lemma 4.1, it yield that 1 2 ‖∆y⃗⃗⃗⃗ ‖ (L2(Ω)) 3 2 + ∝ 2 ‖∆u⃗⃗⃗⃗ ‖ (L2(∂Ω)) 3 2 = є(∆u⃗⃗⃗⃗ )‖∆u⃗⃗⃗⃗ ‖ (L2(∂Ω)) 3 2 (36) Where є(∆u⃗⃗⃗⃗ ) ⟶ 0 , as ‖∆u⃗⃗⃗⃗ ‖ (L2(∂Ω)) 3 2 ⟶ 0 with є(∆u⃗⃗⃗⃗ ) = є1(∆u⃗⃗⃗⃗ ) + є2(∆u⃗⃗⃗⃗ ) Then from the FDe of G0 , and (35-36), once get: (G0 ′ (u⃗ ), ∆u⃗⃗⃗⃗ ) = (z +∝ u⃗ , ∆u⃗⃗⃗⃗ ) L2(∂Ω) . Theorem 5.2: The (CCBOC) of (1-6) is: G0 ′ (u⃗ ) = z +∝ u⃗ = 0 with y⃗ = y⃗ u⃗⃗ and z = z u⃗⃗ . Proof:If u⃗ is an optimal control of the problem, then G0(u⃗ ) = minw⃗⃗⃗ ∈W⃗⃗⃗⃗ G0( w⃗⃗⃗ ) , ∀ w⃗⃗⃗ ∈ (L2(∂Ω)) 3 i.e., G0 ′ (u⃗ ) = 0 ⟹ z = −∝ u⃗ , ∆u⃗⃗⃗⃗ = w⃗⃗⃗ − u⃗ The necessary condition for optimality is: (z +∝ u⃗ , u⃗ ) ≤ (z +∝ u⃗ , w⃗⃗⃗ ) , ∀ w⃗⃗⃗ ∈ (L2(∂Ω)) 3 . 6. Conclusions The existence and uniqueness theorem for the SSV of the TLEPDEs is proved successfully using the GME when the CCBCVr is given. The proof of the existence CCBOCVr ruled by the considered TLEPDEs is demonstrated. The studding of the existence solution of the TAJEs related with the TLEPDEs is demonstrated. The FDe is derived. Finally the NCTh of optimality for the considered problem is demonstrated. References 1. Grigorenko, N.L.; Grigorieva, Ѐ.V.; Roi, P.K.; Khailov, E.N. Optimal control problems for a mathematical model of the treatment of psoriasis. 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