Variational discretization for parabolic optimal control problems with control constraints

Abstract

This paper studies variational discretization for the optimal control problem governed by parabolic equations with control constraints. First of all, the authors derive a priori error estimates where \(\left\| {\left| {u - U_h } \right|} \right\|_{L^\infty \left( {J;L^2 \left( \Omega \right)} \right)} = O\left( {h^2 + k} \right)\). It is much better than a priori error estimates of standard finite element and backward Euler method where \(\left\| {\left| {u - U_h } \right|} \right\|_{L^\infty \left( {J;L^2 \left( \Omega \right)} \right)} = O\left( {h + k} \right)\). Secondly, the authors obtain a posteriori error estimates of residual type. Finally, the authors present some numerical algorithms for the optimal control problem and do some numerical experiments to illustrate their theoretical results.