135 This work is licensed under a Creative Commons Attribution 4.0 International License. The Classical Continuous Optimal Control for Quaternary Nonlinear Parabolic Boundary Value Problems with State Vector Constraints 1Jamil A. Ali Al-Hawasy 2Wissam A. Abdul-Hussien Al-Anbaki 1-2Department of Mathematics, College of Science, Mustansiriyah University, Baghdad, Iraq. Jhawassy17@uomustansiriyah.edu.iq wissamali14595@uomustansiriyah.edu.iq Abstract This paper aims to study the quaternary classical continuous optimal control problem consisting of the quaternary nonlinear parabolic boundary value problem, the cost function, and the equality and inequality constraints on the state and the control. Under appropriate hypotheses, it is demonstrated that the quaternary classical continuous optimal control ruling by the quaternary nonlinear parabolic boundary value problem has a quaternary classical continuous optimal control vector that satisfies the equality constraint and inequality state and control constraint. Moreover, mathematical formulation of the quaternary adjoint equations related to the quaternary state equations is discovered, and then the weak form of the quaternary adjoint equations is obtained. Lastly, both the necessary conditions for optimality and sufficient conditions for optimality of the proposed problem are stated and proved. The derivation for the Fréchet derivative of the Hamiltonian is attained. Keywords: Quaternary Classical Optimal Control, Quaternary Nonlinear Parabolic Boundary Value Problems, Necessary and Sufficient for Optimality Theorems. 1. Introduction It is a well-known fact that optimal control problems (OCPs) are widely used in a variety of scientific fields, including biology [1], economics [2], robotics [3], Aircraft [4], and many others. OCPs are typically ruled by nonlinear ODEs (NLODEs) [5] or nonlinear PDEs (NLPDEs) [6]. During the last decade, great attention has been made to studying OCPs for system ruling by NLPDEs of elliptic, hyperbolic, and parabolic types [7-9]. Later, the study of this subject is expanded to include classical continuous optimal control problem (CCOCP) for systems ruling by couple NLPDEs and then recently by triple NLPDEs for the above three Ibn Al-Haitham Journal for Pure and Applied Sciences http://jih.uobaghdad.edu.iq/index.php/j/index: Journal homepage Doi: 10.30526/35.3.2816 Article history: Received 1 March 2022, Accepted 22 March 2022, Published in July 2022. https://creativecommons.org/licenses/by/4.0/ file:///D:/رسالة%20ماجستير/ibn%20al-haitham/Jhawassy17@uomustansiriyah.edu.iq file:///D:/رسالة%20ماجستير/ibn%20al-haitham/wissamali14595@uomustansiriyah.edu.iq file:///F:/العدد%20الثاني%202022/:%20http:/jih.uobaghdad.edu.iq/index.php/j/index IHJPAS. 