Error Analysis of $hp$ Spectral Element Approximation for Optimal Control Problems with Control Constraint

Abstract

In this paper, an hp spectral element approximation for distributed optimal control problem governed by an elliptic equation is investigated, whose objective functional does not include the control variable. And the constraint set on control variable is stated with $L^2$ -norm. Optimality condition of the continuous and discretized systems are deduced. In order to solve the equivalent systems with high accuracy, $hp$ spectral element method is employed to discretize the constrained optimal control systems. Based on the property of some interpolation operators, a posteriori error estimates are also established by using some properties of some interpolation operators carefully. Finally, a projection gradient algorithm and a numerical example are provided, which confirm our analytical results. Such estimators guarantee the construction of reliable adaptive methods for optimal control problems.