341 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 J. A. Al-Hawasy Doaa Kateb Jasim Abstract In this paper the Galerkin method is used to prove the existence and uniqueness theorem for the solution of the state vector of the triple linear elliptic partial differential equations for fixed continuous classical optimal control vector. Also, the existence theorem of a continuous classical optimal control vector related with the triple linear equations of elliptic types is proved. The existence of a unique solution for the triple adjoint equations related with the considered triple of the state equations is studied. The Fréchet derivative of the cost function is derived. Finally the theorem of necessary conditions for optimality of the considered problem is proved. Keyword: Triple linear equations of elliptic type, optimal control (vector) of continuous classical type. 1. Introduction Optimal control problems are a fundamental tool in many fields of applied mathematics and taken an important role in many aspects of life, for example in an electric power [1]. In robotics [2]. In biology [3]. In economic [4]. In medicine as [5]. In heat condition [6]. And in many others aspects. This importance encouraged researchers to study problems for the optimal control related with nonlinear ordinary differential equations [7]. Or related with different types of nonlinear partial differential equation as hyperbolic, parabolic, elliptic [8- 10]. Or related with couple of nonlinear hyperbolic, parabolic and elliptic partial differential equation [11-13]. While many others researchers studied the Numann boundary optimal control problems related with couple of nonlinear hyperbolic, parabolic and elliptic partial differential equation [14-16]. This article deals with; the existence theorem for a unique solution (continuous state vector (CSV)) for the triple linear elliptic partial differential equations (TLEPDEqs) is sated, studied and proved by using the Galerkin Method (GM) for fixed continuous classical optimal control vector (CCOCV). The existence theorem for a continuous classical optimal control vector (CCOCV) related with the TLEPDEqs is state and proved. The existence for the unique solution of the triple adjoint equations (TAEqs) which corresponds to the TLEPDEqs is studied. The Fréchet derivative (FD) of the cost function is Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/33.1.2380 Department of Mathematics, College of Science, University of Mustansiriyah Jhawassy17@uomustansiriyah.edu.iq hawasy20@yahoo.com The Continuous Classical Optimal Control Problems for Triple Elliptic Partial Differential Equations Article history: Received 13 May 2019, Accepted 11 June 2019, Publish January 2020. file:///C:/Users/المجلة/Desktop/عدد%20خاص/New%20folder/العدد%20كامل%20ومعدل/Jhawassy17@uomustansiriyah.edu.iq, file:///C:/Users/المجلة/Desktop/عدد%20خاص/New%20folder/العدد%20كامل%20ومعدل/Jhawassy17@uomustansiriyah.edu.iq, file:///C:/Users/المجلة/Desktop/عدد%20خاص/New%20folder/العدد%20كامل%20ومعدل/hawasy20@yahoo.com file:///C:/Users/المجلة/Desktop/عدد%20خاص/New%20folder/العدد%20كامل%20ومعدل/hawasy20@yahoo.com 344 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 derived; finally the theorem for necessary conditions of optimality (NCO) is stated and proved. 