The Necessary Condition for Optimal Boundary Control Problems for Triple Elliptic Partial Differential Equations - A. Ali Al-Hawasy, Jamil; A. Thyab Al-Ajeeli, Nabeel (Ibn AL-Haitham Journal For Pure and Applied Sciences, 2021)
In this work, we prove that the triple linear partial differential equations (PDEs) of elliptic type (TLEPDEs) with a given classical continuous boundary control vector (CCBCVr) has a unique "state" solution vector (SSV)  by utilizing the Galerkin's method (GME). Also, we prove the existence of a classical continuous boundary optimal control vector (CCBOCVr) ruled by the TLEPDEs. We study the existence solution for the triple adjoint equations (TAJEs) related with the triple state equations (TSEs). The FrÃ©chet derivative (FDe) for the objective function is derived. At the end we prove the necessary "conditions" theorem (NCTh) for optimality for the problem.