Microsoft Word - 113-126   113 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020       The Continuous Classical Boundary Optimal Control Vector Governing by Triple Linear Partial Differential Equations of Parabolic Type Department of Mathematics, College of Scien c e , University of Mustansiriyah, Baghdad, Iraq. Abstract In this paper, the continuous classical boundary optimal control problem (CCBOCP) for triple linear partial differential equations of parabolic type (TLPDEPAR) with initial and boundary conditions (ICs & BCs) is studied. The Galerkin method (GM) is used to prove the existence and uniqueness theorem of the state vector solution (SVS) for given continuous classical boundary control vector (CCBCV). The proof of the existence theorem of a continuous classical boundary optimal control vector (CCBOCV) associated with the TLPDEPAR is proved. The derivation of the Fréchet derivative (FrD) for the cost function (CoF) is obtained. At the end, the theorem of the necessary conditions for optimality (NCsThOP) of this problem is stated and proved. Keywords: Continuous Classical Boundary Optimal Control, Triple Linear Partial Differential Equations, Galerkin Method, Necessary Conditions for Optimality. 1. Introduction Different applications for real life problems take a main place in the optimal control problems, for example 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 of OCP which are governined by parabolic, hyperbolic or elliptic PDEs are studied by [5,-7]. Respectively, while OCP which are governing by couple of PDEs (CPDEs) of Parabolic, hyperbolic or elliptic type are studied by [8-10]. On the other hand, [11-13]. Are studied boundary OCP associated with CPDEs of these three types; while [14, 15]. Are studied the OCP for triple PDEs (TPDEs) of parabolic and elliptic type respectively. All these works push us to seek about the CCBOCV for the TLPDEPAR. This work starts with the state and prove the existence theorem of a unique solution (SVS) for the triple state equations (TSEs) of PDEs of parabolic type (TLPDEPAR) by using the GM when the Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/33.3.2478 Jamil A. Al-Hawasy Article history: Received 24 September 2019, Accepted 4 November 2019, Published in July 2020. Mohammed A. K. Jaber Jhawassy17@uomustansiriyah.edu.iq  hawasy20@ yahoo.com   114 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 CCBCV is fixed, then we deals with the proof of the existence theorem of a CCBOCV, the solution vector of the triple adjoint equations (TAPEs) associated the TLPDEPAR is studied. The derivation of the FrD for the CoF is obtained. At the end, the NCsThOP of this OCP is sated and proved.. 2. Description of the problem: Let Ω ⊂ ℝ , 𝓍 𝑥 ,, 𝑥 , 𝑄 0, T Ω, Ĩ 0, T , Γ 𝜕Ω, Σ Γ Ĩ. The CCBOCP consists of the TSEs which are given by the following TLPDEPAR 𝑦 ∆𝑦 𝑦 𝑦 𝑦 𝑓 𝑥, 𝑡 , in 𝑄 (1) 𝑦 ∆𝑦 𝑦 𝑦 𝑦 𝑓 𝑥, 𝑡 , in 𝑄 (2) 𝑦 ∆𝑦 𝑦 𝑦 𝑦 𝑓 𝑥, 𝑡 , in 𝑄 (3) with the BCs and ICs. ∑ cos 𝑛 , 𝑥 𝑢 𝑥, 𝑡 , on Σ (4) ∑ cos 𝑛 , 𝑥 𝑢 𝑥, 𝑡 , on Σ (5) ∑ cos 𝑛 , 𝑥 𝑢 𝑥, 𝑡 , on Σ (6) 𝑦 𝑥, 0 𝑦 𝑥 , on Ω (7) 𝑦 𝑥, 0 𝑦 𝑥 , on Ω (8) 𝑦 𝑥, 0 𝑦 𝑥 , on Ω (9) Where 𝑛ℊ, ∀ ℊ 1,2,3, is an outer normal vector on the boundary Σ, and (𝑛ℊ, 𝑥 is the angle between 𝑛ℊ and the 𝑥 axis, (𝑓 ,𝑓 ,𝑓 ) is a vector