id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
iajs-2477	Al-Hawasy, Jamil A.; Jasim, Doaa Kateb	The Continuous Classical Optimal Control Problems for Triple Nonlinear Elliptic Boundary Value Problem	2020	12	.pdf	application/pdf	6127	256	78	Then from FD, we have that �⃗� �⃗� 𝛿�⃗� 𝐻 ⃗ . Proof: suppose 𝑎 𝑥, 𝜉 , 𝑣 𝑎 𝑥 𝜉 𝑎 𝑥 𝑣 𝑎 𝑥 , 𝑎 𝑥, , 𝑣 𝑎 𝑥 𝑣 𝑎 𝑥 , 𝑝 𝑥, 𝜉 , 𝑣 𝑝 𝑥 𝜉 𝑝 𝑥 𝑣 𝑝 𝑥 , 𝑝 𝑥, , 𝑣 𝑝 𝑥 𝑣 𝑝 𝑥 , 𝑘 𝑥, 𝜉 , 𝑣 𝑘 𝑥 𝜉 𝑘 𝑥 𝑣 𝑘 𝑥 , 𝑘 𝑥, , 𝑣 𝑘 𝑥 𝑣 𝑘 𝑥 , And that�⃗� ∈ �⃗� , �⃗� is satisfied the K.T.L. and the Transversality condition i.e. 𝐻 ⃗ 𝑥, 𝜉, 𝜁, �⃗� ⋅ 𝛿�⃗�𝑑𝑥 0, ∀𝑢 ∈ �⃗� and 𝜆 𝑦 �⃗� 0 Let 𝑌 �⃗� ∑ ℓ 𝜆ℓ𝑦ℓ �⃗� , then�⃗� �⃗� 𝛿�⃗� ∑ ℓ 𝜆ℓ�⃗�ℓ �⃗� 𝛿�⃗� ∑ 𝜆ℓℓ 𝜁 ℓ 𝑎 𝑎 𝑦 ℓ 𝛿𝑣 𝜁 ℓ 𝑝 𝑝 𝑦 ℓ 𝛿𝑣 𝜁 ℓ 𝑘 𝑘 𝑦 ℓ 𝛿𝑣 𝑑𝑥 𝐻 ⃗ 𝑥, 𝜉, 𝜁, �⃗� ⋅ 𝛿�⃗�𝑑𝑥 0 Let 𝑣 , 𝑣 , 𝑣 and �̅� , �̅� , �̅� are two given controls, then 𝜉 𝜉 , 𝜉 𝜉 , 𝜉 𝜉 and 𝜉̅ 𝜉̅ , 𝜉̅ 𝜉̅ , 𝜉̅ 𝜉̅ ) are their corresponding solutions, substituting the pair (�⃗�, 𝜉 in (1)-(4) and multiplying the obtained equation by 𝜅 ∈ 0,1 once and once again the pair (�⃗̅�, 𝜉̅⃗ in (1)-(4) multiplying the obtained equation by 1 𝜅 , finally then blending together the obtained equations from each corresponding equations once get: 𝐵 𝜅𝜉 1 𝜅 𝜉̅ 𝜅𝜉 1 𝜅 𝜉̅ 𝜅𝜉 1 𝜅 𝜉̅ 𝜅𝜉 1 𝜅 𝜉̅ 𝑎 𝑥 𝜅𝜉 1 𝜅 𝜉̅ 𝑎 𝑥 𝜅𝑣 1 𝜅 �̅� 𝑎 𝑥 𝑎 𝑥 𝜅𝑣 1 𝜅 �̅� 𝑎 𝑥 (30a)	cache/iajs-2477.pdf	txt/iajs-2477.txt