Optimal mixed control problem dominating by triple system of parabolic equations

This paper is devoted to study one of the important subject in the branch of applied mathematics, namely the optimal mixed control vector problem (OMCVP) dominating by triple system of parabolic PDEs (for brief TSPE). The Galerkin approach is employed to demonstrate the existence and uniqueness theorem of a triple state vector solution (TSVS) of the weak formulation system (SWF) of the TSPE when the mixed control vector (MCV) is specified. The continuity of the operator between the MCV and its conforming TSVS is established. The demonstration of the existence of an optimal mixed control vector (OMCV) controlling by TSVS is done. The Fréchet derivative (FD) of the objective function (OF) is secured. The derivation of the adjiont triple system (ATS) associated to the TSVS is found. The theorem for necessary optimality conditions for the problem is demonstrated.