Convergence of Discontinuous Finite Volume Discretizations for a Semilinear Hyperbolic Optimal Control Problem

Abstract

In this paper, we discuss discontinuous finite volume approximations of the distributed optimal control problems governed by a class of semilinear hyperbolic partial differential equations with control constraints. The spatial discretization of the state and costate variables follows discontinuous finite volume schemes with piecewise linear elements, whereas three different strategies are used for the control approximation: variational discretization, piecewise constant and piecewise linear discretization. As the resulting semi-discrete optimal system is non-symmetric, we have employed optimize then discretize approach to approximate the control problem. A priori error estimates for control, state and costate variables are derived in suitable natural norms. The present analysis is an extension of the analysis given in Kumar and Sandilya [Int. J. Numer. Anal. Model. (2016), 13: 545-568]. Numerical experiments are presented to illustrate the performance of the proposed scheme and to confirm the predicted accuracy of the theoretical convergence rates.