129 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 Doi: 10.30526/33.1.2379 Abstract This paper deals with the continuous classical optimal control problem for triple partial differential equations of parabolic type with initial and boundary conditions; the Galerkin method is used to prove the existence and uniqueness theorem of the state vector solution for given continuous classical control vector. The proof of the existence theorem of a continuous classical optimal control vector associated with the triple linear partial differential equations of parabolic type is given. The derivation of the Fréchet derivative for the cost function is obtained. At the end, the theorem of the necessary conditions for optimality of this problem is stated and is proved. Keywords: continuous classical optimal control, triple parabolic partial differential equations, Galerkin Method, the necessary conditions for optimality. 1. Introduction Different applications for real life problems take a main place in the optimal control problems, for examples in medicine [1]. Robots [2]. Engineering [3]. Economic [4]. And many others fields. In the field of mathematics, optimal control problem ( OCP ) usually governing either by ordinary differential equations ( ODEs ) or partial differential equations( PDEs ), examples for OCP which are governing by parabolic or hyperbolic or elliptic PDEs are studied by [5-7]. Respectively, while which are governing by couple of PDEs ( CPDEs ) of parabolic or of hyperbolic or of elliptic type are studied by [8-10]. On the other hand [11-13]. Rre studied boundary OCP associated with CPDEs of parabolic, hyperbolic and elliptic; while [14]. Studied the OCP for triple PDEs (TPDEs) of elliptic type. These works push us to seek about the OCP for TPDEs of parabolic type. This work consists of the study of The Continuous Classical Optimal Control governing by Triple Linear Parabolic Boundary Value Problem Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Department of Mathematics, College of Science, Mustansiriyah of University, Baghdad, Iraq Jamil Amir Ali Al-Hawasy Mohammed A. K. Jaber Jhawassy17@uomustansiriyah.edu.iq Article history: Received 3 May 2019, Accepted 17 June 2019, Publish January 2020. hawasy20@ yahoo.com file:///D:/فزياء%20العدد%20الثالث%202019/Moaayed99@gmail.com file:///C:/Users/Mustafa/Desktop/2020/رياضيات/New%20folder/Jhawassy17@uomustansiriyah.edu.iq file:///C:/Users/Mustafa/Desktop/2020/رياضيات/New%20folder/جميل%20ا.docx 130 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 the continuous classical optimal control problem ( CCOCP ) , starting with the state and prove the existence theorem of a unique solution ( state vector solution SVS ) for the triple state equations (TSE) of PDEs of parabolic type ( TPPDEs ) by using the Galerkin method (GM) when the continuous classical control vector ( CCCV ) is fixed then it deals with the state and proof of the existence theorem of a continuous classical optimal control vector (CCOCV) , the solution vector of the triple adjoint equations ( TAPEs ) associated the (TPPDEs) is studied . The derivation of the Fréchet derivative (FD) for the cost function is obtained, at the end; the theorem of the necessary conditions for optimality (NCO) of this OCP is sated and proved. 