Accuracy Estimates for Bilinear Optimal Control Problems Governed by Ordinary Differential Equations

First- and second-order accuracy estimates for an optimal control problem governed by a system of ordinary differential equations with a bilinear control mechanism are presented. The numerical time discretization scheme under consideration is the finite element method with continuous piecewise linear functions. Central to this work is a first- and second-order analysis of optimality of the continuous and approximated optimal control problems. In the case of box constraints on the control, first-order error estimates for the control function are obtained assuming a piecewise constant approximation of the control, whereas second-order accuracy can be obtained in the case of a continuous, piecewise polynomial approximation. Numerical evidence is presented that supports the theoretical findings.