53 (3)2022 136 indicated types of NLPDEs [10 - 15]. As a result, these concerns made us study the quaternary classical continuous optimal control problem (QCCOCP) ruling by quaternary nonlinear parabolic boundary value problems (QNLPBVPs) with equality constraint (EQC) and inequality constraint (INEQC). This paper is concerned with studying the QCCOCP ruling by a QNLPBVP; it begins with stating and demonstrating the existence theorem of a quaternary classical continuous optimal control vector (QCCOCV) ruling by the QNLPBVP with EQC and INEQC under suitable hypotheses. In addition, the mathematical formulation of the quaternary adjoint equations (QAEs) related to the quaternary state equations (QSEs) is discovered so as the weak form (WF). Moreover, the Fréchet derivative (FrD) of the Hamiltonian is attained. Lastly, both the necessary conditions for optimality (NCsTh) and sufficient conditions (SCsTh)) for optimality are stated and demonstrated. Description of the problem Let Ω ⊂ ℝ2 be an open and bounded region with boundary Γ = 𝜕Ω, 𝑥 = (𝑥1,𝑥2), 𝑄 = 𝐼 ×Ω, 𝐼 = [0,𝑇] , Γ = 𝜕Ω , Σ = Γ×I. The QCCOC consists of the continuous quaternary state vector solution (CQSVS), which is expressed by the following QNLPBVP: 𝑦1𝑡 −∆𝑦1 +𝑦1 −𝑦2 +𝑦3 +𝑦4 = 𝑓1(𝑥,𝑡,𝑦1,𝑢1) , in 𝑄 (1) 𝑦2𝑡 −∆𝑦2 +𝑦1 +𝑦2 −𝑦3 −𝑦4 = 𝑓2(𝑥,𝑡,𝑦2,𝑢2), in 𝑄 (2) 𝑦3𝑡 −∆𝑦3 −𝑦1 +𝑦2 +𝑦3 +𝑦4 = 𝑓3(𝑥,𝑡,𝑦3,𝑢3) , in 𝑄 (3) 𝑦4𝑡 −∆𝑦4 −𝑦1 +𝑦2 −𝑦3 +𝑦4 = 𝑓4(𝑥,𝑡,𝑦4,𝑢4) , in 𝑄 (4) With the following boundary conditions (BCs) and initial conditions (ICs): 𝑦𝑖(𝑥,𝑡) = 0 , ∀ 𝑖 = 1,2,3,4. on Σ (5) 𝑦𝑖(𝑥,0) = 𝑦𝑖 0(𝑥) , ∀ 𝑖 = 1,2,3,4. on Ω (6) Where �⃗� = (𝑦1,𝑦2,𝑦3,𝑦4) = (𝑦1(𝑥,𝑡),𝑦2(𝑥,𝑡),𝑦3(𝑥,𝑡),𝑦4(𝑥,𝑡)) ∈ (𝐻 2(Q̅)) 4 is the quaternary state vector solution (QSVS), �⃗⃗� = (𝑢1,𝑢2,𝑢3,𝑢4) = (𝑢1(𝑥,𝑡),𝑢2(𝑥,𝑡),𝑢3(𝑥,𝑡),𝑢4(𝑥,𝑡)) ∈ (𝐿 2(Q)) 4 is the QCCCV and (𝑓1,𝑓2,𝑓3,𝑓4) = (𝑓1(𝑥,𝑡),𝑓2(𝑥,𝑡),𝑓3(𝑥,𝑡),𝑓4(𝑥,𝑡)) ∈ (𝐿 2(Q)) 4 is given, for all 𝑥 = (𝑥1,𝑥2) ∈ Ω. The set of admissible control (SAC) is: �⃗⃗⃗⃗�𝐴= {�⃗⃗⃗� ∈ (𝐿 2(Q)) 4 |�⃗⃗⃗� ∈ �⃗⃗⃗� ⊂ ℝ4 a.e. in Q ,𝐺1(�⃗⃗⃗�) = 0 ,𝐺2(�⃗⃗⃗�) ≤ 0 }. The CF is: 𝐺0(�⃗⃗�) = ∫ 𝑔01(𝑥,𝑡,𝑦1,𝑢1)𝑑𝑥𝑑𝑡 +∫ 𝑔02(𝑥,𝑡,𝑦2,𝑢2)𝑑𝑥𝑑𝑡 𝑄 𝑄 +∫ 𝑔03(𝑥,𝑡,𝑦3,𝑢3)𝑑𝑥𝑑𝑡 𝑄 + ∫ 𝑔04(𝑥,𝑡,𝑦4,𝑢4)𝑑𝑥𝑑𝑡 𝑄 , (7.a) The constraints on the state and the control (CSSC) are: 𝐺1(�⃗⃗�) = ∫ 𝑔11(𝑥,𝑡,𝑦1,𝑢1)𝑑𝑥𝑑𝑡 +∫ 𝑔12(𝑥,𝑡,𝑦2,𝑢2)𝑑𝑥𝑑𝑡 𝑄 𝑄 +∫ 𝑔13(𝑥,𝑡,𝑦3,𝑢3)𝑑𝑥𝑑𝑡 𝑄 + ∫ 𝑔14(𝑥,𝑡,𝑦4,𝑢4)𝑑𝑥𝑑𝑡 𝑄 = 0, (7.b) 𝐺2(�⃗⃗�) = ∫ 𝑔21(𝑥,𝑡,𝑦1,𝑢1)𝑑𝑥𝑑𝑡 +∫ 𝑔22(𝑥,𝑡,𝑦2,𝑢2)𝑑𝑥𝑑𝑡 𝑄 𝑄 +∫ 𝑔23(𝑥,𝑡,𝑦3,𝑢3)𝑑𝑥𝑑𝑡 𝑄 + ∫ 𝑔24(𝑥,𝑡,𝑦4,𝑢4)𝑑𝑥𝑑𝑡 𝑄 ≤ 0, (7.c) Where (𝑦1,𝑦2,𝑦3,𝑦4) = (𝑦𝑢1,𝑦𝑢2,𝑦𝑢3,𝑦𝑢4) is the QSVS of ((1) – (6)) corresponding to the QCCCV (𝑢1,𝑢2,𝑢3,𝑢4). IHJPAS. 53 (3)2022 137 Let �⃗⃗� = 𝑉1 ×𝑉2 ×V3 ×V4 = (ℋ0 1(Ω)) 4 and �⃗� = (𝑣1,𝑣2,𝑣3,𝑣4) = (𝑣1(𝑥), 𝑣2(𝑥),𝑣3(𝑥),𝑣4(𝑥)). �⃗⃗� = {�⃗�: �⃗� ∈ (ℋ0 1(Ω)) 4 ,with 𝑣1 = 𝑣2 = 𝑣3 = 𝑣4 = 0 on 𝜕Ω}. The WF of the QSVEs: The wf of ((1)−(6)) with �⃗� ∈ (ℋ0 1(Ω)) 4 is given by 〈𝑦1𝑡,𝑣1〉+(∇𝑦1,∇𝑣1)+(𝑦1,𝑣1)−(𝑦2,𝑣1)+(𝑦3,𝑣1)+(𝑦4,𝑣1) = (𝑓1(𝑦1,𝑢1),𝑣1), (8.a) (𝑦1 0,𝑣1) = (𝑦1(0),𝑣1), (8.b) 〈𝑦2𝑡,𝑣2〉+(∇𝑦2,∇𝑣2)+(𝑦1,𝑣2)+(𝑦2,𝑣2)−(𝑦3,𝑣2)−(𝑦4,𝑣2) = (𝑓2(𝑦2,𝑢2),𝑣2), (9.a) (𝑦2 0,𝑣2) = (𝑦2(0),𝑣2), (9.b) 〈𝑦3𝑡,𝑣3〉+(∇𝑦3,∇𝑣3)−(𝑦1,𝑣3)+(𝑦2,𝑣3)+(𝑦3,𝑣3)+(𝑦4,𝑣3) = (𝑓3(𝑦3,𝑢3),𝑣3), (10.a) (𝑦3 0,𝑣3) = (𝑦3(0),𝑣3), (10.b) 〈𝑦4𝑡,𝑣4〉+(∇𝑦4,∇𝑣4)−(𝑦1,𝑣4)+(𝑦2,𝑣4)−(𝑦3,𝑣4)+(𝑦4,𝑣4) = (𝑓4(𝑦4,𝑢4),𝑣4), (11.a) (𝑦4 0,𝑣4) = (𝑦4(0),𝑣4), (11.b) The following hypotheses are important to study the QCCOC. Hypotheses (𝐀): Assume ∀𝒊 = 𝟏,𝟐,𝟑,𝟒 that: (i) 𝒇𝒊 is Carathéodory type (Cara. T.) on 𝑸×(ℝ) 𝟒, and satisfies the following conditions w.r.t. 