  We show how a posteriori goal oriented error estimation can be used to
efficiently solve the subproblems occurring in a Model Predictive Control (MPC)
algorithm. In MPC, only an initial part of a computed solution is implemented
as a feedback, which motivates grid refinement particularly tailored to this
context. To this end, we present a truncated cost functional as objective for
goal oriented adaptivity and prove under stabilizability assumptions that error
indicators decay exponentially outside the support of this quantity. This leads
to very efficient time and space discretizations for MPC, which we will
illustrate by means of various numerical examples.