of a given function on Ω,𝑢 𝑢 , 𝑢 , 𝑢 ∈ 𝐿 Σ is the CCBCV and �⃗� 𝑦 , 𝑦 , 𝑦 ∈ 𝐻 𝑄 is their corresponding SVS. The set of admissible CCBCV is defined by 𝑊 𝑢 𝑢 , 𝑢 , 𝑢 ∈ 𝐿 Σ ∣ 𝑢 , 𝑢 , 𝑢 ∈ �⃗� 𝑈 𝑈 𝑈 ⊂ 𝑅 a.e. in Σ , where a.e. denotes to almost everywhere. The CoF is defined by 𝐺 𝑢 ‖𝑦 𝑦 ‖ ‖𝑦 𝑦 ‖ ‖𝑦 𝑦 ‖ ‖𝑢 ‖ ‖𝑢 ‖ ‖𝑢 ‖ , 𝑢 ∈ 𝑊 , 𝛽 0 (10) Let �⃗� 𝑉 𝑉 𝑉 𝐻 Ω , �⃗� �⃗�: �⃗� 𝑣 𝑥 , 𝑣 𝑥 , 𝑣 𝑥 ∈ 𝐻 Ω . The weak form (wf) of the boundary value problem (BVP) (1)-(9), when �⃗� ∈ 𝐻 𝑄 is given by 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑓 , 𝑣 𝑢 , 𝑣 , ∀ 𝑣 ∈ 𝑉 (11.a) 𝑦 0 , 𝑣 𝑦 , 𝑣 , ∀ 𝑣 ∈ 𝑉 (11.b) 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑓 , 𝑣 𝑢 , 𝑣 , ∀ 𝑣 ∈ 𝑉 (12.a) 𝑦 0 , 𝑣 𝑦 , 𝑣 , ∀ 𝑣 ∈ 𝑉 (12.b) 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑓 , 𝑣 𝑢 , 𝑣 , ∀ 𝑣 ∈ 𝑉 (13.a) 𝑦 0 , 𝑣 𝑦 , 𝑣 , ∀ 𝑣 ∈ 𝑉 (13.b) The following assumption is important to study the existence theorem of the SVS for the wf (11)−(13). 2.1. Assumption (A): The function 𝑓 (∀ 𝑖 1,2,3 ) is satisfied the following condition with respect to (w.r.t.) 𝑥 & 𝑡, that is (i.e.) |𝑓 | 𝜂 𝑥, 𝑡 , where 𝑥, 𝑡 ∈ 𝑄, 𝜂 ∈ 𝐿 𝑄, ℝ . 3. The Existence Solution for the wf:   115 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 Theorem 3.1: Existence of a Unique Solution for the wf of the SEs: With assumption (A), for each given CCBCV 𝑢 ∈ 𝐿 Σ , the wf (11)–(13) of the TSEs has a unique SVS �⃗� with �⃗� ∈ 𝐿 Ĩ, 𝑉 , �⃗� 𝑦 , 𝑦 , 𝑦 ∈ 𝐿 Ĩ, 𝑉 ∗ . Proof: Let for each n, �⃗� 𝑉 𝑉 𝑉 ⊂ �⃗� be the set of continuous and piecewise affine functions in Ω, let 𝑣 ∈ 𝑉 𝑉 , 𝑖 1,2,3, and 𝑗 1,2, … , 𝑛, be a basis of 𝑉 , let �⃗� 𝑦 , 𝑦 , 𝑦 be an approximate solution for the solution �⃗�, then by GM: 𝑦 ∑ 𝑐 𝑡 𝑣 𝑥 , (14) 𝑦 ∑ 𝑐 𝑡 𝑣 𝑥 , (15) 𝑦 ∑ 𝑐 𝑡 𝑣 𝑥 , (16) where 𝑐 𝑡 are unknown functions of 𝑡, ∀ 𝑖 1,2,3 , 𝑗 1,2, … , 𝑛. The wf (11 13) is approximated by using (14) 16) as follows: 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑓 , 𝑣 𝑢 , 𝑣 , ∀ 𝑣 ∈ 𝑉 (17.a ) 𝑦 , 𝑣 𝑦 , 𝑣 , ∀ 𝑣 ∈ 𝑉 (17.b) 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑓 , 𝑣 𝑢 , 𝑣 , ∀ 𝑣 ∈ 𝑉 (18.a) 𝑦 , 𝑣 𝑦 , 𝑣 , ∀ 𝑣 ∈ 𝑉 (18.b) 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑓 , 𝑣 𝑢 , 𝑣 , ∀ 𝑣 ∈ 𝑉 (19.a) 𝑦 , 𝑣 𝑦 , 𝑣 , ∀ 𝑣 ∈ 𝑉 (19.b) Where 𝑦 𝑦 𝑥 𝑦 𝑥, 0 ∈ 𝑉 ⊂ 𝑉 ⊂ 𝐿 Ω is the projection of 𝑦 , thus 𝑦 , 𝑣 𝑦 , 𝑣 , ∀ 𝑣 ∈ 𝑉 ⟺ ∣∣ 𝑦 𝑦 ∣∣ ∣∣ 𝑦 𝑣 ∣∣ , ∀ 𝑣 ∈ 𝑉 . Substituting (14 16) in (17) 19) respectively, and then setting 𝑣 𝑣 , 𝑣 𝑣 & 𝑣 𝑣 ∀ 𝑙 1,2, … , 𝑛, the obtained equations are equivalent to the following linear system (LS) of 1st order ODEs with ICs (which has a unique solution), i.e. 𝐴𝐶 ´ 𝑡 𝐵𝐶 𝑡 𝐷𝐶 𝑡 𝐸𝐶 𝑡 𝑏 , (20.a) 𝐴𝐶 0 𝑏 , (20.b) 𝐹𝐶 ´ 𝑡 𝐺𝐶 𝑡 𝐻𝐶 𝑡 𝐾𝐶 𝑡 𝑏 , (21.a) 𝐹𝐶 0 𝑏 , (21.b) 𝑀𝐶 ´ 𝑡 𝑁𝐶 𝑡 𝑅𝐶 𝑡 𝑊𝐶 𝑡 𝑏 , (22.a) 𝑀𝐶 0 𝑏 , (22.b) Where 𝐴 𝑎 , 𝑎 𝑣 , 𝑣 , 𝐵 𝑏 , 𝑏 (∇𝑣 , ∇𝑣 𝑣 , 𝑣 ), 𝐷 𝑑 , 𝑑 𝑣 , 𝑣 , 𝐸 𝑒 , 𝑒 𝑣 , 𝑣 ), 𝐹 𝑓 , 𝑓 𝑣 , 𝑣 , 𝐺 𝑔 , 𝑔 𝛻𝑣 , 𝛻𝑣 𝑣 , 𝑣 ), 𝐻 ℎ , ℎ 𝑣 , 𝑣 , 𝐾 𝑘 , 𝑘 𝑣 , 𝑣 ), 𝑀 𝑚 , 𝑚 𝑣 , 𝑣 , 𝑁 𝑛 , 𝑛 𝛻𝑣 , 𝛻𝑣 𝑣 , 𝑣 , 𝑅 𝑟 , 𝑟 𝑣 , 𝑣 ), 𝑊 𝑤 , 𝑤 𝑣 , 𝑣 ), 𝑏 𝑏 ), 𝑏 𝑦 , 𝑣 , 𝑏 𝑏 , 𝑏 𝑓 , 𝑣 𝑢 , 𝑣 , C´ t c´ t , 𝐶 t 𝑐 t , 𝐶 0 𝑐 0 , ∀ 𝑙 1,2,3 … 𝑛, 𝑖 1,2,3. To show the norm 𝐲𝐧𝟎 𝟎 is bounded: Since 𝑦 ∈ 𝐿 Ω , there exists a sequence 𝑣 with 𝑣 ∈ 𝑉 , such that 𝑣 ⟶ 𝑦 strongly in 𝐿 Ω , then from the projection Theorem [16]. And (17.b), ‖𝑦 𝑦 ‖ ‖𝑦 𝑣 ‖ , ∀ 𝑣 ∈ 𝑉, and then ‖𝑦 𝑦 ‖ ‖𝑦 𝑣 ‖ , ∀ 𝑣 ∈ 𝑉 ⊂ 𝑉, ∀𝑛 ⇒ 𝑦 ⟶ 𝑦 , strongly in 𝐿 Ω , implies to   116 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 ‖𝑦 ‖ 𝑏 , similarly ‖𝑦 ‖ 𝑏 & ‖𝑦 ‖ 𝑏 , thus 𝑦 ⃗ 𝟎 is bounded in 𝐿 Ω . The norms ‖�⃗�𝒏 𝒕 ‖𝑳 Ĩ,𝑳𝟐 Ω and ‖�⃗�𝒏 𝒕 ‖𝑸 are bounded: Setting 𝑣 𝑦 , 𝑣 𝑦 and 𝑣 𝑦 in (17), (18) & (19) respectively, integrating w.r.t. 