Solvability of Quaternary Mixed Optimal Control Governed by Elliptic System

This paper studies the solvability of quaternary mixed optimal control problem (QMOCP) governed by the quaternary elliptic system (QES) in an infinite dimensional space. Under suitable hypotheses, the mathematical formulation of the weak form (WF) for the QES is found. Then, the proof of the existence of a unique QSSV of this WF is done by employing the Galerkin approach (GAP), when the quaternary mixed control vector (QMCV) is known. The Lipschitz operator (LO) from the control space into the state space is proved continuous. Under appropriate hypotheses, the existence of a QMOCV which is controlled by the WF of the QES is demonstrated. The mathematical formulation and the study of the quaternary adjoint equation (QAE) ruled by the QES are done. The Fréchet derivative (FD) for the cost function (CF) is determined, and the necessity for optimality of the MQOCP is proved.