The Continuous Classical Optimal Control Problems for Quaternary Elliptic Partial Differential Equations

Abstract

In this paper, the Quaternary Continuous Classical Optimal Control Problem (QCCOCP) for the Quaternary Linear Elliptic Partial Differential Equations (QLEPDEqs) is studied. The mathematical model for the proposed problem is formulated, and it consists of the QLEPDEqs, the Objective Function (OF), and the set of state controls. The method of Galerkin (MG) is used to prove the existence theorem of a unique state vector solution (QSVS) of the Weak Form (WF) for the QLEPDEqs when the Quaternary Classical Continuous Control Vector (QCCCV) is fixed. Furthermore, the existence of a Quaternary Classical Continuous Optimal Control Vector (QCCOCV) ruled by the QLEPDEqs is stated and proved. The Quaternary Adjoint Equations (QAJEqs) associated with the QLEPDEqs are formulated and then studied. The Fréchet Derivative (FD) for the OF is derived. Finally, the necessary condition theorem (NCTH) for the optimality of the QCCOCP is proved.

Keywords

• Quaternary Continuous Classical Optimal Control Vector
• Quaternary Linear Partial Differential Equations
• Objective functions
• Adjoint Equations