𝒚𝒊 & 𝒖𝒊 , i.e.: |𝒇𝒊(𝒙,𝒕,𝒚𝒊,𝒖𝒊)| ≤ 𝜼𝒊(𝒙,𝒕)+𝒄𝒊|𝒚𝒊|+ �́�𝒊|𝒖𝒊| , where (𝒙,𝒕) ∈ 𝑸, 𝒄𝒊, �́�𝒊 > 𝟎 and 𝜼𝒊 ∈ 𝑳𝟐(𝑸,ℝ). (ii) 𝑓𝑖 satisfies Lipschitz condition (LC) w.r.t. 𝑦𝑖, i.e.: |𝑓𝑖(𝑥,𝑡,𝑦𝑖,𝑢𝑖)−𝑓𝑖(𝑥,𝑡, �̅�𝑖,𝑢𝑖)| ≤ 𝐿𝑖|𝑦𝑖 −�̅�𝑖|. Where (𝑥,𝑡) ∈ 𝑄 , 𝑦𝑖, �̅�𝑖,𝑢𝑖 ∈ ℝ and 𝐿𝑖 > 0. Theorem (2.1) [13]: (EUTh for the wf of the QSVEs) With hypotheses (A), for each given QCCCV �⃗⃗� ∈ (𝐿2(𝑄)) 4 , the wf ((8) – (11)) has a unique QSVS �⃗� ∈ (𝐿2(𝐼,𝑉)) 4 , with �⃗�𝑡 ∈ (𝐿 2(𝐼,𝑉∗)) 4 . Hypotheses (B): Suppose that for each 𝑙 = 0,1,2 and 𝑖 = 1,2,3,4, that 𝑔𝑙𝑖 is of Cara. T. on 𝑄 ×(ℝ) 4 and satisfies the following conditions w.r.t. 𝑦𝑖 and 𝑢𝑖: |𝑔𝑙𝑖(𝑥,𝑡,𝑦𝑖,𝑢𝑖)| ≤ 𝜂𝑙𝑖(𝑥,𝑡) +𝑐𝑙𝑖1(𝑦𝑖) 2 +𝑐𝑙𝑖2(𝑢𝑖) 2 . Where 𝑦𝑖,𝑢𝑖 ∈ ℝ with 𝜂𝑙𝑖 ∈ 𝐿 1(𝑄). Lemma (2.1): With hypotheses (B), for each 𝑙 = 0,1,2, the functional �⃗⃗� ⟼ 𝐺𝑙(�⃗⃗�) is cont. on (𝐿 2(𝑄))4 . Proof: The requirement result is gotten (∀𝑙 = 0,1,2) directly from hypotheses (B) and Lemma 1.12 in [16]. Theorem (2.2) [16]: Consider the set W⃗⃗⃗⃗A ≠ ∅, for each 𝑖 = 1,2,3,4, the functions 𝑓𝑖 has the form 𝑓𝑖(𝑥,𝑡,𝑦𝑖,𝑢𝑖) = 𝑓𝑖1(𝑥,𝑡,𝑦𝑖)+ 𝑓𝑖2(𝑥,𝑡)𝑢𝑖 With |𝑓𝑖1 (𝑥,𝑡,𝑦𝑖)| ≤ ƞ𝑖(𝑥,𝑡) + 𝑐𝑖| 𝑦𝑖| where ƞ𝑖 ∈ 𝐿 2(𝑄) and |𝑓𝑖2(𝑥,𝑡)| ≤ 𝑘𝑖, If ∀𝑖 = 1,2,3,4, 𝑔0𝑖 is convex (CO) w.r.t. 𝑢𝑖 for fixed (𝑥,𝑡,𝑦𝑖). Then there is a QCCOCV. Hypotheses (C): Assume that, for 𝑙 = 0,2 and 𝑖 = 1,2,3,4 𝑔𝑙𝑖𝑦𝑖 and 𝑔𝑙𝑖𝑢𝑖 are of Cara. T. on ×(ℝ) 4 , |𝑔𝑙𝑖𝑦𝑖(𝑥,𝑡,𝑦𝑖,𝑢𝑖)| ≤ 𝜂𝑙𝑖5(𝑥,𝑡) +𝑐𝑙𝑖5|𝑦𝑖|+ �́�𝑙𝑖5|𝑢𝑖|, and |𝑔𝑙𝑖𝑢𝑖(𝑥,𝑡,𝑦𝑖,𝑢𝑖)| ≤ 𝜂𝑙𝑖6(𝑥,𝑡)+𝑐𝑙𝑖6|𝑦𝑖|+ �́�𝑙𝑖6|𝑢𝑖|. Where(𝑥,𝑡) ∈ 𝑄, 𝑦𝑖,𝑢𝑖 ∈ ℝ , 𝜂𝑙𝑖5 ,𝜂𝑙𝑖6 ∈ 𝐿 2(𝑄). IHJPAS. 53 (3)2022 138 Theorem (2.3) [16]: In addition to hypotheses (A), if �⃗� and �⃗� + 𝛿𝑦⃗⃗⃗⃗⃗ are the QSVS corresponding to the QCCCV �⃗⃗� , �⃗⃗� + 𝛿𝑢⃗⃗⃗⃗⃗ ∈ (𝐿2(𝑄))4, resp., then ‖𝛿𝑦⃗⃗⃗⃗⃗‖ 𝑳∞(𝑰,𝑳𝟐(𝜴) ≤ M ‖𝛿𝑢⃗⃗⃗⃗⃗‖ 𝑳𝟐(𝑸) , ‖𝛿𝑦⃗⃗⃗⃗⃗‖ 𝑳𝟐(𝑸) ≤ M ‖𝛿𝑢⃗⃗⃗⃗⃗‖ 𝑳𝟐(𝑸) , ‖𝛿𝑦⃗⃗⃗⃗⃗‖ 𝑳𝟐(𝑰,𝑽) ≤ M ‖𝛿𝑢⃗⃗⃗⃗⃗‖ 𝑳𝟐(𝑸) . Theorem (2.4) (The Kuhn-Tucker-Lagrange conditions (KTL)) [10]: Let 𝑈 be a nonempty CO subset of a vector space 𝑋, 𝐾 be a nonempty CO positive cone in a normed space 𝑍, and 𝑊 = {𝑢 ∈ 𝑈|𝐺1(𝑢) = 0,𝐺1(𝑢) ∈ −𝐾}. The functional 𝐺0:𝑈 → ℝ,𝐺1:𝑈 → ℝ 𝑚 ,𝐺2:𝑈 → 𝑍 are (𝑚 +1)− locally continuous at 𝑢 ∈ 𝑈, and have (𝑚 +1)− derivatives at 𝑢 where 𝑚 ≠ 0. And if 𝑚 = 0 , we assume that 𝐷𝐺𝑙(𝑢), 𝑙 = 0,1,2 , are 𝐾 -linear at the point 𝑢. If 𝐺0(𝑢) has a minimum at 𝑢 in 𝑊, then it satisfies the following KUTULA conditions, ∀𝑤 ∈ 𝑊: There exists 𝜆0 ∈ ℝ,𝜆1 ∈ ℝ 𝑚,𝜆2 ∈ ℤ ∗ , with 𝜆0 ≥ 0,𝜆2 ≥ 0 ,∑ |𝜆𝑙| = 1 2 𝑙=0 s.t 𝜆0𝐷𝐺0(𝑢,𝑤 −𝑢)+𝜆1 𝑇𝐷𝐺1(𝑢,𝑤 −𝑢)+〈𝜆2,𝐷𝐺2(𝑢,𝑤 −𝑢)〉 ≥ 0, 〈𝜆2,𝐺2(𝑢)〉 = 0. Main Results 3. Existence of the QCCOCV and the FrD: This section deals with the existence