Classical Continuous Boundary Optimal Control Vector Problem for Triple Nonlinear Parabolic System

Abstract

In this paper, our purpose is to study the classical continuous boundary optimal triple control vector problem (CCBOTCVP) dominating by nonlinear triple parabolic boundary value problem (NLTPBVP). Under suitable assumptions and with given classical continuous boundary triple control vector (CCBTCV), the existence theorem for a unique state triple vector solution (STVS) of the weak form W.F for the NLTPBVP is stated and demonstrated via the Method of Galerkin (MGa), and the first compactness theorem. Furthermore, the continuity operator between the STVS of the WFO for the NLTPBVP and the corresponding CCBTCV is stated and demonstrated. The continuity of the Lipschitz (LIP.) operator between the STVS of the WFo for the QNLPBVP and the corresponding CCBTCV is proved. The existence of a CCBOTCV is stated and demonstrated under suitable conditions.

Classical Continuous Constraint Boundary Optimal Control Vector Problem for Triple Nonlinear Parabolic System

“…Where 𝑘 1 (𝑥, 𝑡, 𝑦 1 , 𝑦 2 , 𝑦 3 ) = 𝑔 1 (𝑥, 𝑡, 𝑦 1 ) + 𝑔 2 (𝑥, 𝑡, 𝑦 2 )+𝑔 3 (𝑥, 𝑡, 𝑦 3 ), and 𝑘 2 (𝑥, 𝑡, 𝑢 1 , 𝑢 2 ) = ℎ 1 (𝑥, 𝑡, 𝑢 1 ) + ℎ 2 (𝑥, 𝑡, 𝑢 2 ) + ℎ 3 (𝑥, 𝑡, 𝑢 3 ), From the definition of the FD and the result of Theorem (2-(a)) [16] and from the assumptions on 𝑔 𝑖 (∀𝑖 = 1,2,3), and then using the inequality of Minkowski once obtains:…”

“…Where 3,4,5,6) and 𝛼 ̅ are real positive constants. Theorem 1 [16]: With assumptions (A), for each "fixed" 𝑢 ⃗ ∈ (𝐿 2 (Σ)) 3 , the WFO (( 13)-( 15)) has a unique TSVS 𝑦 = (𝑦 1 , 𝑦 2 , 𝑦 3 ) s.t. 𝑦 ∈,𝑦 𝑡 = (𝑦 1𝑡 , 𝑦 2𝑡, 𝑦 3𝑡, ) ∈ (𝐿 2 (I, V)) 3 .…”

“…respectively, and satisfies the following sub quadratic condition with respect to 𝑦 𝑖 and 𝑢 𝑖 |𝑔 𝑙𝑖 (𝑥, 𝑡, 𝑦 𝑖 )| ≤ 𝛾 𝑙𝑖 (𝑥, 𝑡) + 𝑐 𝑙𝑖 (𝑦 𝑖 )2 , |ℎ 𝑙𝑖 (𝑥, 𝑡, 𝑢 𝑖 )| ≤ 𝛿 𝑙𝑖 (𝑥, 𝑡) + 𝑑 𝑙𝑖 (𝑢 𝑖 )2 Where𝑦 𝑖 , 𝑢 𝑖 ∈ ℝ with 𝛾 𝑙𝑖 ∈ 𝐿 1 (𝑄) , 𝛿 𝑙𝑖 ∈ 𝐿 1 (Σ) Lemma 1[16]: If assumptions (B) are held, the functional 𝐺 𝑙 (𝑢 ⃗ ) is continuous on (𝐿 2 (Σ))3 , ∀ 𝑙 = 0,1,2 . Theorem 2[16]: Beside the assumptions (A) and (B). If 𝑈 ⃗ ⃗ is compact, 𝑊 ⃗⃗⃗ 𝐴 ≠ ∅, 𝐺 0 (𝑢 ⃗ ) is convex.…”