Error estimates and superconvergence of semidiscrete mixed methods for optimal control problems governed by hyperbolic equations

In this paper, we investigate $L^\infty(L_2)$-error estimates and superconvergence of semidiscrete mixed finite element methods for quadratic optimal control problems governed by linear hyperbolic equations. The state and the co-state are discretized by order $k$ Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order $k$ ($k\ge0$). We derive error estimates for both the state and the control approximation. Moreover, we present superconvergence analysis for mixed finite element approximation of the optimal control problems.