𝑡 from 0 to 𝑇, and then adding the obtained three equations, one gets 〈�⃗� , �⃗� 〉𝑑𝑡 ‖�⃗� ‖ 𝑑𝑡 𝑓 , 𝑦 𝑢 , 𝑦 𝑓 , 𝑦 𝑢 , 𝑦 𝑓 , 𝑦 𝑢 , 𝑦 𝑑𝑡, (23) Using Lemma (1.2) in [11]. For the 1st term in the L.H.S. of (23), and since the 2nd term is positive, taking 𝑇 𝑡 ∈ 0, 𝑇 . Finally, using assumption (A) for the R.H.S. of (23), it yields to ‖ ⃗ ‖ 𝑑𝑡 𝜂 |𝑦 | 𝑑𝑥𝑑𝑡 |𝑢 | |𝑦 | 𝑑𝛾𝑑𝑡 𝜂 |𝑦 | 𝑑𝑥𝑑𝑡 |𝑢 | |𝑦 | 𝑑𝛾𝑑𝑡 𝜂 |𝑦 | 𝑑𝑥𝑑𝑡 |𝑢 | |𝑦 | 𝑑𝛾𝑑𝑡 ]. Using the Trace Theorem [17]. Of the R.H.S., to get ‖�⃗� 𝑡 ‖ ‖�⃗� 0 ‖ ‖𝜂 ‖ ‖𝜂 ‖ ‖𝜂 ‖ ‖𝑢 ‖ ‖𝑢 ‖ ‖𝑢 ‖ ‖𝑦 ‖ 𝑑𝑡 𝑐 ‖𝑦 ‖ 𝑑𝑡 ‖𝑦 ‖ 𝑑𝑡 𝑐 ‖𝑦 ‖ 𝑑𝑡 ‖𝑦 ‖ 𝑑𝑡 𝑐 ‖𝑦 ‖ 𝑑𝑡. Since ‖𝜂 ‖ 𝑏 , ‖𝑢 ‖ �́� , ∀ 𝑖 1,2,3, ‖�⃗� 0 ‖ 𝑏, then ‖�⃗� 𝑡 ‖ 𝑐∗ ℏ ‖�⃗� ‖ 𝑑𝑡, where 𝑐∗ 𝑏 𝑏 𝑏 �́� �́� �́� 𝑏 and ℏ max ℏ∗ , ℏ∗ , ℏ∗ ; ℏ∗ 1 𝑐 , ℏ∗ 1 𝑐 , ℏ∗ 1 𝑐 . Using the continuous Bellman Gronwall Inequality ( BGI ), one gets ‖�⃗� 𝑡 ‖ 𝑐∗𝑒ℏ 𝑏 𝑐 , ∀ 𝑡 ∈ 0, 𝑇 or ‖�⃗� 𝑡 ‖ 𝑳 Ĩ,𝑳𝟐 Ω 𝑏 𝑐 ⇒ ‖�⃗� 𝑡 ‖𝑸 𝑏 𝑐 . The norm ‖�⃗� 𝑡 ‖𝑳𝟐 Ĩ,𝑽 is bounded : Again for (23) by using Lemma (1.2) in [11]. For the L.H.S. the same results may be obtained (from the above steps) and since ‖�⃗� 0 ‖ is bounded, equation (23) with 𝑡 =T , becomes ‖�⃗� 𝑡 ‖ ‖�⃗� 0 ‖ 2 ‖�⃗� ‖ 𝑑𝑡 ‖𝜂 ‖ ‖𝜂 ‖ ‖𝜂 ‖ ‖𝑢 ‖ ‖𝑢 ‖ ‖𝑢 ‖ ℏ‖�⃗� ‖𝑸 . Which gives ‖�⃗� ‖ 𝑑𝑡 𝑏 𝑐 , with 𝑏 𝑐 ́ ́ ́ ℏ , thus ‖�⃗� ‖𝑳𝟐 Ĩ,𝑽 𝑏 𝑐 . The convergence of the solution: Let �⃗� be a sequence of subspaces of �⃗�, s.t. ∀ �⃗� ∈ 𝑉,⃗ there exists a sequence �⃗� } with �⃗� ∈ �⃗� , ∀ 𝑛 and �⃗� ⟶ �⃗�, strongly in �⃗� ⇒ �⃗� ⟶ �⃗�, strongly in 𝐿 Ω , since for each 𝑛, with �⃗� ⊂ �⃗�, (17 19) has a unique solution �⃗� 𝑦 , 𝑦 , 𝑦 , hence corresponding to the sequence of subspaces �⃗� , there exists a sequence of (approximation) problems like (17 19), thus by substituting �⃗� �⃗� 𝑣 , 𝑣 , 𝑣 , one has for 𝑛 1,2, … 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 (𝑓 ,𝑣 ) 𝑢 , 𝑣 , ∀ 𝑣 ∈ 𝑉 (24.a) 𝑦 , 𝑣 𝑦 , 𝑣 , ∀ 𝑣 ∈ 𝑉 (24.b)   117 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 (𝑓 ,𝑣 ) 𝑢 , 𝑣 , ∀ 𝑣 ∈ 𝑉 (25.a) 𝑦 , 𝑣 𝑦 , 𝑣 , ∀ 𝑣 ∈ 𝑉 (25.b) 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 (𝑓 ,𝑣 ) 𝑢 , 𝑣 , ∀ 𝑣 ∈ 𝑉 (26.a) 𝑦 , 𝑣 𝑦 , 𝑣 , ∀ 𝑣 ∈ 𝑉 (26.b) which has a sequence of solutions �⃗� , with �⃗� , but from the above steps we have ‖�⃗� ‖𝑳𝟐 𝑸 and ‖�⃗� ‖𝑳𝟐 Ĩ,𝑽 are bounded, then by Alaoglu’s Theorem (ATh), there exists a subsequence of �⃗� ∈ , say again �⃗� ∈ s.t. �⃗� ⟶ �⃗� weakly in 𝐿 𝑄 and in 𝐿 Ĩ, 𝑉 . Multiplying both sides of (24.a), (25.a) & (26.a) by 𝜑 𝑡 ∈ 𝐶 0, 𝑇 , ∀𝑖 1,2,3, respectively, s.t. 𝜑 𝑇 0 , 𝜑 0 0, integrating both sides w.r.t. 𝑡 on 0, 𝑇 , then integrating by parts the 1st terms in the L.H.S. of each one obtained equation, one gets 𝑦 , 𝑣 𝜑ˊ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 0 , (27) 𝑦 , 𝑣 𝜑ˊ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 0 , (28) 𝑦 , 𝑣 𝜑ˊ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 0 , (29) Since 𝑣 ⟶ 𝑣 strongly in 𝐿 Ω 𝑣 ⟶ 𝑣 strongly in 𝑉 ⇒ 𝑣 𝜑´ ⟶ 𝑣 𝜑´ strongly in 𝐿 𝑄 𝑣 𝜑 ⟶ 𝑣 𝜑 strongly in 𝐿 Ĩ, 𝑉 , Also, since 𝑦 ⟶ 𝑦 weakly in 𝐿 𝑄 , and 𝑦 ⟶ 𝑦 weakly in 𝐿 𝑄 , ∀ 𝑖 1,2,3, then 