Finite Element Methods for Optimal Control Problems Governed by Linear Quasi-parabolic Integro-differential Equations

Linear quasi-parabolic integro-differential equations and their control appear in many scientific problems and engineering applications such as biology mechanics, nuclear reaction dynamics, heat conduction in materials with memory, and visco-elasticity, etc.. The existence and uniqueness of the solution of the linear quasi-parabolic integro-differential equations have been studied by Wheeler M. F. in [17]. Furthermore the finite element methods for linear quasi-parabolic integrodifferential equations with a smooth kernel have been discussed in, e.g., X. Cui [2]. However there exists little research on optimal control problems governed by quasi-parabolic integro-differential equations, in spite of the fact that such control problems are often encountered in practical engineering applications and scientific computations. Furthermore the finite element methods of the optimal control problem governed by such equations have not been studied although there has existed much research on the finite element approximations of quasi-parabolic integrodifferential equations. Finite element approximations of optimal control problems governed by various partial differential equations have been extensively studied in the literature. There have been extensive studies in convergence of the standard finite element approximation of optimal control problems, for examples, see [1, 4, 5, 10, 11, 12, 13, 14, 15, 16]. For optimal control problems governed by linear PDEs, the optimality conditions and their finite element approximation and the a prior error estimates were established long ago, for example, see [4, 7]. The purpose of this paper is to study the mathematical formulation and its finite element approximation of the optimal control problem governed by a linear quasi-parabolic integro-differential equation. In particular we establish the optimality conditions and analyze the a priori error estimates for these constrained optimal control problems. We also present some numerical tests to verify the theoretical analysis.