The Continuous Classical Optimal Control for a Couple Fourth Order Linear Elliptic Equations

Abstract

In this paper, the finite element Galerkin method (FEGM) with piecewise cubic Hermite basis function is applied to prove the existence and uniqueness of a couple state vector solution for a system of fourth-order linear partial differential equations (PDEs) of elliptic type with Dirichlet-Neumann boundary conditions (DNBCs), when the continuous classical couple control vector (CCCPCV) is considered. An existence theorem for a coupled continuous classical optimal control vector associated with the fourth-order linear PDEs of elliptic type is formulated and proved under appropriate conditions. The paper also discusses the existence and uniqueness of the solution to the coupled adjoint equations involving the couple state vector, when the classical couple optimal control vector is given.
Finally, the derivation of the Fr´echet derivative (FrD) of the cost function (CFn) to establish the theorem of the necessary condition (NEC) for optimality of the considered problem is demonstrated.

Keywords: A continuous classical couple optimal control; The finite element Galerkin method ;The necessary condition for optimality; Fourth order linear elliptic PDEs.

2010 Mathematics Subject Classification. 26A25; 26A35