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 ⟶ 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 , (30.a) 𝑦 , 𝑣 𝜑 ´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 ⟶ 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 , (31.a) 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 ⟶ 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 , (32.a) 𝑦 , 𝑣 𝜑 0 ⟶ 𝑦 , 𝑣 𝜑 0 , (30.b) 𝑦 , 𝑣 𝜑 0 ⟶ 𝑦 , 𝑣 𝜑 0 , (31.b) 𝑦 , 𝑣 𝜑 0 ⟶ 𝑦 , 𝑣 𝜑 0 , (32.b)   118 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 since 𝑣 ⟶ 𝑣 , weakly in 𝐿 Ω , then 𝑓 , 𝑣 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 ⟶ 𝑓 , 𝑣 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡, (30.c) 𝑓 , 𝑣 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 ⟶ 𝑓 , 𝑣 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡, (31.c) 𝑓 , 𝑣 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 ⟶ 𝑓 , 𝑣 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡, (32.c) which means (30 32), converge to (33 35), respectively 𝑦 , 𝑣 𝜑ˊ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 0 , (33) 𝑦 , 𝑣 𝜑ˊ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 0 , (34) 𝑦 , 𝑣 𝜑ˊ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 0 , (35) Case1: Choose 𝜑 ∈ 𝐷 0, 𝑇 , i.e. 𝜑 0 𝜑 𝑇 0, ∀𝑖 1,2 ,3. Substituting in (33) 35), using integration by parts for the 1st terms in the L.H.S. of each one of the obtained equations, one has 〈𝑦 , 𝑣 〉𝜑 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 , (36) 〈𝑦 , 𝑣 〉𝜑 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 , (37) 〈𝑦 , 𝑣 〉𝜑 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 , (38) i.e. �⃗� is a solution of the wf (11 13) . Case 2: Choose 𝜑 ∈ 𝐶 0, 𝑇 , s.t. 𝜑 𝑇 0 & 𝜑 0 0, ∀𝑖 1,2,3. Using integration by parts for 1st term in the L.H.S. of (36), one gets 𝑦 , 𝑣 𝜑ˊ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 0 , 𝑣 𝜑 0 , (39) Subtracting (33) from (39), one obtains that 𝑦 , 𝑣 𝜑 0 𝑦 0 , 𝑣 𝜑 0 , 𝜑 0 0,∀ 𝜑 ∈ 𝐶 0, 𝑇 ⇒ 𝑦 , 𝑣 𝑦 0 , 𝑣 , i.e. the ICs (11.b) holds. By the same above way one can show that 𝑦 , 𝑣 𝑦 0 , 𝑣 , 𝑦 , 𝑣 𝑦 0 , 𝑣 which means the ICs (12.b) & (13.b) are holds. The strongly convergence for �⃗�𝐧: Substituting 𝑣 𝑦 ,𝑣 𝑦 and 𝑣 𝑦 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 𝑣 𝑦 ,𝑣 𝑦 & 𝑣 𝑦 in (11.a), (12.a) &(13.a) respectively, adding them together, integrating the obtained equations from 0 to 𝑇, one gets 〈�⃗� , �⃗� 〉 𝑑𝑡 𝑎 �⃗� , �⃗� 𝑑𝑡 𝑓 , 𝑦 𝑢 , 𝑦 𝑓 , 𝑦 𝑢 , 𝑦 𝑓 , 𝑦 𝑢 , 𝑦 𝑑𝑡, (40.a) 〈�⃗� , �⃗�〉 𝑑𝑡 𝑎 �⃗� , �⃗� 𝑑𝑡 𝑓 , 𝑦 𝑢 , 𝑦 𝑓 , 𝑦 𝑢 , 𝑦 𝑓 , 𝑦 𝑢 , 𝑦 𝑑𝑡, (40.b) using Lemma(1.2) in [11] for the 1st terms in the L.H.S. of (40.a&b), they become ‖�⃗� 𝑇 ‖ ‖�⃗� 0 ‖ 𝑎 �⃗� , �⃗� 𝑑𝑡 𝑓 , 𝑦 𝑢 , 𝑦 𝑓 , 𝑦 𝑢 , 𝑦 𝑓 , 𝑦 𝑢 , 𝑦 𝑑𝑡, (41.a)   119 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 ‖�⃗� 𝑇 ‖ ‖�⃗� 0 ‖ 𝑎 �⃗�, �⃗� 𝑑𝑡 𝑓 , 𝑦 𝑢 , 𝑦 𝑓 , 𝑦 𝑢 , 𝑦 𝑓 , 𝑦 𝑢 , 𝑦 𝑑𝑡, (41.b) since ‖�⃗� 𝑇 �⃗� 𝑇 ‖ ‖�⃗� 0 �⃗� 0 ‖ 𝑎 �⃗� �⃗�, �⃗� �⃗� 𝑑𝑡 𝐴 𝐵 𝐶 , (42) where 𝐴 ‖�⃗� 𝑇 ‖ ‖�⃗� 0 ‖ 𝑎 �⃗� 𝑡 , �⃗� 𝑡 𝑑𝑡 𝐵 �⃗� 𝑇 , �⃗� 𝑇 �⃗� 0 , �⃗� 0 𝑎 �⃗� 𝑡 , �⃗� 𝑡 𝑑𝑡, 𝐶 �⃗� 𝑇 , �⃗� 𝑇 �⃗� 𝑇 �⃗� 0 , �⃗� 0 �⃗� 0 𝑎 �⃗� 𝑡 , �⃗� 𝑡 �⃗� 𝑡 𝑑𝑡, Since �⃗� �⃗� 0 ⟶ �⃗� �⃗� 0 strongly in 𝐿 Ω , (42.a) �⃗� 𝑇 ⟶ �⃗� 𝑇 strongly in 𝐿 Ω , (42.b) Then �⃗� 0 , �⃗� 0 �⃗� 0 ⟶ 0 �⃗� 𝑇 , �⃗� 𝑇 �⃗� 𝑇 ⟶ 0 , (42.c) ‖�⃗� 0 �⃗� 0 ‖ ⟶ 0 ‖�⃗� 𝑇 �⃗� 𝑇 ‖ ⟶ 0 , (42.d) Since �⃗� ⟶ �⃗� weakly in 𝐿 Ĩ, 𝑉 , then 𝑎 �⃗� 𝑡 , �⃗� 𝑡 �⃗� 𝑡 𝑑𝑡 ⟶ 0, (42.e) As well as, since �⃗� ⟶ �⃗� weakly in 𝐿 𝑄 , then 𝑓 , 𝑦 𝑢 , 𝑦 𝑓 , 𝑦 𝑢 , 𝑦 𝑓 , 𝑦 𝑢 , 𝑦 𝑑𝑡 ⟶ 𝑓 , 𝑦 𝑢 , 𝑦 𝑓 , 𝑦 𝑢 , 𝑦 𝑓 , 𝑦 𝑢 , 𝑦 𝑑𝑡, (42.f) i.e. when 𝑛 → ∞ in both sides of (42), one has the following results: (1) The first two terms in the L.H.S. of (42) are tending to zero from (42.d). (2) From (41.a) Eq. 