2. Description of the problem Let Ω ⊂ 𝑅 2 , =( ), =[0,T]×Ω , Ĩ= [0,T] , =𝜕Ω, , the CCOCP consists of TSE are given by the following TPDEs : ( ) (1) ( ) in (2) ( ) in (3) with the following boundary conditions (BCs) and the initial conditions ( ICs) ( ) , on ∑ (4) ( ) , on ∑ (5) ( ) , on ∑ (6) ( ) ( ) , on Ω (7) ( ) ( ) , on Ω (8) ( ) ( ) , on Ω (9) where ( , , ) is a vector of given function for each ( ) , ⃗ ( , , ) ( ( )) is a CCCV and ( , , ) ( ( ) ) , is its corresponding SVS The set of admissible CCCV is defined by ⃗⃗⃗ ={ ( , , ) ( ( )) ∣( , , ) ⃗⃗ ⊂ 𝑅 a.e. in Q }, ⃗⃗ is convex. The cost function is defined for by ( ⃗ ) (‖ ‖ ‖ ‖ ‖ ‖ ) (‖ ‖ ‖ ‖ ‖ ‖ ) (10) ⃗ ; ( ) ⃗ { : =( , , ) ( ( )) , on𝜕 }. The weak form(wf) of problem (1- 9) when ( ( )) is given by 〈 〉 ( ) ( ) ( ) ( ) = ( ), (11.a) ( ) ( ( ) ), (11.b) 〈 〉 ( ) ( ) ( ) ( ) ( ) (11.a) ( ) ( ( ) ) , (11.b) 〈 〉 ( ) ( ) ( ) ( ) ( ) (11.a) ( ) ( ( ) ), (11.b) The following assumption is important to study the CCOCV problem (CCOCVP) 2.1. Assumption (A): The function ( ) is satisfied the following condition w.r.t. & , i.e. | | ( ) , where ( ) , ( ). 3. The Solution for the wf: Theorem 3.1: Existence of a Unique Solution for the wf: With assumption (A), for each given CCCV ⃗ ( ( )) , the wf(11−13 (has a unique solution ( ) with ( ( )) and ( ) ( ( )) 131 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 Proof: Let for each , ⃗ ⊂ ⃗ be the set of continuous and piecewise affine functions in let , be a basis of = ,and let be an approximate solution for the solution , then by Gm: ∑ ( ) ( ) (14) ∑ ( ) ( ) (15) ∑ ( ) ( ) (16) where ( ) is unknown function of , , . The wf (11–13) is approximated by using ( 14−16) as , 〈 〉 ( ) ( ) ( ) ( ) ( ), (11.a ) ( ) ( ), (11.b) 〈 〉 ( ) ( ) ( ) ( ) ( ), (11.a) ( ) ( ), (11.b) 〈 〉 ( ) ( ) ( ) ( ) ( ), (11.a) ( ) ( ), (11.b) where ( ) ( ) ⊂ ⊂ ( ) is the projection of , thus ( ) ( ) ∣∣ ∣∣ ∣∣ ∣∣ , Substituting (14−16) in (17−19) respectively and then setting , & Then the obtained equations are equivalent to the following linear system (LS) of order ODEs with ICs (which has a unique solution), i.e. ( ) ( ) ( ) ( ) (20.a) ( ) (20.b) ( ) ( ) ( ) ( ) (21.a ) ( ) (21.b ) ( ) ( ) 𝑅 ( ) ( ) (22.a ) ( ) (22.b ) Where A ( ) , ( , ),B = ( ) , = ( , ) + ( , ),D = ( ) , ( , ) , E = ( ) , ( , ) , F = ( ) , ( , ),G = ( ) = ( , ) + ( , ) , H = ( ) , ( , ) , K = ( ) , ( , ) ,M = ( ) , ( , ), N = ( ) , = ( , )+( , ) R ( ) , ( , ), W = ( ) , ( , ) , ( ), = ( ) , = ( ) , = ( , ) , ( ) = ( ( )) , (t) = ( ( )) , (0) = ( ( )) , = 1,2,3,…,n , = 1,2,3. To show the norm ‖ ⃗⃗⃗⃗ ‖ is bounded Since ( ) , then there exists a sequence * + with , such that strongly in ( ) , then from the projection theorem [15]. And (11.b), ‖ ‖ ‖ ‖ Then ‖ ‖ ‖ ‖ ⊂ , strongly in ( ) with ‖ ‖ , by the same way, one can show that ‖ ‖ & ‖ ‖ , then ‖ ⃗⃗⃗⃗ ‖ is bounded in ( ( )) . 