2. Problem Description Let Λ be a bounded and open connected subset in with Lipschitz boundary ∂Λ. Consider the CCOCV of the TLEPDEqs (1) (2) (3) with the Dirchlet boundary condition , in ∂Λ (4) where ∑ ( ) , , ( ) ( ) , ( ) ( ( ) ( ) ( )) ( ( ̅)) ( the system (1-4)), ( ) ( ( ) ( ) ( )) ( ( )) is the classical control vector and ( ) ( ( ) ( ) ( )) ( ( )) is a vector of a given function , for all ( ) . The Set of Admissible Control is ⃗⃗ ( ( )) , such that ⃗⃗ {( ) ( ( )) |( ) ⃗⃗ } where is convex set. The Cost Functional is ( ⃗ ) ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ⃗ ⃗⃗ ( ) Where α is a positive real number, ⃗⃗ is the solution vector of (1 -4) corresponding to the continuous classical control vector (CCV) ⃗ and ( ) is a vector of desired date. The CCOCV Problem is to minimize ( ⃗ ) (5) subject to ⃗⃗ ( ) ⃗⃗ . Let ⃗⃗⃗ ( ) ( ) ( ).We denote by ( ) and ‖ ‖ the inner product and the norm in ( ), by ( ⃗⃗⃗ ⃗⃗⃗ ) , ‖ ⃗⃗⃗ ‖ the inner product and the norm in ( ) by ( ⃗⃗⃗ ⃗⃗⃗ ) ( ) ( ) ( ) and ‖ ⃗⃗⃗ ‖ ‖ ‖ ‖ ‖ ‖ ‖ the inner product and the norm in ⃗⃗⃗ and ⃗⃗ ⃗⃗ ⃗ (the dual of ⃗⃗⃗ ). 3. Weak Formulation of the TLEPDEqs The weak form (WF) of problem (1-4) are obtained by multiplying both sides of Equations (1-3) by , and respectively, integrating the obtained Equations and finally using the generalize Green's theorem for the 1 s t term in the Left hand side (L.H.S) of the three obtained equations , to get ( ) ( ) ( ) ( ) ( ) ( ) (6) ( ) ( ) ( ) ( ) ( ) ( ) (7) 341 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 ( ) ( ) ( ) ( ) ( ) ( ) (8) where ( ) ∬ ∑ , ( ) ∬ =( ), blending to gather Equations (6), (7) and (8), once get ( ⃗⃗⃗ ) ̆( ⃗⃗⃗ ) (9) where ( ⃗⃗⃗ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and for fixed ⃗ , ̆( ⃗⃗⃗ ) ( ) ( ) ( ) ( ) ( ) ( ) The following hypotheses are useful to study the existence of unique solution for the WF (9) . Hypotheses: a) ( ⃗⃗⃗ ) is coercive, i .e. ( ) ‖ ‖ , b) | ( ⃗⃗⃗ )| ‖ ‖ ‖ ⃗⃗⃗ ‖ , . c) ̆( ⃗⃗⃗ ) ⃗⃗⃗ , ⃗⃗⃗ ⃗⃗⃗ , . The GM is used here to find the solution of (9), This is doing through choosing a finite subspace ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ( ⃗⃗⃗ contains the continuous and piecewise affine functions in Λ), hence the problem reduces to find an approximate solution of the following an approximation problem ( ⃗⃗ ⃗ ⃗⃗⃗ ) ̆( ⃗⃗⃗ ), ⃗⃗⃗ ⃗⃗⃗ (10) Theorem 3.1: For every fixed control vector ⃗ ( ( )) , the WF (10) has a unique approximation solution ⃗⃗⃗ . Proof: Let { ⃗⃗ ⃗⃗ ⃗⃗ } be a finite basis of ⃗⃗⃗ and let ( ) ∑ ⃗⃗ ( ) (∑ ∑ ∑ ) (11) Where ⃗⃗ (( ) ( ) ( ) ), for , , , [(( ) ) ] [ ( ) ] , and with are unknown constants. By using ∑ ⃗⃗ ⃗⃗⃗ ⃗⃗ , in (10), to get (∑ ⃗⃗ ⃗⃗ ) ̆( ⃗⃗ ), (12) which can be rewritten as a linear algebraic system, i .e. (13) From hypothesis (a), easily once obtained the uniqueness of the solution of problem (13), which gives also the uniqueness of the solution of problem (10). 