𝐴 𝑓 , 𝑦 𝑢 , 𝑦 𝑓 , 𝑦 𝑢 , 𝑦 (𝑓 , 𝑦 ) 𝑢 , 𝑦 𝑑𝑡 from ⟶ 42. f 𝑓 , 𝑦 𝑢 , 𝑦 𝑓 , 𝑦 𝑢 , 𝑦 𝑓 , 𝑦 𝑢 , 𝑦 𝑑𝑡 , (3) Eq. 𝐵 ⟶ L.H.S. of (3.41.b) 𝑓 , 𝑦 𝑢 , 𝑦 𝑓 , 𝑦 𝑢 , 𝑦 𝑓 , 𝑦 𝑢 , 𝑦 𝑑𝑡, (4) The 1st two terms in Eq. 𝐶 are tending to zero from (42.c), and the last one term also tend to zero from (42.e), from these results (42) gives when 𝑛 → ∞ ‖�⃗� �⃗�‖ 𝑑𝑡 𝑎 �⃗� �⃗�, �⃗� �⃗� 𝑑𝑡 ⟶ 0 ⇒ �⃗� ⟶ �⃗� strongly in 𝐿 Ĩ, 𝑉 . Uniqueness of the solution: Let �⃗�, �⃗� are two solutions of the wf (11) 13) i.e. 𝑦 and 𝑦 are satisfied (11.a), or 〈𝑦 , 𝑣 〉 𝑎 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑓 , 𝑣 𝑢 , 𝑣 , ∀ 𝑣 ∈ 𝑉 〈𝑦 , 𝑣 〉 𝑎 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑓 , 𝑣 𝑢 , 𝑣 , ∀ 𝑣 ∈ 𝑉 Subtracting the 2 equation from the 1st one and substituting 𝑣 𝑦 𝑦 in the obtained equation, one gets that 〈 𝑦 𝑦 , 𝑦 𝑦 〉 𝑎 𝑦 𝑦 , 𝑦 𝑦 𝑦 𝑦 , 𝑦 𝑦 𝑦 𝑦 , 𝑦 𝑦 0, (43) by the same way, one gets 〈 𝑦 𝑦 , 𝑦 𝑦 〉 𝑎 𝑦 𝑦 , 𝑦 𝑦 𝑦 𝑦 , 𝑦 𝑦 𝑦 𝑦 , 𝑦 𝑦 0, (44)   120 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 〈 𝑦 𝑦 , 𝑦 𝑦 〉 𝑎 𝑦 𝑦 , 𝑦 𝑦 𝑦 𝑦 , 𝑦 𝑦 𝑦 𝑦 , 𝑦 𝑦 0, (45) adding (43 45), using Lemma(1.2) in [11]. In the 1st term of the obtained equations, to get �⃗� �⃗� �⃗� �⃗� 0, (46) The 2nd term of the L.H.S. of (46) is positive, integrating both sides of (46) w.r.t. 𝑡 from 0 to 𝑡 , one gets ⃗ ⃗ 𝑑𝑡 0 ⇒ �⃗� �⃗� 𝑡 0 ⇒ �⃗� �⃗� 0 ∀𝑡 ∈ Ĩ , integrating both sides of (46) from 0 to 𝑇, using the given ICs, one has �⃗� �⃗� 𝑑𝑡 0 ⇒ �⃗� �⃗� 𝑳𝟐 Ĩ,𝑽 0 ⇒ �⃗� �⃗� . 4. Existence of a CCBOCP: Theorem 4.1: In addition to assumptions (A), assume that �⃗� and �⃗� δ�⃗� are the SVS corresponding to the CVS 𝑢 and 𝑢 δ𝑢 respectively with 𝑢 and δ𝑢 are bounded in 𝐿 Σ , then ‖δ�⃗� ‖ 𝑳 Ĩ,𝑳𝟐 Ω 𝐿‖δ𝑢‖ , 𝐿 ∈ ℝ ‖δ�⃗�‖𝑳𝟐 𝑸 𝐿‖δ𝑢‖ , 𝐿 ∈ ℝ ‖δ�⃗�‖𝑳𝟐 Ĩ,𝑽 𝐿 ‖δ𝑢‖ , 𝐿 ∈ ℝ Proof: Let 𝑢 𝑢 , 𝑢 , 𝑢 ∈ 𝐿 Σ be given, then by Theorem3.1, there exists �⃗� 𝑦 𝑦 , 𝑦 𝑦 , 𝑦 𝑦 which is satisfied (11) 13) and also let �⃗� 𝑦 , 𝑦 , 𝑦 be the solution of (11 13), corresponds to the CV 𝑢 𝑢 , 𝑢 , 𝑢 ∈ 𝐿 Σ i.e. 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑓 , 𝑣 𝑢 , 𝑣 , (47.a) 𝑦 0 , 𝑣 𝑦 , 𝑣 , (47.b) 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑓 , 𝑣 𝑢 , 𝑣 , (48.a) 𝑦 0 , 𝑣 𝑦 , 𝑣 , (48.b) 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑓 , 𝑣 𝑢 , 𝑣 , (49.a) 𝑦 0 , 𝑣 𝑦 , 𝑣 , (49.b) Subtracting (11.a&b) from (47.a&b), (12.a&b) from (48.a&b) ,and (13.a&b) from (49.a&b) and setting 𝛿𝑦 𝑦 𝑦 , 𝛿𝑦 𝑦 𝑦 , 𝛿𝑦 𝑦 𝑦 , 𝛿𝑢 𝑢 𝑢 , 𝛿𝑢 𝑢 𝑢 and 𝛿𝑢 𝑢 𝑢 in the obtained equations, they give 〈𝛿𝑦 , 𝑣 〉 ∇δ𝑦 , ∇𝑣 𝛿𝑦 , 𝑣 𝛿𝑦 , 𝑣 𝛿𝑦 , 𝑣 𝛿𝑢 , 𝑣 , (50.a) 𝛿𝑦 0 , 𝑣 0, (50.b) 〈𝛿𝑦 , 𝑣 〉 ∇δ𝑦 , ∇𝑣 𝛿𝑦 , 𝑣 𝛿𝑦 , 𝑣 𝛿𝑦 , 𝑣 𝛿𝑢 , 𝑣 , (51.a) 𝛿𝑦 0 , 𝑣 0, (51.b) 〈𝛿𝑦 , 𝑣 〉 ∇δ𝑦 , ∇𝑣 𝛿𝑦 , 𝑣 𝛿𝑦 , 𝑣 𝛿𝑦 , 𝑣 𝛿𝑢 , 𝑣 , (52.a) 𝛿𝑦 0 , 𝑣 0, (52.b) substituting 𝑣 𝛿𝑦 , 𝑣 𝛿𝑦 & 𝑣 𝛿𝑦 in (3.50), (3.51) & (3.52) respectively, adding the obtained equations, using Lemma(1.2) in[11]. They give ‖𝛿�⃗�‖ ‖𝛿�⃗�‖ 𝛿𝑢 , 𝛿𝑦 𝛿𝑢 , 𝛿𝑦 𝛿𝑢 , 𝛿𝑦 , (53) Since the 2nd term of (53) is positive, integrating w.r.t. 