132 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 The norms‖ ⃗⃗⃗⃗ ( )‖ . ( )/ and ‖ ⃗⃗⃗⃗ ( )‖ are bounded Setting , and in (11 –19) respectively, integrating w.r.t. from to , adding the obtained three equations , one gets ∫ 〈 〉 ∫ ‖ ‖ ∫ ,( ) ( ) ( )- (23) Using Lemma (1.2) in [11]. For the term in the L.H.S. of (23) and since the term is positive, using assumptions (A) for the R.H.S. of (23), it yields ∫ ‖ ( )‖ ∫ ∫ | | ∫ ∫ | || | ∫ ∫ | | ∫ ∫ | || | +∫ ∫ | | ∫ ∫ | || | ∫ ‖ ( )‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ∫ ‖ ‖ ∫ ‖ ‖ ∫ ‖ ‖ since‖ ‖ ́ ‖ ‖ , , ‖ ( )‖ . ‖ ( )‖ ∫ ‖ ‖ ́ ́ ́ , using the Continuous Bellman Gronwall Inequality ( BGI ) , one gets ‖ ( )‖ ( ) , , - ‖ ( )‖ . ( )/ ( ) ‖ ( )‖ ( ) . The norm ‖ ( )‖ ( ) is bounded Again for (23) by using Lemma (1.2) in [11]. For the R.H.S. The same results will be obtained (from the above steps ) and since ‖ ( )‖ is positive, equation (23) with =T , becomes ‖ ( )‖ ‖ ( )‖ ∫ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ , which gives ∫ ‖ ‖ ( ) , with ( ) . ( )/ , thus ‖ ‖ ( ) ( ). The solution convergence Let { ⃗ } be a sequence of subspaces of ⃗ , s.t. ⃗⃗ ⃗ there exists a sequence { } with ⃗ , n and strongly in ⃗ strongly in ( ( )) , since for each , with ⃗ ⊂ ⃗ , ( 17 19) has a unique solution ( , , ) , hence corresponding to the sequence of subspaces { ⃗ } , there exist a sequence of (approximation) problems like (17 19) now, by substituting ( ), in these equations for , one has 〈 〉 ( ) ( ) ( ) ( ) ( ) (24.a) ( ) ( ) (24.b) 〈 〉 ( ) ( ) ( ) ( ) ( ) (25.a) ( ) ( )), (25.b) 〈 〉 ( ) ( ) ( ) ( ) ( ) (26.a) ( ) ( ) (26.b) which has a sequence of solutions * + , ( ) but from the above steps we have ‖ ‖ ( ) and‖ ‖ ( ) are bounded , then by Alaoglu’s theorem (AT), there exists a subsequence of * + ,say again * + s.t. weakly in ( ( )) and 133 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 weakly in . ( )/ , multiplying both sides of (24.a ) , ( 25.a) (26.a) by ( ) , - respectively, s.t. ( ) , ,3, then integrating both sides w.r.t. on, - , then integrating by parts the terms in the L.H.S. of each one obtained equation, one gets ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ( ) ( ) ( ) ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ( ) ( ) ( ) ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ( ) ( ) ( ) Since ( ) } { ( ) ( ) since weakly in ( ) , also ( ) ,3. Then ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) (30.a) ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) (31.a) ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) (32.a) ( ) ( ) ( ) ( ) (30.b) ( ) ( ) ( ) ( ) (31.b) ( ) ( ) ( ) ( ) (32.b) since ( ) , then ∫ ( ) ( ) ∫ ( ) ( ) (30.c) ∫ ( ) ( ) ∫ ( ) ( ) (31.c) ∫ ( ) ( ) ∫ ( ) ( ) (32.c) which means (30 32) converge to ( 33–35) respectively , with ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) = ∫ ( ) ( ) ( ) ( ) (33) ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ( ) ( ) (34) 134 