A Priori And A Posteriori Error Estimates Of H~1-Galerkin Mixed Finite Element Methods For Optimal Control Problems Governed By Pseudo-Hyperbolic Integro-Differential Equations

At present,several efficient numerical methods have been developed for solving optimal control problems governed by partial differential equations.Among these meth-ods,finite element method has been used and studied widely,including design of effec-tive algorithms and theoretical analysisHowever,when the objective functional in the control problems contains the gra-dient of the state variable,mixed finite element methods should be the best choice for discretization of the state equation.At present,some experts and scholars have began to solve optimal control problems governed by partial differential equations by use of various mixed finite element methods,such as standard mixed finite element methods,Hl-Galerkin mixed finite element methods,splitting positive definite mixed finite ele-ment methods,least-squares mixed finite element methods and expanded mixed finite element methods.As far as we know,there exist a few articles on mixed finite element approximation of optimal control problems governed by integro-differential equations in the literature.In this paper,we shall use H1-Galerkin mixed finite element method to solve a class of optimal control problem governed by pseudo-hyperbolic integro-differential e-quations.The state variables and co-state variables are approximated by linear finite element and the lowest order Raviart-Thomas mixed finite element,and the control variable is approximated by piecewise constant functions.We mainly consider a priori and a posteriori error estimates for all variables.