A Priori Error Estimates for Finite Element Discretizations of Parabolic Optimal Control Problems with Measure Data

摘要

We study finite element approximations of parabolic optimal control problem with measure data in space in a bounded convex domain. The main mathematical difficulty of this kind of problem is low regularity of the solution of the state equation due to the presence of measure data in source term. This introduces some difficulties in both theory and numerics of finite element error analysis. We first prove the existence, uniqueness and regularity of the solution to the control problem. A priori error estimates for the state and control variables are derived for both the spatially discrete and fully discrete approximations of optimal control problems. Moreover, convergence properties for the state and the control variables are established. We use piecewise linear and continuous finite elements for the approximations of the state and co-state variables whereas the control variable is approximated by piecewise constant functions. Numerical experiment is provided to illustrate our theoretical results.