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ( ) ( ) (35) Case1: Choose , -, i.e. ( ) ( ) , ,3. Substituting in (33 35), using integration by parts for the terms in L.H.S. of each one of the obtained equations, to get ∫ 〈 〉 ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) (36) ∫ 〈 〉 (t)dt ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) (t) ∫ ( ) ( ) (37) ∫ 〈 〉 ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) (38) i.e. ( , ) is solution of the wf (11 13). case 2 : Choose , - s.t. ( ) & ( ) , using integration by parts for the term in the L.H.S. of (36) , one gets ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ( ( ) ) ( ) (39) subtracting (33) from (39) , one obtains that ( ) ( ) = ( ( ) ) ( ) , ( ) , , - ( ) ( ( ) ) i.e. the IC (11.b) holds . By the same above way one can show that ( ) ( ( ) ) & ( ) ( ( ) ) that means the ICs (12.b)&(13.b) are hold. The strongly convergence for : Substituting , and in (17.a),(18.a)&(19.a) respectively, adding the three obtained equations together , and then integrating the obtained equation from 0 to T, on the other hand substituting , & in (11.a),(12.a)&(13.a) respectively, adding them and then integrating the three obtained equations from 0 to T , to get ∫ 〈 〉 ∫ ( ) ∫ ,( ) ( ) ( )- (40) and ∫ 〈 〉 ∫ ( ) ∫ ,( ) ( ) ( )- (41) using Lemma (1.2) in [11]. For the terms in the L.H.S. of (40) and (41), they become ‖ ( )‖ ‖ ( )‖ ∫ ( ) ∫ ,( + , ) ( + , ) ( + , )]dt (42) ‖ ( )‖ ‖ ( )‖ ∫ ( ) ∫ ,( + , ) ( + , ) ( + , )]dt (43) Since ‖ ( ) ( )‖ ‖ ( ) ( )‖ ∫ ( ) (44) where ‖ ( )‖ ‖ ( )‖ ∫ ( ( ) ( )) ( ( ) ( )) ( ( ) ( )) ∫ ( ( ) ( )) ( ( ) ( ) ( )) ( ( ) ( ) ( )) ∫ ( ( ) ( ) ( )) 135 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 since ( ) ( ) strongly in ( ( )) (44.a) ( ) ( ) strongly in( ( )) (44.b) Then ( ( ) ( ) ( )) ( ( ) ( ) ( )) (44.c) ‖ ( ) ( )‖ and ‖ ( ) ( )‖ (44.d) Since weakly in ( ( )) ,then ∫ ( ( ) ( ) ( )) (44.e) also since weakly in ( ( )) , then ∫ ,( ) ( ) ( )- ∫ , ( ) ( ) ( )- (44.f ) i.e. when in both sides of (44), one has the following results : (1)The first two terms in the L.H.S. of (44) are tending to zero (from (44.d)), (2) Eq. ( ) ∫ , ( ) ( ) ( )- ( ) ∫ , ( ) ( ) ( )- (3) Eq. L.H.S. of (43) ∫ , ( ) ( ) ( )- (4) The two terms in Eq. are tending to zero from (44.c), and the last one term also tend to zero from (44.e), from these results (44) gives when ∫ ‖ ‖ ∫ ( ) strongly in ( ( )) . Uniqueness of the solution: Let ⃗ , ̅ are two solutions of the wf ( 11–13), i.e. and ̅ are satisfied (11.a), or 〈 〉 ( ) ( ) ( ) ( ) , 〈 ̅ 〉 ( ̅ ) ( ̅ ) ( ̅ ) ( ) , Subtracting the equation from the one and substituting ̅ in the obtained equation, one gets that 〈( ̅ ) ̅ 〉 ( ̅ ̅ ) ( ̅ ̅ ) ( ̅ ̅ ) (45) By the similar manner, one gets 〈( ̅ ) ̅ 〉 ( ̅ ̅ ) ( ̅ ̅ ) ( ̅ ̅ ) (46) 〈( ̅ ) ̅ 〉 ( ̅ ̅ ) ( ̅ ̅ ) ( ̅ ̅ ) (47) Adding (45 47), using Lemma(1.2) in [11]. In the term of the obtained equation , to get ‖ ̅ ‖ ‖ ̅ ‖ (48) since the term of the L.H.S. of (48) is positive, integrating both sides of (48 ) w.r.t. from to , one gets ∫ ‖ ̅ ‖ ‖( ̅ )( )‖ ‖ ̅ ‖ , integrating both sides of (48) from to , using the given ICs and the above result, one has 136 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 ∫ ‖ ̅ ‖ ‖ ̅ ‖ ( ) ̅ . 