theorem of the QCCOCV, the discovery of the mathematical formulation for the QAEs and their WF is obtained, and the derivation of the FrD is derived under some appropriate hypotheses. Theorem (3.1): Consider the set W⃗⃗⃗⃗A ≠ ∅, the functions 𝑓𝑖, ∀𝑖 = 1,2,3,4, has the form 𝑓𝑖(𝑥,𝑡,𝑦𝑖,𝑢𝑖) = 𝑓𝑖1(𝑥,𝑡,𝑦𝑖)+ 𝑓𝑖2(𝑥,𝑡)𝑢𝑖 With |𝑓𝑖1 (𝑥,𝑡,𝑦𝑖)| ≤ ƞ𝑖(𝑥,𝑡) + 𝑐𝑖| 𝑦𝑖| and |𝑓𝑖2(𝑥,𝑡)| ≤ 𝑘𝑖, where ƞ𝑖 ∈ 𝐿 2(𝑄). If ∀𝑖 = 1,2,3,4 , 𝑔1𝑖 is independent of 𝑢𝑖, 𝑔0𝑖 and 𝑔2𝑖 are convex w.r.t. 𝑢𝑖 for fixed (𝑥,𝑡,𝑦𝑖). Then there is a QCCOCV. Proof: From the hypotheses on 𝑊𝑖 and 𝑔1𝑖 (∀𝑖 = 1,2,3,4), with using lemma (2.1) and the. 2.2, one can get that there is a QCCOCV with the EQC and INEQC. Theorem (3.2): We drop the index 𝑙 in 𝑔𝑙 and 𝐺𝑙 . In addition to hypotheses (A), (B), and (C), the following adjoint (𝑧1 ,𝑧2 ,𝑧3 ,𝑧4) = (𝑧𝑢1 ,𝑧𝑢2 ,𝑧𝑢3 ,𝑧𝑢4) equations corresponding to the state (𝑦1 ,𝑦2 ,𝑦3 ,𝑦4) = (𝑦𝑢1 ,𝑦𝑢2 ,𝑦𝑢3 ,𝑦𝑢4) equations ((1) – (6)) are expressed by: −𝑧1𝑡 −∆𝑧1 +𝑧1 +𝑧2 −𝑧3 −𝑧4 = 𝑧1𝑓𝑦1(𝑥,𝑡,𝑦1,𝑢1)+𝑔𝑦1(𝑥,𝑡,𝑦1,𝑢1) , (12) −𝑧2𝑡 −∆𝑧2 +𝑧2 −𝑧1 +𝑧3 +𝑧4 = 𝑧1𝑓𝑦1(𝑥,𝑡,𝑦1,𝑢1)+𝑔𝑦1(𝑥,𝑡,𝑦1,𝑢1) , (13) −𝑧3𝑡 −∆𝑧3 +𝑧3 +𝑧1 −𝑧2 −𝑧4 = 𝑧1𝑓𝑦1(𝑥,𝑡,𝑦1,𝑢1)+𝑔𝑦1(𝑥,𝑡,𝑦1,𝑢1) , (14) −𝑧4𝑡 −∆𝑧4 +𝑧4 +𝑧1 −𝑧2 +𝑧3 = 𝑧1𝑓𝑦1(𝑥,𝑡,𝑦1,𝑢1)+𝑔𝑦1(𝑥,𝑡,𝑦1,𝑢1) , (15) 𝑧𝑖(𝑥,𝑡) = 0, ∀𝑖 = 1,2,3,4, on Σ , (16) 𝑧𝑖(𝑇) = 0, ∀𝑖 = 1,2,3,4, on Γ , (17) Also, the Hamiltonian is defined: 𝐻(𝑥,𝑡,𝑦𝑖, 𝑧𝑖,𝑢𝑖) = ∑ 𝑧𝑖𝑓𝑖(𝑥,𝑡,𝑦𝑖,𝑢𝑖)+𝑔𝑖(𝑥,𝑡,𝑦𝑖,𝑢𝑖) 4 𝑖=1 , Then the FrD of 𝐺 is given by �́�(�⃗⃗�) ∙ 𝛿𝑢⃗⃗⃗⃗⃗ = ∫ ( 𝑧1𝑓𝑢1 +𝑔𝑢1 𝑧2𝑓𝑢2 +𝑔𝑢2 𝑧3𝑓𝑢3 +𝑔𝑢3 𝑧4𝑓𝑢4 +𝑔𝑢4) 𝑄 ∙( 𝛿𝑢1 𝛿𝑢2 𝛿𝑢3 𝛿𝑢4 )𝑑𝑥. IHJPAS. 53 (3)2022 139 Proof: Firstly, let �⃗⃗� be a QCCCV, and �⃗� be its QSVS, and let 𝐺(�⃗⃗�) = ∑ ∫ 𝑔𝑖(𝑥,𝑡,𝑦𝑖,𝑢𝑖)𝑑𝑥𝑑𝑡 𝑄 4 𝑖=1 , From the hypotheses on 𝑔𝑙 (𝑙 = 1,2,3,4), the FrD definition, the result of The. 2.3, and then using the Minkowski’s inequality (MKIN), one can get that: 𝐺(�⃗⃗� +𝛿𝑢⃗⃗⃗⃗⃗)−𝐺(�⃗⃗�) = ∫ (𝑔𝑦1𝛿𝑦1 +𝑔𝑢1𝛿𝑢1) 𝑄 𝑑𝑥𝑑𝑡 +∫ (𝑔𝑦2𝛿𝑦2 +𝑔𝑢2𝛿𝑢2) 𝑄 𝑑𝑥𝑑𝑡 + ∫ (𝑔𝑦3𝛿𝑦3 +𝑔𝑢3𝛿𝑢3) 𝑄 𝑑𝑥𝑑𝑡 +∫ (𝑔𝑦4𝛿𝑦4 +𝑔𝑢4𝛿𝑢4) 𝑄 𝑑𝑥𝑑𝑡 + 6(𝛿𝑢⃗⃗⃗⃗⃗)‖𝛿𝑢⃗⃗⃗⃗⃗ ‖𝑳𝟐(𝑸) (18) Where 6(𝛿𝑢⃗⃗⃗⃗⃗) = 2(𝛿𝑢⃗⃗⃗⃗⃗)+ 3(𝛿𝑢⃗⃗⃗⃗⃗)+ 4(𝛿𝑢⃗⃗⃗⃗⃗)+ 5(𝛿𝑢⃗⃗⃗⃗⃗) ⟶ 0 as ‖𝛿𝑢⃗⃗⃗⃗⃗ ‖𝑳𝟐(𝑸) ⟶ 0. On the other hand, the wf of the QAEs for 𝑣𝑖 ∈ 𝑉, ∀ 𝑖 = 1,2,3,4 is given by: −〈𝑧1𝑡,𝑣1〉+(∇𝑧1,∇𝑣1)+(𝑧1,𝑣1)+(𝑧2,𝑣1)−(𝑧3,𝑣1)−(𝑧4,𝑣1) = (𝑧1𝑓1𝑦1,𝑣1)+ (𝑔1𝑦1,𝑣1), (19) −〈𝑧2𝑡,𝑣2〉+(∇𝑧2,∇𝑣2)+(𝑧2,𝑣2)−(𝑧1,𝑣2)+(𝑧3,𝑣2)+(𝑧4,𝑣2) = (𝑧2𝑓2𝑦2,𝑣2) + (𝑔2𝑦2,𝑣2), (20) −〈𝑧3𝑡,𝑣3〉+(∇𝑧3,∇𝑣3)+(𝑧3,𝑣3)+(𝑧1,𝑣3)−(𝑧2,𝑣3)−(𝑧4,𝑣3) = (𝑧3𝑓3𝑦3,𝑣3) + (𝑔3𝑦3,𝑣3), (21) −〈𝑧4𝑡,𝑣4〉 +(∇𝑧4,∇𝑣4)+(𝑧4,𝑣4)+(𝑧1,𝑣4)−(𝑧2,𝑣4)+(𝑧3,𝑣4) = (𝑧4𝑓4𝑦4,𝑣4) + (𝑔4𝑦4,𝑣4), (22) The existence of a unique solution of ((19) – (22)) can be proved by the same manner which is used in the proof of the unique of the QSVS. Now, substituting 𝑣𝑖 = 𝛿𝑦𝑖, ∀𝑖 = 1,2,3,4 in ((19) − (22)) resp., then taking the integrating both sides (IBS) from 0 to 𝑇, lastly, applying integration by part (IBP) for the 1st terms of each resulting equation, to get that: ∫ 〈𝛿𝑦1,𝑧1 〉 𝑇 0 𝑑𝑡 +∫ [(∇𝑧1,∇𝛿𝑦1 )+(𝑧1,𝛿𝑦1 )+(𝑧2,𝛿𝑦1 )−(𝑧3,𝛿𝑦1 )−(𝑧4,𝛿𝑦1 )] 𝑇 0 𝑑𝑡 = ∫ [(𝑧1𝑓1𝑦1,𝛿𝑦1)+(𝑔1𝑦1,𝛿𝑦1)]𝑑𝑡 𝑇 0 , (23) ∫ 〈𝛿𝑦2,𝑧2〉 𝑇 0 𝑑𝑡 +∫ [(∇𝑧2,∇𝛿𝑦2)+(𝑧2,𝛿𝑦2)−(𝑧1,𝛿𝑦2)+(𝑧3,𝛿𝑦2)+(𝑧4,𝛿𝑦2)]𝑑𝑡 𝑇 0 = ∫ [(𝑧2𝑓2𝑦2,𝛿𝑦2)+(𝑔2𝑦2,𝛿𝑦2)]𝑑𝑡 𝑇 0 , (24) ∫ 〈𝛿𝑦3,𝑧3〉 𝑇 0 𝑑𝑡 +∫ [(∇𝑧3,∇𝛿𝑦3)+(𝑧3,𝛿𝑦3)+(𝑧1,𝛿𝑦3)−(𝑧2,𝛿𝑦3)−(𝑧4,𝛿𝑦3)]𝑑𝑡 𝑇 0 = ∫ [(𝑧3𝑓3𝑦3,𝛿𝑦3)+(𝑔3𝑦3,𝛿𝑦3)]𝑑𝑡 𝑇 0 , (25) ∫ 〈𝛿𝑦4,𝑧4〉 𝑇 0 𝑑𝑡 +∫ [(∇𝑧4,∇𝛿𝑦4)+(𝑧4,𝛿𝑦4)+(𝑧1,𝛿𝑦4)−(𝑧2,𝛿𝑦4)+(𝑧3,𝛿𝑦4)]𝑑𝑡 𝑇 0 = ∫ [(𝑧4𝑓4𝑦4,𝛿𝑦4)+(𝑔4𝑦4,𝛿𝑦4)]𝑑𝑡 𝑇 0 , (26) Also, substituting 𝑣𝑖 = 𝑧𝑖, ∀𝑖 = 1,2,3,4 in ((8.a) − (11.a)) resp., then IBS w.r.t. 𝑡 from 0 to 𝑇, to obtain: ∫ 〈𝛿𝑦1𝑡,𝑧1〉 𝑇 0 𝑑𝑡 +∫ [(∇δ𝑦1,∇𝑧1)+(𝛿𝑦1,𝑧1)−(𝛿𝑦2,𝑧1)+(𝛿𝑦3,𝑧1)+(𝛿𝑦4,𝑧1)] 𝑇 0 𝑑𝑡 = ∫ (𝑓1(𝑦1 +𝛿𝑦1,𝑢1 +𝛿𝑢1),𝑧1)𝑑𝑡 𝑇 0 −∫ (𝑓1(𝑦1,𝑢1),𝑧1)𝑑𝑡 𝑇 0 , (27) ∫ 〈𝛿𝑦2𝑡,𝑧2〉 𝑇 0 𝑑𝑡 +∫ [(∇𝛿𝑦2,∇𝑧2)+(𝛿𝑦1,𝑧2)+(𝛿𝑦2,𝑧2)−(𝛿𝑦3,𝑧2)−(𝛿𝑦4,𝑧2)] 𝑇 0 𝑑𝑡 = ∫ (𝑓2(𝑦2 +𝛿𝑦2,𝑢2 +𝛿𝑢2),𝑧2)𝑑𝑡 𝑇 0 −∫ (𝑓2(𝑦2,𝑢2),𝑧2)𝑑𝑡 𝑇 0 , (28) IHJPAS. 53 (3)2022 140 ∫ 〈𝛿𝑦3𝑡,𝑧3〉 𝑇 0 𝑑𝑡 +∫ [(∇𝛿𝑦3,∇𝑧3)−(𝛿𝑦1,𝑧3)+(𝛿𝑦2,𝑧3)+(𝛿𝑦3,𝑧3)+(𝛿𝑦4,𝑧3)] 𝑇 0 𝑑𝑡 = ∫ (𝑓3(𝑦3 +𝛿𝑦3,𝑢3 +𝛿𝑢3),𝑧3)𝑑𝑡 𝑇 0 −∫ (𝑓3(𝑦3,𝑢3),𝑧3)𝑑𝑡 𝑇 0 , (29) ∫ 〈𝛿𝑦4𝑡,𝑧4〉 𝑇 0 𝑑𝑡 +∫ [(∇𝛿𝑦4,∇𝑧4)−(𝛿𝑦1,𝑧4)+(𝛿𝑦2,𝑧4)−(𝛿𝑦3,𝑧4)+(𝛿𝑦4,𝑧4)] 𝑇 0 𝑑𝑡 = ∫ (𝑓4(𝑦4 +𝛿𝑦4,𝑢4 +𝛿𝑢4),𝑧4)𝑑𝑡 𝑇 0 −∫ (𝑓4(𝑦4,𝑢4),𝑧4)𝑑𝑡 𝑇 0 , (30) Using the hypotheses on 𝑓𝑖 (for each 𝑖 = 1,23,4) , the FrD of it exists, then from the result of The. 2.3, and the MKIN, the followings are yielded: ∫ 〈𝛿𝑦1𝑡,𝑧1〉 𝑇 0 𝑑𝑡 +∫ [(∇δ𝑦1,∇𝑧1)+(𝛿𝑦1,𝑧1)−(𝛿𝑦2,𝑧1)+(𝛿𝑦3,𝑧1)+(𝛿𝑦4,𝑧1)] 𝑇 0 𝑑𝑡 = ∫ ( 𝑇 0 𝑓1𝑦1𝛿𝑦1 +𝑓1𝑢1𝛿𝑢1,𝑧1)𝑑𝑡 + 12(𝛿𝑢 ⃗⃗⃗⃗⃗)‖𝛿𝑢⃗⃗⃗⃗⃗‖ 𝑳𝟐(𝑸) , (31) ∫ 〈𝛿𝑦2𝑡,𝑧2〉 𝑇 0 𝑑𝑡 +∫ [(∇𝛿𝑦2,∇𝑧2)+(𝛿𝑦1,𝑧2)+(𝛿𝑦2,𝑧2)−(𝛿𝑦3,𝑧2)−(𝛿𝑦4,𝑧2)] 𝑇 0 𝑑𝑡 = ∫ ( 𝑇 0 𝑓2𝑦2𝛿𝑦2 +𝑓2𝑢2𝛿𝑢2,𝑧2)𝑑𝑡 + 22(𝛿𝑢 ⃗⃗⃗⃗⃗)‖𝛿𝑢⃗⃗⃗⃗⃗‖ 𝑳𝟐(𝑸) , (32) ∫ 〈𝛿𝑦3𝑡,𝑧3〉 𝑇 0 𝑑𝑡 +∫ [(∇𝛿𝑦3,∇𝑧3)−(𝛿𝑦1,𝑧3)+(𝛿𝑦2,𝑧3)+(𝛿𝑦3,𝑧3)+(𝛿𝑦4,𝑧3)] 𝑇 0 𝑑𝑡 = ∫ ( 𝑇 0 𝑓3𝑦3𝛿𝑦3 +𝑓3𝑢3𝛿𝑢3,𝑧3)𝑑𝑡 + 32(𝛿𝑢 ⃗⃗⃗⃗⃗)‖𝛿𝑢⃗⃗⃗⃗⃗‖ 𝑳𝟐(𝑸) , (33) ∫ 〈𝛿𝑦4𝑡,𝑧4〉 𝑇 0 𝑑𝑡 +∫ [(∇𝛿𝑦4,∇𝑧4)−(𝛿𝑦1,𝑧4)+(𝛿𝑦2,𝑧4)−(𝛿𝑦3,𝑧4)+(𝛿𝑦4,𝑧4)] 𝑇 0 𝑑𝑡 = ∫ ( 𝑇 0 𝑓4𝑦4𝛿𝑦4 +𝑓4𝑢4𝛿𝑢4,𝑧4)𝑑𝑡 + 42(𝛿𝑢 ⃗⃗⃗⃗⃗)‖𝛿𝑢⃗⃗⃗⃗⃗‖ 𝑳𝟐(𝑸) , (34) By subtracting ((31) – (34)) from ((23) − (26)) resp., and adding the attained equations, one obtains: ∫ [ 𝑇 0 (𝑓1𝑢1𝛿𝑢1,𝑧1)+(𝑓2𝑢2𝛿𝑢2,𝑧2)+(𝑓3𝑢3𝛿𝑢3, 𝑧3)+(𝑓4𝑢4𝛿𝑢4,𝑧4)]𝑑𝑡 + 5(𝛿𝑢 ⃗⃗⃗⃗⃗)‖𝛿𝑢⃗⃗⃗⃗⃗‖ 𝑳𝟐(𝑸) = ∫ [ 𝑇 0 (𝑔1𝑦1,𝛿𝑦1)+(𝑔2𝑦2,𝛿𝑦2)+(𝑔3𝑦3,𝛿𝑦3)+(𝑔4𝑦4,𝛿𝑦4)]𝑑𝑡, (35) Where 5(𝛿𝑢⃗⃗⃗⃗⃗) = 12(𝛿𝑢⃗⃗⃗⃗⃗)+ 22(𝛿𝑢⃗⃗⃗⃗⃗)+ 32(𝛿𝑢⃗⃗⃗⃗⃗)+ 42(𝛿𝑢⃗⃗⃗⃗⃗) ⟶ 0 , as ‖𝛿𝑢⃗⃗⃗⃗⃗ ‖𝑳𝟐(𝑸) ⟶ 0 , Now, by substituting (35) in (18), one gets: 𝐺(�⃗⃗� +𝛿𝑢⃗⃗⃗⃗⃗)−𝐺(�⃗⃗�) = ∫ [ 𝑄 (𝑧1𝑓1𝑢1 +𝑔1𝑢1)𝛿𝑢1 +(𝑧2𝑓2𝑢2 +𝑔2𝑢2)𝛿𝑢2 +(𝑧3𝑓3𝑢3 + 𝑔3𝑢3)𝛿𝑢3 +(𝑧4𝑓4𝑢4 +𝑔4𝑢4)𝛿𝑢4]𝑑𝑥𝑑𝑡 + 7(𝛿𝑢 ⃗⃗⃗⃗⃗)‖𝛿𝑢⃗⃗⃗⃗⃗ ‖ 𝑳𝟐(𝑸) 36) Where 7(𝛿𝑢⃗⃗⃗⃗⃗) = 5(𝛿𝑢⃗⃗⃗⃗⃗)+ 6(𝛿𝑢⃗⃗⃗⃗⃗) ⟶ 0 , as ‖𝛿𝑢⃗⃗⃗⃗⃗ ‖𝑳𝟐(𝑸) ⟶ 0 , Using the FrD definition of 𝐺, one gets: 𝐺(�⃗⃗� +𝛿𝑢⃗⃗⃗⃗⃗)−𝐺(�⃗⃗�) = (�́�(�⃗⃗�),𝛿𝑢⃗⃗⃗⃗⃗)+ 7(𝛿𝑢⃗⃗⃗⃗⃗)‖𝛿𝑢⃗⃗⃗⃗⃗ ‖𝑳𝟐(𝑸) (37) From (36) and (37), one can get: (�́�(�⃗⃗�),𝛿𝑢⃗⃗⃗⃗⃗) = ∫ ( 𝑧1𝑓1𝑢1 +𝑔1𝑢1 𝑧2𝑓2𝑢2 +𝑔2𝑢2 𝑧3𝑓3𝑢3 +𝑔3𝑢3 𝑧4𝑓4𝑢4 +𝑔4𝑢4) ∙( 𝛿𝑢1 𝛿𝑢2 𝛿𝑢3 𝛿𝑢4 ) 𝑄 𝑑𝑥 . 4. The NCsTh and The SCsTh for Optimality: This section deals with the state and demonstration of the NCsTh, so as the SCsTh, under some additional hypotheses. IHJPAS. 