𝑡 from 0 to 𝑡 , and then using the Cauchy Schwartz inequality (CSI), it becomes ‖𝛿�⃗�‖ 𝑑𝑡 ‖𝛿𝑢 ‖ ‖𝛿𝑦 ‖ 𝑑𝑡 ‖𝛿𝑢 ‖ ‖𝛿𝑦 ‖ 𝑑𝑡 ‖𝛿𝑢 ‖ ‖𝛿𝑦 ‖ 𝑑𝑡, which gives by using the Trace Theorem [17]. ‖𝛿�⃗� 𝑡 ‖ ‖𝛿𝑢‖𝚺 𝑐 ‖𝛿�⃗�‖ 𝑑𝑡, ∀ 𝑡 ∈ 0, 𝑇 using the BGI, one gets ‖𝛿�⃗� 𝑡 ‖ 𝑒 ‖𝛿𝑢‖𝚺 𝐿 ‖𝛿𝑢‖𝚺 , 𝑒 𝐿 , 𝐿 0   121 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 or ‖𝛿�⃗� ‖𝑳 Ĩ,𝑳𝟐 Ω 𝐿‖𝛿𝑢‖𝚺 Since ‖𝛿�⃗�‖𝑳𝟐 𝑸 𝑇𝐿 ‖𝛿𝑢‖𝚺 , thus ‖𝛿�⃗�‖𝑳𝟐 𝑸 𝐿‖𝛿𝑢‖𝚺 , 𝑇𝐿 𝐿 Using a similar way which is used in the above steps, gives ‖𝛿�⃗� ‖ 2 ‖𝛿�⃗� ‖ 𝑑𝑡 ‖𝛿𝑢‖𝚺 𝑐 ‖𝛿�⃗�‖ 𝑑𝑡 ⇒ ‖𝛿�⃗�‖𝑳𝟐 Ĩ,𝑽 𝐿 ‖𝛿𝑢‖𝚺 where 𝐿 1 𝐿 𝑐 /2 or ‖𝛿�⃗� ‖𝑳𝟐 Ĩ,𝑽 𝐿 ‖𝛿𝑢‖𝚺 . Theorem 4.2: With assumption (A), the operator 𝑢 ↦ �⃗� ⃗ is continuous from 𝐿 Σ in to 𝐿 Ĩ,𝑳𝟐 Ω or in to 𝐿 Ĩ, 𝑉 𝟑 or in to 𝐿 𝑄 𝟑 . Proof: Let 𝛿𝑢 𝑢 𝑢 and �⃗� �⃗� �⃗�, where �⃗� and �⃗� are the corresponding SVS to the CVS 𝑢 and 𝑢 respectively, using the first result in Theorem4.1, we get �⃗� Ĩ, Ω ⎯⎯⎯⎯⎯⎯⎯ �⃗� if 𝑢 ⎯⎯ 𝑢 , i.e. the operator 𝑢 ↦ �⃗� ⃗ is Lipschitz continuous (LC) from 𝑳 𝟐 𝚺 in to 𝐋 Ĩ, 𝐋𝟐 Ω . Easily, one can get this operator is also LC from 𝑳𝟐 𝚺 into 𝑳𝟐 𝑸 and into 𝑳𝟐 Ĩ, 𝑽 . Lemma 4.1: [10]. The norm ∥. ∥𝟎 is weakly lower semi continuous ( W.L.S.C. ). Lemma 4.2: The CoF which is given by (10) is W.L.S.C. Proof: From Lemma(4.1), we got that the norm ‖𝑢 ‖𝚺 is W.L.S.C., 𝑢 → 𝑢 weakly in 𝑳𝟐 𝚺 , then by (Theorem 4.2) �⃗� → �⃗� �⃗� ⃗ is weakly in 𝑳 𝟐 𝚺 , which gives that the norm ‖�⃗� �⃗� ‖𝚺 is W.L.S.C. ( by Lemma 4.1 ), hence 𝐺 𝑢 is W.L.S.C. . Theorem 4.3: Consider the cost function (10), if 𝐺 𝑢 is coercive, then there exists a CCBOCV. Proof: Since 𝐺 𝑢 0 and 𝐺 𝑢 is coercive, then there exists a minimizing sequence 𝑢 𝑢 , 𝑢 , 𝑢 ∈ 𝑊 , ∀𝑘 such that lim → 𝐺 𝑢 inf ⃗∈ ⃗ 𝐺 𝑢 , then ‖𝑢 ‖𝚺 Ĉ , Ĉ 0, then by ATh there exists a subsequence of 𝑢 , for simplicity say again 𝑢 s.t. 𝑢 ⟶ 𝑢 weakly in 𝐿 Σ , 𝑎s 𝑘 → ∞, from Theorem 3.1, corresponding to the sequence of controls 𝑢 , there exists a sequence of solutions �⃗� , but the norms ‖�⃗� ‖ 𝑳 Ĩ,𝑳𝟐 Ω , ‖�⃗� ‖𝑳𝟐 𝑸 & ‖�⃗� ‖𝑳𝟐 Ĩ,𝑽 are bounded, then by ATh there exists a subsequence of �⃗� , for simplicity, say again �⃗� , such that �⃗� ⟶ �⃗� weakly in 𝐿 Ĩ, 𝐿 Ω , in 𝐿 𝑄 , and in 𝐿 Ĩ, 𝑉 , Suppose that (17.a), (18.a) & (19.a) can be rewritten as 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑓 , 𝑣 𝑢 , 𝑣 , 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑓 , 𝑣 𝑢 , 𝑣 , 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑓 , 𝑣 𝑢 , 𝑣 , Adding the above three equations and integrating both sides of the obtained equation from 0 to 𝑇, taking the absolute value, then using CSI. Finally, using assumption (A), it yields 〈�⃗� , �⃗�〉𝑑𝑡 ‖∇𝑦 ‖ ‖∇𝑣 ‖ ‖𝑦 ‖ ‖𝑣 ‖ ‖𝑦 ‖ ‖𝑣 ‖ ‖𝑦 ‖ ‖𝑣 ‖ ‖∇𝑦 ‖ ‖∇𝑣 ‖ ‖𝑦 ‖ ‖𝑣 ‖ ‖𝑦 ‖ ‖𝑣 ‖ ‖𝑦 ‖ ‖𝑣 ‖ ‖∇𝑦 ‖ ‖∇𝑣 ‖ ‖𝑦 ‖ ‖𝑣 ‖ ‖𝑦 ‖ ‖𝑣 ‖ ‖𝑦 ‖ ‖𝑣 ‖ ‖𝜂 ‖ ‖𝑣 ‖ ‖𝜂 ‖ ‖𝑣 ‖ ‖𝜂 ‖ ‖𝑣 ‖ ‖𝑢 ‖ ‖𝑣 ‖ ‖𝑢 ‖ ‖𝑣 ‖ ‖𝑢 ‖ ‖𝑣 ‖ . Since for each 𝑖 1,2,3, the following inequalities are satisfied   122 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 ‖∇𝑦 ‖ ‖∇�⃗� ‖𝑸 ‖�⃗� ‖𝑳𝟐 Ĩ,𝑽 , ‖∇𝑣 ‖ ‖∇�⃗�‖𝑸 ‖�⃗�‖𝑳𝟐 Ĩ,𝑽 , ‖𝑦 ‖ ‖�⃗� ‖𝑸 ‖�⃗� ‖𝑳𝟐 Ĩ,𝑽 , ‖𝑣 ‖ ‖�⃗�‖𝑸 ‖�⃗�‖𝑳𝟐 Ĩ,𝑽 , ‖𝑢 ‖ ‖𝑢 ‖𝚺 Ĉ , ‖𝑣 ‖𝚺 ‖�⃗�‖𝚺 ĥ ‖�⃗�‖𝑳𝟐 Ĩ,𝑽 , ‖𝜂 ‖ 𝑏 ´ , then 〈�⃗� , �⃗�〉 𝑑𝑡 12‖�⃗� ‖𝑳𝟐 Ĩ,𝑽 ‖�⃗�‖𝑳𝟐 Ĩ,𝑽 𝑏 ´ 𝑏´ 𝑏´ 3Ĉ ĥ ‖�⃗�‖𝑳𝟐 Ĩ,𝑽 Or 〈�⃗� , �⃗�〉𝑑𝑡 12𝑏 𝑐 𝑏´ 𝑐 ‖�⃗�‖𝑳𝟐 Ĩ,𝑽 With ‖�⃗� ‖𝑳𝟐 Ĩ,𝑽 𝑏 𝑐 &𝑏 ´ 𝑐 𝑏´ 𝑏´ 𝑏´ 3Ĉ ĥ ⇒ 〈 ⃗ , ⃗〉 ‖ ⃗‖ 𝑳𝟐 Ĩ,𝑽 𝑏 𝑐 , with 𝑏 𝑐 12𝑏 𝑐 𝑏´ 𝑐 ⇒ ‖�⃗� ‖𝑳𝟐 Ĩ,𝑽∗ 𝑏 𝑐 Since for each 𝑘, �⃗� is a solution of the TSEs (1 9), then 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 (𝑓 , 𝑣 )+ 𝑢 , 𝑣 , (54) 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 (𝑓 , 𝑣 )+ 𝑢 , 𝑣 , (55) 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 (𝑓 , 𝑣 )+ 𝑢 , 𝑣 , (56) Let 𝜑 ∈ 𝐶 0, 𝑇 , s.t. 