4. Existence of a CCOCP Theorem 4.1: In addition to assumptions (A), assume that and are the SVS corresponding to the CVS ⃗ and ⃗ ⃗ respectively with ⃗ and ⃗ are bounded in ( ( )) then ‖ ‖ . ( )/ ‖ ⃗ ‖ , ‖ ‖ ( ) ‖ ⃗ ‖ , ‖ ‖ ( ) ‖ ⃗ ‖ , Proof : let ⃗ ( ) ( ( )) then by Theorem 3.1 there exists ( ) which is satisfied (11 13) and also let ̅ ( ̅ ̅ ̅ )be the solution of (11 13) corresponds to the cv ⃗̅ ( ̅ ̅ ̅ ) ( ( )) i.e. 〈 ̅ 〉 ( ̅ ) ( ̅ ) ( ̅ ) ( ̅ ) ( ̅ ) (49.a) ( ̅ ( ) ) ( ) (49.b) 〈 ̅ 〉 ( ̅ ) ( ̅ ) ( ̅ ) ( ̅ ) ( ̅ ) (50.a) ( ̅ ( ) ) ( ) (50.b) 〈 ̅ 〉 ( ̅ ) ( ̅ ) ( ̅ ) ( ̅ ) ( ̅ ) (51.a) ( ̅ ( ) ) ( ) (51.b) subtracting (11.a&b) from (49.a&b) , (12.a&b) from (50.a&b) ,and (13.a&b) from (51.a&b) and setting ̅ ̅ , ̅ , ̅ ̅ and ̅ in the obtained equations , they give 〈 〉 ( ) ( ) ( ) ( ) ( ) (52.a) ( ( ) ) (52.b) 〈 〉 ( ) ( ) ( ) ( ) ( ) (53.a) ( ( ) ) (53.b) 〈 〉 ( ) ( ) ( ) ( ) ( ) (54.a) ( ( ) ) (54.b) substituting , & in (52.a&b) , (53.a&b) and (54.a&b) respectively, adding the obtained equations, using Lemma (1.2) in [11]. They give ‖ ‖ ‖ ‖ ( ) ( ) ( ) (55) since the term of (55) is positive , integrating w.r.t. From to , and then using the Cauchy Schwartz inequality ( CSI ), it becomes ∫ ‖ ‖ dt 2 ∫ ∫ | || | 2∫ ∫ | || | ∫ ∫ | || | ∫ ‖ ⃗ ‖ ∫ ‖ ‖ , , -, by the BGI , once get ‖ ( )‖ ‖ ⃗ ‖ ‖ ( )‖ ‖ ⃗ ‖ , where , - ‖ ‖ . ( )/ ‖ ⃗ ‖ since ‖ ‖ ( ) ‖ ⃗ ‖ , then ‖ ‖ ( ) ‖ ⃗ ‖ , ̅ Using a similar way which is used in above steps, gives ∫ ‖ ‖ ∫ ‖ ‖ ‖ ⃗ ‖ ∫ ‖ ‖ ̅ ‖ ⃗ ‖ 137 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 ‖ ‖ ( ) ̅ ‖ ⃗ ‖ where ̅ ( ̅ ) ‖ ‖ ( ) ‖ ⃗ ‖ . Theorem 4.2: With assumptions (A), the operator ⃗ ⃗⃗ is continuous from ( ( )) in to ( ( ( ))) or in to ( ( )) or in to ( ( )) . Proof: Let ⃗ ⃗̅ ⃗ and ̅ , where ̅ are the correspond SVS to the CVS ⃗̅ and ⃗ using the first result in Theorem 4.1 , one has ‖ ̅ ‖ . ( )/ ‖ ⃗̅ ⃗ ‖ , If ⃗̅ ( ) → ⃗ then ̅ . ( )/ → i.e. The operator ⃗ ⃗⃗ is Lipschitz continuous (LC) from ( ( )) in to( ( ( ))) By a similar way this operator is also LC from ( ( )) into ( ( )) and into ( ( )) . Lemma 4.1 [10]: The norm is weakly lower semi continuous (W.L.S.C. ). Lemma (4.2) : The cost function which is given by ( ) is W.L.S.C. Proof: From Lemma (3.1) ‖ ⃗ ‖ ( ) , is W.L.S.C. when ⃗ ⃗ weakly in ( ( )) then weakly in ( ( )) by Theorem 4.1 then ‖ ⃗⃗ ⃗⃗ ‖ ‖ ⃗⃗ ⃗⃗ ‖ Then ‖ ⃗ ⃗ ‖ is W.L.S.C., hence ( ⃗ ) is W.L.S.C. Lemma 4.3 [13]: The norm ‖ ‖ is strictly convex. Theorem 4.3: Consider the cost function ( ) , if ( ⃗ ) is coercive, then there exists CCOC . Proof: Since ( ⃗ ) and ( ⃗ ) is coercive, then there exists a minimizing sequence* ⃗ + *( )+ ⃗⃗⃗ , such that ( ⃗ ) ⃗⃗̅ ⃗⃗⃗ ( ⃗̅ ) , and ‖ ⃗ ‖ , then by AT there exists a subsequence of * ⃗ + , for simplicity say again * ⃗ + s.t. ⃗ ⃗ weakly in ( ( )) , as . From Theorem 3.1 we got that for each control ⃗ there exists a unique solution ⃗⃗ then corresponding