53 (3)2022 141 Theorem (4.1): NCsTh for Optimality: (a) In addition to hypotheses (A), (B) and (C), if �⃗⃗� ∈ �⃗⃗⃗⃗�𝐴 is QCCOCV, then there is "multiplier" 𝜆𝑙 ∈ ℝ,𝑙 = 0,1,2, with 𝜆0 ≥ 0, 𝜆2 ≥ 0, ∑ |𝜆𝑙| = 1 2 𝑙=0 , s.t. the following Lagrange -Kuhn-Tucker conditions (LKT) conditions are held: ∫ 𝐻�⃗⃗⃗�(𝑥,𝑡, �⃗�,𝑧, �⃗⃗�) 𝑄 ∙ 𝛿𝑢⃗⃗⃗⃗⃗𝑑𝑥𝑑𝑡 ≥ 0, ∀�⃗⃗⃗� ∈ �⃗⃗⃗⃗�, 𝛿𝑢⃗⃗⃗⃗⃗ = �⃗⃗⃗� − �⃗⃗� (38.a) Where 𝑔𝑖 = ∑ 𝜆𝑙𝑔𝑙𝑖 2 𝑙=0 ,∀𝑖 = 1,2,3,4 in the definition of 𝐻, and also 𝜆2𝐺2(�⃗⃗�) = 0, (38.b) (b) (Minimum principle in weak form) If �⃗⃗⃗⃗� is of the form �⃗⃗⃗⃗� = {�⃗⃗⃗� ∈ (𝐿2(𝑄,ℝ))4|�⃗⃗⃗�(𝑥,𝑡) ∈ �⃗⃗⃗�a.e.on 𝑄} , with �⃗⃗⃗� ⊂ ℝ2. Then, (37.a) is equivalent to the minimum principle in point-wise form (MPPWF) 𝐻�⃗⃗⃗�(𝑥,𝑡, �⃗�,𝑧, �⃗⃗�)�⃗⃗�(𝑡) = min �⃗⃗⃗�∈�⃗⃗⃗� 𝐻�⃗⃗⃗�(𝑥,𝑡, �⃗�,𝑧, �⃗⃗�)�⃗⃗⃗� a.e. on 𝑄 (39) Proof: (a) The functional 𝐺𝑙(�⃗⃗�) is 𝜌 −locall cont. at each �⃗⃗� ∈ �⃗⃗⃗⃗� , ∀𝑙 = 0,1,2 and for every 𝜌 (by hypotheses (A) , (B) and (C) and Lemma 2.1), and 𝐺𝑙(�⃗⃗�) is 𝜌 −differentiable at each �⃗⃗� ∈ �⃗⃗⃗⃗� , ∀𝜌 (by hypotheses (A) , (B) and (C) and The. 3.2) and since �⃗⃗⃗⃗� ⊂ (𝐿2(𝑄)) 4 , 𝐿2(𝑄) is open, then 𝐷𝐺𝑙(�⃗⃗�, �⃗⃗⃗� − �⃗⃗�) = �́�𝑙(�⃗⃗�)(�⃗⃗⃗� − �⃗⃗�) , 𝑙 = 0,1,2, And since �⃗⃗� ∈ �⃗⃗⃗⃗�𝐴 is QCCOCV, by The. 2.4, there is 𝜆𝑙 ∈ ℝ , 𝑙 = 0,1,2 , with 𝜆0 ≥ 0 , 𝜆2 ≥ 0 , ∑ |𝜆𝑙| = 1 2 𝑙=0 s.t. (38a&b) are held. Again by The. 3.2, setting 𝛿𝑢⃗⃗⃗⃗⃗ = �⃗⃗⃗� − �⃗⃗� and substituting the FrD of �́�𝑙, 𝑙 = 0,1,2 in (38.a), one has: ∑ ∫ [(𝜆0𝑧0𝑖 +𝜆1𝑧1𝑖 +𝜆2𝑧2𝑖)𝑓𝑖𝑢𝑖 𝑄 +(𝜆0𝑔0𝑖𝑢𝑖 +𝜆1𝑔1𝑖𝑢𝑖 +𝜆2𝑔2𝑖𝑢𝑖)]𝛿𝑢𝑖𝑑𝑥𝑑𝑡 ≥ 0 4 𝑖=1 , ⇒ ∑ ∫ [(𝑧𝑖𝑓𝑖𝑢𝑖 +𝑔𝑖𝑢𝑖) 𝑄 ]4𝑖=1 𝛿𝑢𝑖𝑑𝑥𝑑𝑡 ≥ 0, Where 𝑔𝑖 = ∑ 𝜆𝑙 2 𝑙=0 𝑔𝑙𝑖 , 𝑧𝑖 = ∑ 𝜆𝑙 2 𝑙=0 𝑧𝑙𝑖 , ∀𝑖 = 1,2,3,4. ⇒ ∫ ( 𝑧1𝑓1𝑢1 +𝑔1𝑢1 𝑧2𝑓2𝑢2 +𝑔2𝑢2 𝑧3𝑓3𝑢3 +𝑔3𝑢3 𝑧4𝑓4𝑢4 +𝑔4𝑢4) ∙( 𝛿𝑢1 𝛿𝑢2 𝛿𝑢3 𝛿𝑢4 ) 𝑄 𝑑𝑥𝑑𝑡 ≥ 0 , or ∫ 𝐻�⃗⃗⃗�(𝑥,𝑡, �⃗�,𝑧, �⃗⃗�) 𝑄 ∙ 𝛿𝑢⃗⃗⃗⃗⃗𝑑𝑥𝑑𝑡 ≥ 0. (ii) To prove that (38.a) is equivalent to (39): Let �⃗⃗⃗⃗��⃗⃗⃗� = {�⃗⃗⃗� ∈ (𝐿 2(𝑄,ℝ))4|�⃗⃗⃗�(𝑥,𝑡) ∈ �⃗⃗⃗�a.e. in 𝑄} with �⃗⃗⃗� ⊂ ℝ2, let {�⃗⃗⃗�𝑘} dense in a set �⃗⃗⃗⃗��⃗⃗⃗� , 𝜇 is “Lebesgue measure” on 𝑄 and let 𝑆 ⊂ 𝑄 be a measurable set s.t.: �⃗⃗⃗�(𝑥,𝑡) = { �⃗⃗⃗�𝑘(𝑥,𝑡) , if (𝑥,𝑡) ∈ 𝑆 �⃗⃗�(𝑥,𝑡) , if (𝑥,𝑡) ∉ 𝑆 Therefore (38.a) becomes: ∫ 𝐻�⃗⃗⃗�(𝑥,𝑡, �⃗�,𝑧, �⃗⃗�) 𝑆 (�⃗⃗⃗�𝑘 −�⃗⃗�) ≥ 0 , ∀𝑆. Using the 3.2, to obtain: 𝐻�⃗⃗⃗�(𝑥,𝑡, �⃗�,𝑧, �⃗⃗�)(�⃗⃗⃗�𝑘 −�⃗⃗�) ≥ 0 , a.e. in 𝑄, 𝐻�⃗⃗⃗�(𝑥,𝑡, �⃗�,𝑧, �⃗⃗�)(�⃗⃗⃗�𝑘 −�⃗⃗�) ≥ 0 , in 𝑃 = ⋂ 𝑃𝑘𝑘 , where 𝑃𝑘 = 𝑄 −𝑄𝑘 with 𝜇(𝑄𝑘) = 0 , ∀𝑘 , since 𝑃 is independent of 𝑘 , hence 𝜇(𝑄 −𝑃) = 𝜇(⋃ 𝑄𝑘𝑘 ) = 0 , from the density of {�⃗⃗⃗�𝑘} in �⃗⃗⃗⃗��⃗⃗⃗� , one has 𝐻�⃗⃗⃗�(𝑥,𝑡, �⃗�,𝑧, �⃗⃗�)(�⃗⃗⃗� −�⃗⃗�) ≥ 0 , a.e. in 𝑄. ⇒ 𝐻�⃗⃗⃗�(𝑥,𝑡, �⃗�,𝑧, �⃗⃗�)�⃗⃗� = min �⃗⃗⃗�∈�⃗⃗⃗� 𝐻�⃗⃗⃗�(𝑥,𝑡, �⃗�,𝑧, �⃗⃗�)�⃗⃗⃗� , ∀�⃗⃗⃗� ∈ �⃗⃗⃗⃗� , a.e. in 𝑄. Conversely, if 𝐻�⃗⃗⃗�(𝑥,𝑡, �⃗�,𝑧, �⃗⃗�)�⃗⃗� = min �⃗⃗⃗�∈�⃗⃗⃗� 𝐻�⃗⃗⃗�(𝑥,𝑡, �⃗�,𝑧, �⃗⃗�)�⃗⃗⃗� , a.e. on 𝑄 IHJPAS. 