341 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 Theorem 3.2 ( ), - For each ⃗⃗⃗ ⃗⃗⃗ there exists a sequence { ⃗⃗ } with ⃗⃗ ⃗⃗⃗ for each n, such that ⃗⃗ → ⃗⃗⃗ strongly in ⃗⃗⃗ . Now from the WF (10) and theorem(3.2),once get that there exists a sequence of WF ( ⃗⃗ ⃗ ⃗⃗ ) ̆( ⃗⃗ ) , ⃗⃗ ⃗⃗⃗ (14) which has a sequence of solutions { } and the sequence ⃗⃗ → ⃗⃗⃗ strongly in ⃗⃗⃗ . Theorem 3.3: The sequence of solutions { } converges strongly to the solution ( ). Proof: Since for each n, is a solution of (14), then from hypotheses (a&c), ‖ ‖ , , with Then by using Alaoglu theorem, there exists a subsequence of { } (for simplicity say again { }), such that weakly in ⃗⃗⃗ . To prove, that the sequence { } of solution of (14) converges to a vector which is the solution of problem (9). First, from hypothesis (b), the above weakly convergences and since ⃗ ⃗⃗ strongly in ⃗⃗⃗ , then | ( ⃗ ) ( ⃗⃗ )| | ( ⃗ ⃗⃗ )| | ( ⃗⃗ )| ‖ ‖ ‖ ⃗ ⃗⃗ ‖ ‖ ‖ ‖ ⃗⃗ ‖ → Which means ( ⃗ ) → ( ⃗⃗ ) Second, from theorem (3.2) ⃗ ⃗⃗ weakly in ⃗⃗⃗ , then ̆( ⃗ ) ̆( ⃗⃗ ) to prove → strongly in ⃗⃗⃗ , from hypothesis (1-a), one has ‖ ‖ ( ) ( ) ( ) ( ) ( ) ̌( )= ̌( ) ̌( ) → Which complete the proof of { } converges strongly to with respect to‖ ‖ . The uniqueness of solution is obtained easily through using hypothesis (a) . 4. Existence of a CCOCV: Lemma 4.1: The operator ⃗ from ⃗⃗ to ( ( )) is Lipschitz continuous (LC), i .e. ⃗⃗⃗⃗ ̆ ⃗⃗⃗⃗ , for ̆ Proof: Let ⃗⃗ ⃗ ( ) be a given control vector of the WF(6-8) and ⃗⃗ ( ) be the corresponding state vector solution, we get new equations for ⃗⃗⃗ and ⃗⃗ , then by subtracting these new equations from their corresponding Equations (6 -8) and then substituting δ = , δ 341 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 , δ = δ , = and δ in the obtained equations, to get ( ) ( ) ( ) ( ) ( ) (15) ( ) ( ) ( ) ( ) ( ) (16) ( ) ( ) ( ) ( ) ( ) (17) Next blending together the equations which obtained by substituting , and in (15-17)) respectively, to give ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (18) After using Cauch-Schwarz inequality (C-S-I) and applying hypothesis (1 - a),once has ‖ ⃗⃗⃗⃗ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ (19) Since ‖ ‖ ‖ ⃗⃗⃗⃗ ‖ ‖ ⃗⃗⃗⃗ ‖ and ‖ ‖ ‖ ⃗⃗⃗⃗ ‖ , , then (19) becomes ‖ ‖ ̆‖ ⃗⃗⃗⃗ ‖ , with ̆ So ⃗ is LC on ( ( )) . Lemma 4.2[14]: The norm ‖ ‖ is weakly lower semicontinuous (W.L.S.). Lemma 4.3: The cost function in (5) is W.L.S. . . Proof: the proof easily obtained through applying lemma (4.2), the weakly converge of → in ( ) and lemma (4.1). Lemma 4.4[14]: The norm ‖ ‖ is strictly convex. Remark 4.1: The cost function ( ) is strictly convex by using Lemma (4.4). Theorem 4.1: If ( ) is coercive and ⃗ is convex, then there ex ists CCOCV for the problem (5). Proof: ⃗⃗ is convex since ⃗ is convex with ( ) , and ( ) is coercive then there exist