𝜑 𝑇 0 , 𝜑 0 0 , ∀𝑖 1,2,3, rewriting the 1st terms in the L.H.S. of (54 56) multiplying their both sides by 𝜑 𝑡 , ∀𝑖 1,2,3, respectively, integrating both sides w.r.t. 𝑡 from 0 to 𝑇, and integration by parts for the 1st terms in the L.H.S. of each obtained equations, one gets that 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 0 , 𝑣 𝜑 0 , (57) 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 0 , 𝑣 𝜑 0 , (58) 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 0 , 𝑣 𝜑 0 , (59) Since �⃗� ⟶ �⃗� weakly in 𝐿 𝑄 and in 𝐿 Ĩ, 𝑉 , then the following convergences are held 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 ⟶ 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 , (60) 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 ⟶ 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡, (61) 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 ⟶ 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 , (62) Since 𝑦 0 is bounded in 𝐿 Ω ∀ 𝑖 1,2,3, then 𝑦 , 𝑣 𝜑 0 ⟶ 𝑦 , 𝑣 𝜑 0 , (63) 𝑦 , 𝑣 𝜑 0 ⟶ 𝑦 , 𝑣 𝜑 0 , (64) 𝑦 , 𝑣 𝜑 0 ⟶ 𝑦 , 𝑣 𝜑 0 , (65) and since 𝑢 ⟶ 𝑢 weakly in 𝐿 Σ , then   123 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 𝑓 , 𝑣 𝑑𝑡 𝑢 , 𝑣 𝑑𝑡 → 𝑓 , 𝑣 𝑑𝑡 𝑢 , 𝑣 𝑑𝑡 , (66) 𝑓 , 𝑣 𝑑𝑡 𝑢 , 𝑣 𝑑𝑡 → 𝑓 , 𝑣 𝑑𝑡 𝑢 , 𝑣 𝑑𝑡 , (67) 𝑓 , 𝑣 𝑑𝑡 𝑢 , 𝑣 𝑑𝑡 → 𝑓 , 𝑣 𝑑𝑡 𝑢 , 𝑣 𝑑𝑡 , (68) Finally, using (60) 62),( 63 65),(66 68) in (57 59), one gets 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 0 , (69) 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 0 , (70) 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 0 , (71) Case1: We choose 𝜑 ∈ 𝐷 0, 𝑇 , i.e. 𝜑 0 𝜑 𝑇 0 , ∀𝑖 1,2,3, now by using integration by parts for the 1 terms in the L.H.S. of (69 71), one gets 〈𝑦 , 𝑣 〉𝜑 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 , ∀𝑣 ∈ 𝑉, ∀𝜑 ∈ 𝐷 0, 𝑇 (72) 〈𝑦 , 𝑣 〉𝜑 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 , ∀𝑣 ∈ 𝑉, ∀𝜑 ∈ 𝐷 0, 𝑇 (73) 〈𝑦 , 𝑣 〉𝜑 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡, ∀𝑣 ∈ 𝑉, ∀𝜑 ∈ 𝐷 0, 𝑇 (74) Then 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑓 , 𝑣 𝑢 , 𝑣 , ∀𝑣 ∈ 𝑉 , a.e. on Ĩ 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑓 , 𝑣 𝑢 , 𝑣 , ∀𝑣 ∈ 𝑉 , a.e. on Ĩ 〈𝑦 , 𝑣 〉 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑦 , 𝑣 𝑓 , 𝑣 𝑢 , 𝑣 , ∀𝑣 ∈ 𝑉 , a.e. on Ĩ i.e.�⃗� is satisfied the wf of the TSEs . Case2: We choose 𝜑 ∈ 𝐶 Ĩ , s.t. 𝜑 𝑇 0 & 𝜑 0 0, ∀𝑖 1,2,3 , by using integration by parts for the 1 terms in the L.H.S. of (72 74), one has 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 0 , 𝑣 𝜑 0 , (75) 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 0 , 𝑣 𝜑 0 , (76) 𝑦 , 𝑣 𝜑´ 𝑡 𝑑𝑡 ∇𝑦 , ∇𝑣 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑓 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑢 , 𝑣 𝜑 𝑡 𝑑𝑡 𝑦 0 , 𝑣 𝜑 0 , (77) And subtracting (75 77) from (69 71) respectively, one gets 𝑦 , 𝑣 𝜑 0 𝑦 0 , 𝑣 𝜑 0 , 𝜑 0 0, ∀ 𝜑 ∈ 𝐷 0, 𝑇 ⇒ 𝑦 𝑦 0 𝑦 𝑥 , by the same above way one can show that 𝑦 𝑦 0 𝑦 𝑥 and 𝑦 𝑦 0 𝑦 𝑥 . Then �⃗� is a solutions of the wf of the TSEs, since 𝐺 𝑢 is W.L.S.C. from Lemma 4.2 and 𝑢 ⟶ 𝑢 weakly in 𝐿 Σ , then   124 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 𝐺 𝑢 lim → inf ⃗ ∈ ⃗ 𝐺 𝑢 lim → 𝐺 𝑢 inf ⃗∈ ⃗ 𝐺 𝑢 ⇒ 𝐺 𝑢 inf ⃗∈ ⃗ 𝐺 𝑢 min ⃗∈ ⃗ 𝐺 𝑢 , then 𝑢 is a CCBOCV . 