53 (3)2022 142 ⇒ 𝐻�⃗⃗⃗�(𝑥,𝑡, �⃗�,𝑧, �⃗⃗�)(�⃗⃗⃗� −�⃗⃗�) ≥ 0 , ∀�⃗⃗⃗� ∈ �⃗⃗⃗⃗�, a.e. on 𝑄 ⇒ ∫ 𝐻�⃗⃗⃗�(𝑥,𝑡, �⃗�,𝑧, �⃗⃗�)𝛿𝑢⃗⃗⃗⃗⃗ 𝑄 𝑑𝑥𝑑𝑡 ≥ 0, ∀�⃗⃗⃗� ∈ �⃗⃗⃗⃗�. Theorem (4.2) :( SCsTh for Optimality) In addition to hypotheses (A), (B) and (C), suppose that �⃗⃗⃗⃗� = �⃗⃗⃗⃗��⃗⃗⃗� is CO, 𝑓𝑖 , ∀𝑖 = 1,2,3,4 and 𝑔1𝑖 are affine w.r.t. (𝑦𝑖,𝑢𝑖) in 𝑄, and 𝑔0𝑖 and 𝑔2𝑖 are CO w.r.t. (𝑦𝑖,𝑢𝑖) in 𝑄 ∀𝑖 = 1,2,3,4. Then the NCs in The. 4.1 with 𝜆0 > 0 are also sufficient. Proof: From The. 4.1, 𝐷𝐺𝑙(�⃗⃗�, �⃗⃗⃗� − �⃗⃗�) = �́�𝑙(�⃗⃗�)(�⃗⃗⃗� − �⃗⃗�) , 𝑙 = 0,1,2, Assume that �⃗⃗� satisfies (38) and �⃗⃗� ∈ �⃗⃗⃗⃗�𝐴 , i.e.: ∫ 𝐻�⃗⃗⃗�(𝑥,𝑡, �⃗�,𝑧, �⃗⃗�)𝛿𝑢⃗⃗⃗⃗⃗ 𝑄 𝑑𝑥𝑑𝑡 ≥ 0 , ∀�⃗⃗⃗� ∈ �⃗⃗⃗⃗�. And 𝜆2𝐺2(�⃗⃗�) = 0. Let (�⃗⃗�) = ∑ 𝜆𝑙𝐺𝑙(�⃗⃗�) 2 𝑙=0 , then �́�(�⃗⃗�) ∙ 𝛿𝑢⃗⃗⃗⃗⃗ = ∑ 𝜆𝑙�́�𝑙(�⃗⃗�) 2 𝑙=0 ∙ 𝛿𝑢 ⃗⃗⃗⃗⃗ = 𝜆0 ∫ ∑ (𝑧0𝑖𝑓𝑖𝑢𝑖 +𝑔0𝑖𝑢𝑖) 4 𝑖=1 𝛿𝑢𝑖 𝑄 𝑑𝑥𝑑𝑡 +𝜆1 ∫ ∑ (𝑧1𝑖𝑓𝑖𝑢𝑖 +𝑔1𝑖𝑢𝑖) 4 𝑖=1 𝛿𝑢𝑖 𝑄 𝑑𝑥𝑑𝑡 + 𝜆2 ∫ ∑ (𝑧2𝑖𝑓𝑖𝑢𝑖 +𝑔2𝑖𝑢𝑖) 4 𝑖=1 𝛿𝑢𝑖 𝑄 𝑑𝑥𝑑𝑡, = ∫ 𝐻�⃗⃗⃗�(𝑥,𝑡, �⃗�,𝑧, �⃗⃗�)𝛿𝑢⃗⃗⃗⃗⃗ 𝑄 𝑑𝑥𝑑𝑡 ≥ 0, Now, to demonstrate �⃗⃗� ⟼ �⃗��⃗⃗⃗� is convex – linear (COL), since ∀𝑖 = 1,2,3,4, 𝑓𝑖 is affine from the HYPOTHESES on 𝑓𝑖 , ∀𝑖 = 1,2,3,4: 𝑓𝑖(𝑥,𝑡,𝑦𝑖,𝑢𝑖) = 𝑓𝑖1(𝑥,𝑡)𝑦𝑖 +𝑓𝑖2(𝑥,𝑡)𝑢𝑖 + 𝑓𝑖3(𝑥,𝑡), ∀𝑖 = 1,2,3,4. Let �⃗⃗� = (𝑢1,𝑢2,𝑢3,𝑢4) & �⃗⃗̅� = (�̅�1, �̅�2, �̅�3, �̅�4) be two given QCCCVs and from The. 2.1, �⃗� = (𝑦𝑢1,𝑦𝑢2,𝑦𝑢3,𝑦𝑢4) = (𝑦1,𝑦2,𝑦3,𝑦4) & �⃗̅� = (�̅�𝑢1, �̅�𝑢2, �̅�𝑢3, �̅�𝑢4) = (�̅�1, �̅�2, �̅�3, �̅�4)are their corresponding QSVS, precisely from (1), 𝑦1𝑡 −∆𝑦1 +𝑦1 −𝑦2 +𝑦3 +𝑦4 = 𝑓11(𝑥,𝑡)𝑦1 +𝑓12(𝑥,𝑡)𝑢1 +𝑓13(𝑥,𝑡), 𝑦1(𝑥,0) = 𝑦1 0(𝑥), �̅�1𝑡 −∆�̅�1 +�̅�1 −�̅�2 +�̅�3 +�̅�4 = 𝑓11(𝑥,𝑡)�̅�1 +𝑓12(𝑥,𝑡)�̅�1 +𝑓13(𝑥,𝑡), �̅�1(𝑥,0) = 𝑦1 0(𝑥), By MBS the 1𝑠𝑡 above equation and its IC by 𝛼 ∈ [0,1] , and the 2𝑛𝑑 equation and its IC by (1−𝛼) , and adding the attained equations and their attained ICs , one gets that: (𝛼𝑦1 +(1−𝛼)�̅�1 )𝑡 −∆(𝛼𝑦1 +(1−𝛼)�̅�1)+(𝛼𝑦1 +(1−𝛼)�̅�1)−(𝛼𝑦2 +(1−𝛼)�̅�2)+ (𝛼𝑦3 +(1−𝛼)�̅�3)+(𝛼𝑦4 +(1−𝛼)�̅�4) = 𝑓11(𝑥,𝑡)(𝛼𝑦1 +(1−𝛼)�̅�1)+𝑓12(𝑥,𝑡)(𝛼𝑢1 + (1−𝛼)�̅�1)+𝑓13(𝑥,𝑡) (40.a) 𝛼𝑦1(𝑥,0)+(1−𝛼)�̅�1(𝑥,0) = 𝑦1 0(𝑥) (40.b) By the same way, one obtains that: (𝛼𝑦2 +(1−𝛼)�̅�2 )𝑡 −∆(𝛼𝑦2 +(1−𝛼)�̅�2)+(𝛼𝑦2 +(1−𝛼)�̅�2)+(𝛼𝑦1 +(1−𝛼)�̅�1)− (𝛼𝑦3 +(1−𝛼)�̅�3)−(𝛼𝑦4 +(1−𝛼)�̅�4) = 𝑓21(𝑥,𝑡)(𝛼𝑦2 +(1−𝛼)�̅�2)+𝑓22(𝑥,𝑡)(𝛼𝑢2 + (1−𝛼)�̅�2)+𝑓23(𝑥,𝑡) (41.a) 𝛼𝑦2(𝑥,0)+(1−𝛼)�̅�2(𝑥,0) = 𝑦2 0(𝑥) (41.b) (𝛼𝑦3 +(1−𝛼)�̅�3 )𝑡 −∆(𝛼𝑦3 +(1−𝛼)�̅�3)+(𝛼𝑦3 +(1−𝛼)�̅�3)−(𝛼𝑦1 +(1−𝛼)�̅�1)+ (𝛼𝑦2 +(1−𝛼)�̅�2)+(𝛼𝑦4 +(1−𝛼)�̅�4) = 𝑓31(𝑥,𝑡)(𝛼𝑦3 +(1−𝛼)�̅�3)+𝑓32(𝑥,𝑡)(𝛼𝑢3 + (1−𝛼)�̅�3)+𝑓33(𝑥,𝑡) (42.a) 𝛼𝑦3(𝑥,0)+(1−𝛼)�̅�3(𝑥,0) = 𝑦3 0(𝑥) (42.b) (𝛼𝑦4 +(1−𝛼)�̅�4 )𝑡 −∆(𝛼𝑦4 +(1−𝛼)�̅�4)+(𝛼𝑦4 +(1−𝛼)�̅�4)−(𝛼𝑦1 +(1−𝛼)�̅�1)+ (𝛼𝑦2 +(1−𝛼)�̅�2)−(𝛼𝑦3 +(1−𝛼)�̅�3) = 𝑓41(𝑥,𝑡)(𝛼𝑦4 +(1−𝛼)�̅�4)+𝑓42(𝑥,𝑡)(𝛼𝑢4 + (1−𝛼)�̅�4)+𝑓43(𝑥,𝑡) (43.a) IHJPAS. 53 (3)2022 143 𝛼𝑦4(𝑥,0)+(1−𝛼)�̅�4(𝑥,0) = 𝑦4 0(𝑥) (43.b) From equations ((40) − (43)), the QCCCV �⃗⃗̃� = (�̃�1, �̃�2, �̃�3, �̃�4) , with �⃗⃗̃� = 𝛼�⃗⃗� +(1−𝛼)�⃗⃗̅� has the corresponding QSVS, �⃗̃� = (�̃�1, �̃�2, �̃�3, �̃�4) , �⃗̃� = 𝛼�⃗� +(1−𝛼)�⃗̅� , i.e.: �̃�1𝑡 −∆�̃�1 +�̃�1 −�̃�2 +�̃�3 +�̃�4 = 𝑓11(𝑥,𝑡)�̃�1 +𝑓12(𝑥,𝑡)�̃�1 +𝑓13(𝑥,𝑡), �̃�1(𝑥,0) = 𝑦1 0(𝑥), �̃�2𝑡 −∆�̃�2 +�̃�2 +�̃�1 −�̃�3 −�̃�4 = 𝑓21(𝑥,𝑡)�̃�2 +𝑓22(𝑥,𝑡)�̃�2 +𝑓23(𝑥,𝑡), �̃�2(𝑥,0) = 𝑦2 0(𝑥), �̃�3𝑡 −∆�̃�3 +�̃�3 −�̃�1 +�̃�2 +�̃�4 = 𝑓31(𝑥,𝑡)�̃�3 +𝑓32(𝑥,𝑡)�̃�3 +𝑓33(𝑥,𝑡), �̃�3(𝑥,0) = 𝑦3 0(𝑥), �̃�4𝑡 −∆�̃�4 +�̃�4 −�̃�1 +�̃�2 −�̃�3 = 𝑓41(𝑥,𝑡)�̃�4 +𝑓42(𝑥,𝑡)�̃�4 +𝑓43(𝑥,𝑡), �̃�4(𝑥,0) = 𝑦4 0(𝑥), Therefore �⃗⃗� ⟼ �⃗��⃗⃗⃗� is COL w.r.t. (�⃗�, �⃗⃗�) in 𝑄. From hypotheses on 𝑔1𝑖 in 𝑄 for each 𝑖 = 1,2,3,4: 𝑔1𝑖(𝑥,𝑡,𝑦𝑖,𝑢𝑖) = ℎ1𝑖(𝑥,𝑡)𝑦𝑖 + ℎ2𝑖(𝑥,𝑡)𝑢𝑖 +ℎ3𝑖(𝑥,𝑡). Now, to show 𝑔1𝑖 is COL w.r.t. (𝑦𝑖,𝑢𝑖) , in 𝑄, since 𝐺1(�⃗⃗� +(1−𝛼)�⃗⃗̅� ) = ∑ [ 4 𝑖=1 ∫ 𝑔1𝑖(𝑥,𝑡,𝑦𝑖(𝛼𝑢𝑖+(1−𝛼)𝑢𝑖),𝛼𝑢𝑖 +(1−𝛼)�̅�𝑖) 𝑄 𝑑𝑥𝑑𝑡] = ∑ {4𝑖=1 ∫ [ ℎ1𝑖(𝑥,𝑡)𝑦𝑖(𝛼𝑢𝑖+(1−𝛼)𝑢𝑖) 𝑄 ]𝑑𝑥𝑑𝑡 +∫ [ 𝑄 ℎ2𝑖(𝑥,𝑡)(𝛼𝑢𝑖 +(1−𝛼)�̅�𝑖)+ ℎ3𝑖(𝑥,𝑡)]𝑑𝑥𝑑𝑡 }, Since �⃗⃗� ⟼ �⃗��⃗⃗⃗� is COL. Then 𝐺1(�⃗⃗�) is COL w.r.t. (�⃗�, �⃗⃗�) , in 𝑄,i.e.: 𝐺1(�⃗⃗� +(1−𝛼)�⃗⃗̅� ) = ∑ { 4 𝑖=1 ∫ [ ℎ1𝑖(𝑥,𝑡)(𝛼𝑦𝑖 +(1−𝛼)�̅�𝑖) 𝑄 ]𝑑𝑥𝑑𝑡 +∫ [ 𝑄 ℎ2𝑖(𝑥,𝑡)(𝛼𝑢𝑖 + (1−𝛼)�̅�𝑖)+ℎ3𝑖(𝑥,𝑡)]𝑑𝑥𝑑𝑡 }, = 𝛼∑ ∫ [ 𝑄 4 𝑖=1 ℎ1𝑖(𝑥,𝑡)𝑦𝑖 +ℎ2𝑖(𝑥,𝑡)𝑢𝑖 +ℎ3𝑖(𝑥,𝑡)]𝑑𝑥𝑑𝑡 + (1−𝛼)∑ ∫ [ 𝑄 4 𝑖=1 ℎ1𝑖(𝑥,𝑡)�̅�𝑖 +ℎ2𝑖(𝑥,𝑡)�̅�𝑖 +ℎ3𝑖(𝑥,𝑡)]𝑑𝑥𝑑𝑡, = 𝛼𝐺1(�⃗⃗�)+(1−𝛼)𝐺1(�⃗⃗̅�), Since 𝑔0𝑖 &𝑔2𝑖 are CO w.r.t.(𝑦𝑖,𝑢𝑖) , in 𝑄 , ∀𝑖 = 1,2,3,4 , then ∑ ∫ 𝑔0𝑖 𝑄 4 𝑖=1 𝑑𝑥𝑑𝑡 and ∑ ∫ 𝑔2𝑖 𝑄 4 𝑖=1 𝑑𝑥𝑑𝑡 are CO w.r.t. (𝑦𝑖,𝑢𝑖), in 𝑄, ∀𝑖 = 1,2,3,4, and then 𝐺0(�⃗⃗�) and 𝐺2(�⃗⃗�) are CO w.r.t. (�⃗�, �⃗⃗�), in 𝑄, i.e. 𝐺(�⃗⃗�) is CO w.r.t. (�⃗�, �⃗⃗�) , in 𝑄 . On the other hand, since �⃗⃗⃗⃗� = �⃗⃗⃗⃗��⃗⃗⃗� is CO , and the FrD of 𝐺𝑙(�⃗⃗�), (𝑙 = 0,1,2) exists for each �⃗⃗� ∈ �⃗⃗⃗⃗� and its cont. (By The. 3.2 and hypotheses (A),(B) and (C)), then 𝐺(�⃗⃗�) is CO w.r.t. (�⃗�, �⃗⃗�), in the CO set �⃗⃗⃗⃗� and it has a cont. FrD, and satisfies �́�(�⃗⃗�)𝛿𝑢⃗⃗⃗⃗⃗ ≥ 0, which means 𝐺(�⃗⃗�) has a minimum at �⃗⃗�, i.e.: 𝐺(�⃗⃗�) ≤ 𝐺(�⃗⃗⃗�) , ∀�⃗⃗⃗� ∈ �⃗⃗⃗⃗�, ⇒ 𝜆0𝐺0(�⃗⃗�)+𝜆1𝐺1(�⃗⃗�)+𝜆2𝐺2(�⃗⃗�) ≤ 𝜆0𝐺0(�⃗⃗⃗�)+𝜆1𝐺1(�⃗⃗⃗�)+𝜆2𝐺2(�⃗⃗⃗�) (44) Let �⃗⃗⃗� ∈ �⃗⃗⃗⃗�𝐴, with 𝜆2 ≥ 0, then from (38) and (44) gives: 𝜆0𝐺0(�⃗⃗�) ≤ 𝜆0𝐺0(�⃗⃗⃗�), ∀�⃗⃗⃗� ∈ �⃗⃗⃗⃗� ⇒ 𝐺0(�⃗⃗�) ≤ 𝐺0(�⃗⃗⃗�), ∀�⃗⃗⃗� ∈ �⃗⃗⃗⃗�, since (𝜆0 > 0 ). ⟹ Then �⃗⃗� is QCCOC. Quaternary classical continuous optimal control consists of a quaternary nonlinear parabolic boundary value problem with a cost function and the constraints on state and control (equality constraint and inequality constraint). Under appropriate hypotheses, the quaternary classical continuous optimal control ruling by the quaternary nonlinear parabolic boundary value problem is demonstrated as a quaternary classical continuous optimal control vector that IHJPAS. 53 (3)2022 144 satisfies the equality constraint and inequality constraint. Moreover, mathematical formulation of the quaternary adjoint equations related to the quaternary state equations is discovered so as their weak form. The derivation for the Fréchet derivative of the Hamiltonian is attained. Lastly, both the necessary conditions for optimality and sufficient conditions for optimality of the proposed problem are stated and proved. 5. Conclusion This work studies the quaternary classical continuous optimal control ruling by a quaternary nonlinear parabolic boundary value problem. 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