Necessary conditions for optimal boundary-control of couple linear parabolic partial differential equations

In this research, we will study the classical continuous optimal boundary- control problem(CCOPB-CPR) determine by couple linear PDEs of parabolic type (CCOPB-CCLPDEs) in details. the existence theorem for the uniqueness state vector solution (STVSO) of couple linear parabolic PDEs (CLPPDEs) for given (fixed) continuous classical boundary- control vector (CCB-CV) is considered and proved, the existence theorem of a continuous classical -boundary optimal control vector (CCOPB-CV) associated with the CLPPDEs is developed and proved, while the Frechet derivative (Féde) of the objective function is derived, the theorem for the existence of a unique solution of the adjoint vector equations (ADVEQ) congruous for the STVSO is considered. Lastly the necessary optimality conditions (NOPC) of the CCOPB-CPR is proved.