to the sequence of control * ⃗ + there exists a sequence of solutions * + such that the norms ‖ ‖ . ( )/ , ‖ ‖ ( ) & ‖ ‖ ( ) are bounded ,then by AT there exists a subsequence of * + , for simplicity say again * + , such that weakly in ( . ( )/) , weakly in ( ( )) , and weakly in . ( )/ , to show the norm ‖ ‖ . ( )/ is bounded , let (2.19.a),(2.20.a)&(2.21.a) be rewritten as 〈 〉 ( ) ( ) ( ) ( ) ( ) 〈 〉 ( ) ( ) ( ) ( ) ( ) 〈 〉 ( ) ( ) ( ) ( ) ( ) . by adding the above equations and integrating both sides of the obtained equation from to , taking the absolute value then using the CSI , and finally using assumptions (A) , it yields |∫ 〈 〉 | |∫ , ( ) ( ) ( ) ( ) ( ) ( ) – ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ]dt| , gives |∫ 〈 〉 | S‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ 138 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ . Since for each i = 1,2,3 the following inequalities are satisfied ‖ ‖ ‖ ‖ , ‖ ‖ ‖ ‖ , ‖ ‖ ‖ ⃗ ‖ , ‖ ⃗ ‖ , ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ , ‖ ‖ ‖ ‖ ( ) , ‖ ‖ , ‖ ‖ ‖ ‖ ( ) , ‖ ‖ ‖ ‖ ( ), ‖ ‖ ‖ ‖ ( ) , then |∫ 〈 〉 | , ‖ ‖ ( ) ‖ ‖ ( ) ( )-‖ ‖ ( ) Or |∫ 〈 〉 | . ( ) ( )/ ‖ ‖ ( ) , where ‖ ‖ ( ) ( ) and ( ) |∫ 〈 ⃗ ⃗ 〉 | ‖ ⃗ ‖ ( ) ( ) , with ( ) ( ) ( ) ‖ ‖ ( ) ( ) since is solution of the SEs (1 ) , then 〈 〉 ( ) ( ) ( ) ( ) = ( ) (56) 〈 〉 ( ) ( ) ( ) ( ) =( ) (57) 〈 〉 ( ) ( ) ( ) ( ) =( ) (58) let , -, s.t. ( ) , ,3, rewriting the terms in the L.H.S. of (56–58) multiplying their both sides by ( ) , ( ) & ( ) respectively , integrating both sides w.r.t. from to ,and integration by parts for the terms in the L.H.S. of each obtained equation, one gets that ∫ ( ) ( ) ∫ ,( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ( ( ) ) ( ) (59) ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ( ( ) ) ( ) (60) ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ( ( ) ) ( ) (61) since weakly in ( ( )) and weakly in . ( )/ , then the following convergences are hold ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) (62) ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) (63) 139 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) (64) since ( ( ) ( ) ( ) ) bounded in( ( ) ) and from the projection theorem one has ( ) ( ) ( ) ( ) (65) ( ) ( ) ( ) ( ) (66) ( ) ( ) ( ) ( ) (67) and since ⃗ ⃗ weakly in ( ( ) ) , then ∫ ( ) ( ) ∫ ( ) ( ) (68) ∫ ( ) ( ) ∫ ( ) ( ) (69) ∫ ( ) ( ) ∫ ( ) ( ) (70) finally using(62 64) ,( 65 67), ( 68 70) in (59 61) respectively, one gets ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ( ) ( ) (71) ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ( ) ( ) (72) ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ( ) ( ) (73) Case 1: We choose , - , i.e. ( ) ( ) , .now using integration by parts for the terms in the L.H.S. of (71 73), one gets that ∫ 〈 〉 ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) , - (74) ∫ 〈 〉 ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) , - (75) ∫ 〈 〉 ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) , - (76) Then 〈 〉 ( ) ( ) ( ) ( ) ( ) , , a.e. on 