A priori and a posteriori error estimates of H1-Galerkin mixed finite element methods for optimal control problems governed by pseudo-hyperbolic integro-differential equations

In this paper, we investigate a priori and a posteriori error estimates of H1-Galerkin mixed finite element methods for optimal control problems governed by pseudo-hyperbolic integro-differential equations. The state variables and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element and linear finite element, and the control variable is approximated by piecewise constant functions. Based on two new elliptic projections, we derive a priori error estimates both for the control variable, the state variable and the co-state variable. The related a priori error estimates for the new projections error are also established. Moreover, a posteriori error estimates for all variables are derived via energy method. Such a posteriori error estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.