5. NCsThOP: In order to state the NCcThOP for CCBOCV, the FrD of the CoF is derived and the NCcThOP is proved . Theorem 5.1: Consider CoF (10), then the TAPEs of the TSEs are given by 𝑧 ∆𝑧 𝑧 𝑧 𝑧 𝑦 𝑦 , in 𝑄 (78) 𝑧 ∆𝑧 𝑧 𝑧 𝑧 𝑦 𝑦 , in 𝑄 (79) 𝑧 ∆𝑧 𝑧 𝑧 𝑧 𝑦 𝑦 , in 𝑄 (80) 𝑧 𝑥, 𝑇 0 , in Ω (81) 𝑧 𝑥, 𝑇 0 , in Ω (82) 𝑧 𝑥, 𝑇 0 , in Ω (83) ∑ cos 𝑛 , 𝑥 0 , on Σ (84) ∑ cos 𝑛 , 𝑥 0 , on Σ (85) ∑ cos 𝑛 , 𝑥 0 , on Σ (86) Then 𝑢 , 𝑢 , 𝑢 ∈ 𝑊 and the FrD of the CoF is given by 𝐺 ´ 𝑢 , 𝛿𝑢 𝚺 𝑧 𝛽𝑢, 𝛿𝑢 𝚺 Proof: The wf of (78 86) for 𝑣 ∈ 𝑉 , ∀ 𝑖 1,2,3, is given by 〈𝑧 , 𝑣 〉 ∇𝑧 , ∇𝑣 𝑧 , 𝑣 𝑧 , 𝑣 𝑧 , 𝑣 𝑦 𝑦 , 𝑣 , (87) 〈𝑧 , 𝑣 〉 ∇𝑧 , ∇𝑣 𝑧 , 𝑣 𝑧 , 𝑣 𝑧 , 𝑣 𝑦 𝑦 , 𝑣 , (88) 〈𝑧 , 𝑣 〉 ∇𝑧 , ∇𝑣 𝑧 , 𝑣 𝑧 , 𝑣 𝑧 , 𝑣 𝑦 𝑦 , 𝑣 , (89) The existence of a unique solution of (87 89) can be proved by the same manner which is used in the proof of Theorem 3.1. Now substituting 𝑣 𝑧 , 𝑣 𝑧 and 𝑣 𝑧 in (50.a), (51.a) and (52.a) respectively, to get 〈𝛿𝑦 , 𝑧 〉 ∇δ𝑦 , ∇𝑧 𝛿𝑦 , 𝑧 𝛿𝑦 , 𝑧 𝛿𝑦 , 𝑧 𝛿𝑢 , 𝑧 , (90.a) 〈𝛿𝑦 , 𝑧 〉 ∇δ𝑦 , ∇𝑧 𝛿𝑦 , 𝑧 𝛿𝑦 , 𝑧 𝛿𝑦 , 𝑧 𝛿𝑢 , 𝑧 , (90.b) 〈𝛿𝑦 , 𝑧 〉 ∇δ𝑦 , ∇𝑧 𝛿𝑦 , 𝑧 𝛿𝑦 , 𝑧 𝛿𝑦 , 𝑧 𝛿𝑢 , 𝑧 , (90.c) Also, substituting 𝑣 𝛿𝑦 , 𝑣 𝛿𝑦 and 𝑣 𝛿𝑦 in (87),(88)&(89) respectively, to get 〈𝑧 , 𝛿𝑦 〉 ∇𝑧 , ∇𝛿𝑦 𝑧 , 𝛿𝑦 𝑧 , 𝛿𝑦 𝑧 , 𝛿𝑦 𝑦 𝑦 , 𝛿𝑦 , (91.a) 〈𝑧 , 𝛿𝑦 〉 ∇𝑧 , ∇𝛿𝑦 𝑧 , 𝛿𝑦 𝑧 , 𝛿𝑦 𝑧 , 𝛿𝑦 𝑦 𝑦 , 𝛿𝑦 , (91.b) 〈𝑧 , 𝛿𝑦 〉 ∇𝑧 , ∇𝛿𝑦 𝑧 , 𝛿𝑦 𝑧 , 𝛿𝑦 𝑧 , 𝛿𝑦 𝑦 𝑦 , 𝛿𝑦 , (91.c) Integrating both sides of equations (90.a,b & c) and (91.a,b & c), w.r.t. 𝑡 from 0 to 𝑇, using integration by parts for the 1st terms of the L.H.S. of each of the obtained equations from (91.a), (91.b) & (91.c), then subtracting each one of the obtained equations from its corresponding equation (90.a,b & c), and adding all the obtained equations, give 𝛿𝑢 , 𝑧 𝛿𝑢 , 𝑧 𝛿𝑢 , 𝑧 𝑑𝑡 𝑦 𝑦 , 𝛿𝑦 𝑦 𝑦 , 𝛿𝑦 𝑦 𝑦 , 𝛿𝑦 𝑑𝑡, (92) Now, adding together each of the pair of equations (11.a&50.a), (12.a&51.a) and (13.a&52.a), one has 〈 𝑦 𝛿𝑦 , 𝑣 〉 ∇ 𝑦 𝛿𝑦 , ∇𝑣 𝑦 𝛿𝑦 , 𝑣 𝑦 𝛿𝑦 , 𝑣 𝑦 𝛿𝑦 , 𝑣 𝑓 , 𝑣 𝑢 𝛿𝑢 , 𝑣 , (93) 〈 𝑦 𝛿𝑦 , 𝑣 〉 ∇ 𝑦 𝛿𝑦 , ∇𝑣 𝑦 𝛿𝑦 , 𝑣 𝑦 𝛿𝑦 , 𝑣 𝑦 𝛿𝑦 , 𝑣 𝑓 , 𝑣 𝑢 𝛿𝑢 , 𝑣 , (94)   125 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 〈 𝑦 𝛿𝑦 , 𝑣 〉 ∇ 𝑦 𝛿𝑦 , ∇𝑣 𝑦 𝛿𝑦 , 𝑣 𝑦 𝛿𝑦 , 𝑣 𝑦 𝛿𝑦 , 𝑣 𝑓 , 𝑣 𝑢 𝛿𝑢 , 𝑣 , (95) Which means the CV (𝑢 𝛿𝑢 ,𝑢 𝛿𝑢 , 𝑢 𝛿𝑢 gives that the solution ( 𝑦 𝛿𝑦 , 𝑦 𝛿𝑦 , 𝑦 𝛿𝑦 of (93) – (95). On the other hand, from (92) and the CoF, we have 𝐺 𝑢 𝛿𝑢 𝐺 𝑢 𝛿𝑢 , 𝑧 𝛿𝑢 , 𝑧 𝛿𝑢 , 𝑧 𝛽𝑢 , 𝛿𝑢 𝛽𝑢 , 𝛿𝑢 𝛽𝑢 , 𝛿𝑢 ‖𝛿�⃗�‖ ‖𝛿𝑢‖ 𝛿𝑢, 𝑧 𝛽𝑢, 𝛿𝑢 ‖𝛿�⃗�‖ ‖𝛿𝑢‖ Or 𝐺 𝑢 𝛿𝑢 𝐺 𝑢 𝑧 𝛽𝑢, 𝛿𝑢 𝚺 ‖𝛿�⃗�‖𝑸 ‖𝛿𝑢‖𝚺 . From Theorem 4.1, we have ‖𝛿�⃗�‖𝑸 𝜀 𝛿𝑢 ‖𝛿𝑢‖𝚺 and ‖𝛿𝑢‖𝚺 𝜀 𝛿𝑢 ‖𝛿𝑢‖𝚺 With 𝜀 𝛿𝑢 𝑀 ‖𝛿𝑢‖𝚺 , where 𝜀 𝛿𝑢 , 𝜀 𝛿𝑢 ⟶ 0 as ‖𝛿𝑢‖𝚺 ⟶ 0 Then 𝐺 𝑢 𝛿𝑢 𝐺 𝑢 𝑧 𝛽𝑢, 𝛿𝑢 𝚺 𝜀 𝛿𝑢 ‖𝛿𝑢‖𝚺 Where 𝜀 𝛿𝑢 𝜀 𝛿𝑢 𝜀 𝛿𝑢 ⟶ 0 as ‖𝛿𝑢‖ ⟶ 0 Using the definition of FrD of 𝐺 , one has 𝐺 ´ 𝑢 , 𝛿𝑢 𝚺 𝑧 𝛽𝑢, 𝛿𝑢 𝚺 . Theorem 5.2: The NCsThOP for the CCBOCV of the above problem is 𝐺 ´ 𝑢 𝑧 𝛽𝑢 0 with �⃗� �⃗� ⃗ and 𝑧 𝑧 ⃗ . Proof: If 𝑢 is an CCBOCV of the problem, then 𝐺 𝑢 min ⃗∈ ⃗ 𝐺 𝑢 ∀𝑢 ∈ 𝐿 Σ , i.e. 𝐺 ´ 𝑢 0 ⇒ 𝑧 𝛽𝑢 0 From Theorem 5.1 𝑧 𝛽𝑢 , 𝛿𝑢 0 with 𝛿𝑢 𝑤 𝑢 ⇒ 𝑧 𝛽𝑢 , 𝑤 𝚺 𝑧 𝛽𝑢 , 𝑢 𝚺 , ∀ 𝑤 ∈ 𝐿 Σ . 6. 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