The classical continuous optimal control of coupled fourth order linear parabolic equations

Abstract

In this study, the finite element method based on piecewise cubic Hermite basis functions is employed to establish the existence and uniqueness of a coupled state vector solution for a system of fourth-order linear parabolic partial differential equations with Neumann boundary conditions. An existence theorem for a continuous classical optimal control vector associated with fourth-order linear parabolic partial differential equations is formulated and proved under appropriate conditions. The study also investigates the existence and uniqueness of the solution to the corresponding adjoint system associated with the state vector for a given classical coupled optimal control. Finally, the Fréchet derivative of the quadratic cost functional is derived to establish the necessary optimality condition for the control problem.