End-point constraints guarantee the stability of predicted trajectories and form the basis of stable predictive control algorithms. However, the use of end-point constraints that define sufficient but not necessary conditions for the stability of predicted trajectories can lead to highly tuned controllers. These often possess poor robustness properties and result in overactive input trajectories which, in the presence of input constraints, may lead to instability. Here we develop conditions that are both necessary and sufficient and deploy these to derive stable predictive control algorithms with reduced input activity and improved robustness properties. The efficacy of the new algorithms, in respect of their ability to cope with both input constraints and robustness, are illustrated by means of design studies.