  Recently, suboptimality estimates for model predictive controllers (MPC) have
been derived for the case without additional stabilizing endpoint constraints
or a Lyapunov function type endpoint weight. The proposed methods yield a
posteriori and a priori estimates of the degree of suboptimality with respect
to the infinite horizon optimal control and can be evaluated at runtime of the
MPC algorithm. Our aim is to design automatic adaptation strategies of the
optimization horizon in order to guarantee stability and a predefined degree of
suboptimality for the closed loop solution. Here, we present a stability proof
for an arbitrary adaptation scheme and state a simple shortening and
prolongation strategy which can be used for adapting the optimization horizon.