a minimization sequence * + ⃗⃗ such that ( ) ⃗⃗ ⃗⃗ ( ⃗ ) Therefore ‖ ‖ , (20) Then, the sequence * + has a subsequence for simplicity say again * + such that → weakly in ( ( )) , (by using the Aloglu theorem). But theorem 3.1, tell us that the sequence of problems (9) has the sequence of solutions{ }. To prove{ } , is bounded in ⃗⃗⃗ , the hypotheses (a and c), and the C-S-I, are used to get that : 341 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 ‖ ‖ ( ) ̆( ) (21) ≤‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ +‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ≤ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ≤ ϖ‖ ‖ Where ( ) ( ) ( ) ( ) then ‖ ‖ ,for each n, with By Alaoglu theorem there exists a subsequence of { } (for simplicity say again { }) such that weakly in ⃗⃗ Since for each n, satisfies the weak form (9),then ( ⃗⃗⃗ ) ̆ ( ⃗⃗⃗ ) ( ) ( ) ( ) ( ) ( ) ( ) (22) To show that (22) converges to ( ⃗⃗⃗ ) ̆( ⃗⃗⃗ ) (23) First , since , → ( ). Then by using the C-S-I and hypothesis (b), once gets: | ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )| ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ → Second , the right hand side (R.H.S)of (22) converges to the R..H.S of (23), since → ( ( )) , which gives (22) converges to (23). But ( ) is W.L.S., with ( ( )) , then ( ) ( ) ⃗⃗ ⃗⃗ ( ⃗ ), which gives ( ) ⃗⃗ ⃗⃗ ( ⃗ ) i .e., is a ccocv. One can easily applies remark 4.1, to get the uniqueness of . 5. The Necessary Conditions for Optimality Theorem 5.1: Consider the cost function (5), and the TAEqs ( ) equations of the state Equations (1-4) are given by : (24) (25) (26) 341 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (1) 2020 (27) Then the Fréchet derivative of is ( ( ) ⃗⃗⃗⃗ ) ( ⃗⃗⃗⃗ ) Proof: Writing the TAEqs (19-22) by their WF, then adding them and then substi tuting ⃗⃗ ⃗⃗⃗⃗ in the resulting equation to get the following WF (the proof of the existences of a unique solution for this WF is simpler than the proof of theorem (3.1)): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (28) Now, substituting the solutions and in (6) separately, then subtracting the obtained 1 s t equation from the 2 n d one, finally setting to obtain ( ) ( ) ( ) ( ) ( ) (29) Same steps can be used in Equation (7)for the solutions and with , (in Equation (8) for the solution and with ), to get respectively ( ) ( ) ( ) ( ) ( ) (30) ( ) ( ) ( ) ( ) ( ) (31) Blending together the above triple equations, then subtracting the obtained equation from (28), to get ( ) ( ) ( ) ( ) ( ) ( ) (32) Now, (5) , once get ( ⃗⃗⃗⃗ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ‖ ⃗⃗⃗⃗ ‖ ‖ ⃗⃗⃗⃗ ‖ (33) From (32)and (33), once get ( ⃗⃗⃗⃗ ) ( ) ( ⃗⃗⃗⃗ ) ‖ ⃗⃗⃗⃗ ‖ ‖ ⃗⃗⃗⃗ ‖ (34) from lemma (4.1), once obtain ‖ ⃗⃗⃗⃗ ‖ ‖ ⃗⃗⃗⃗ ‖ ( ⃗⃗⃗⃗ ) ⃗⃗⃗⃗ (35) where ( ⃗⃗⃗⃗ ) ( ⃗⃗⃗⃗ ) ( ⃗⃗⃗⃗ ) → as ⃗⃗⃗⃗ → Then from the definition of FD of , and (34-35), once get ( ( ) ⃗⃗⃗⃗ ) ( ⃗⃗⃗⃗ ) Theorem 5.2 : The CCOCV of (1- 4) is: ( ) with ⃗ and ⃗ . 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