〈 〉 ( ) ( ) ( ) ( ) = ( ) , , a.e. on 〈 〉 ( ) ( ) ( ) ( ) = ( ) , i.e. ( , , ) satisfies the wf of the SEs Case 2 : we choose , - , s.t. ( ) , ( ) , using integration by parts for the terms in the L.H.S. of (74 76),one has ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ( ( ) ) ( ) (77) 140 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ( ( ) ) ( ) (78) ∫ ( ) ( ) ∫ , ( ) ( )- ( ) ∫ ( ) ( ) ∫ ( ) ( ) ∫ ( ) ( ) ( ( ) ) ( ) (79) by subtracting(77 79) from (71 73) respectively, one obtain ( ) ( ) =( ( ) ) ( ) , - ( ) ( ) , by the same way show that ( ) ( ) and ( ) ( ) ( , , ) is a solution of the wf of the SE ,since ( ⃗ ) is W.L.S.C. from Lemma4.1 and since ⃗ ⃗̅ weakly in ( ( )) , then ( ⃗ ) ⃗⃗ ⃗⃗⃗ ( ⃗ ) ( ⃗ ) ( ⃗̅ ) ( ⃗ ) ⃗⃗̅ ⃗⃗⃗ ( ⃗̅ ) ⃗⃗̅ ⃗⃗⃗ ( ⃗̅ ) . Then ⃗ is a CCOC. 5. The NCO: In order to state the NCs for CCOC, the FD of the cost function (10) is derived and the theorem for the NCO is proved. Theorem 5.1: Consider ( ⃗ ) is given by (10) and the TAEs of the STE (1-9) are given by ( ) (80) ( ) (81) ( ) (82) ( ) (83) ( ) (84) ( ) (85) Then ( ) ⃗⃗⃗ and the FD of is given by ( ( ⃗ ) ⃗ ) ( ⃗ ⃗ ) proof: The wf of (80–85) for is given by 〈 〉 ( ) ( ) ( ) ( ) ( ) (86) 〈 〉 ( ) ( ) ( ) ( ) = ( ) (87) 〈 〉 ( ) ( ) ( ) ( ) = ( ) (88) The existence of a unique solution of (86 88) can be proved by the same manner which is used in the proof of Theorem in (52.a) ,( 53.a) and (54.a) respectively, these equations become , 〈 〉 ( ) ( ) ( ) ( ) ( ) (89) 〈 〉 ( ) ( ) ( ) ( ) ( ) (90) 〈 〉 ( ) ( ) ( ) ( ) ( ) (91) also, substituting and in (86 88) respectively, to get 〈 〉 ( ) ( ) ( ) ( ) ( ) (92) 〈 〉 ( ) ( ) ( ) ( ) ( ) (93) 〈 〉 ( ) ( ) ( ) ( ) ( ) (94) Integrating both sides of equations (89 94) w.r.t. t from 0 to T, using integration by parts for the terms of the L.H.S. of each of the obtained equations from (92–94), then subtracting each one of the obtained equations from it's corresponding equation of (89–91), add all three result get 141 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 〈( ) 〉 ( ( ) ) ( ) ( ) ( ) ( ) (95) 〈( ) 〉 ( ( ) ) ( ) ( ) ( ) ( ) (96) 〈( ) 〉 ( ( ) ) ( ) ( ) ( ) ( ) (97) which means the CV ( , , ) gives the solution ( , ) of ( 95 97). Now, from the cost function, we have ( ⃗ ⃗ ) ( ⃗ ) = ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ‖ ‖ ‖ ⃗ ‖ , Or ( ⃗ ⃗ ) ( ⃗ ) = ( ⃗ ⃗ ) ‖ ‖ ‖ ⃗ ‖ from the first results of Theorem 4.1, we have ‖ ‖ ( ⃗ )‖ ⃗ ‖ and ‖ ⃗ ‖ ( ⃗ )‖ ⃗ ‖ with ( ⃗ ) ‖ ⃗ ‖ where ( ⃗ ) ( ⃗ ) , as ‖ ⃗ ‖ Then ( ⃗ ⃗ ) ( ⃗ ) ( ⃗ ⃗ ) ( ⃗ )‖ ⃗ ‖ with ( ⃗ ) ( ⃗ ) ( ⃗ ) where ( ⃗ ) as ‖ ⃗ ‖ using the definition of FD of ,one has ( ( ⃗ ) ⃗ ) ( ⃗ ⃗ ) Theorem 5.2: The CCOC of the above problem is ( ⃗ ) ⃗ with ⃗⃗ and ⃗⃗ . Proof : If ⃗ is an CCOC of the problem, then ( ⃗̅ ) ⃗⃗ ⃗⃗⃗ ( ⃗ ) , ⃗ ( ( )) , i.e. ( ⃗̅ ) ⃗̅ The NCO is ( ⃗̅ ⃗ ) ⃗ ⃗⃗ ⃗̅ ( ⃗̅ ⃗⃗ ) ( ̅ ⃗⃗⃗ ⃗̅ ), ⃗⃗ ( ( )) 6. Conclusions The GM is employed to prove the existence and unique theorem for a SVS of the TSPDEs of parabolic type for fixed CCCV. 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