Stratospheric aerosol injection (SAI) is often discussed in the media and in policy circles as a possible action to limit future increase in global temperatures. Indeed it has been demonstrated in model simulations that in principle injection could be 'controlled', using model information, to meet specific targets on the temperature increase and its spatial distribution. This paper shows that the simulated climate response to SAI is strongly model-dependent, reflecting fundamental uncertainties in model representation of key processes. In particular this means that the SAI determined by the control algorithms as those required to achieve temperature targets different significantly from one model to another. Specific mechanisms, in particular the difference in rapid response in clouds and in precipitation to an imposed radiative perturbation and the ensuing ocean circulation response, are identified that contribute to the strong differences in model response to SAI. There is also a strong sensitivity to the pre-existing sulphate distribution which will be determined by future anthropogenic emissions. The authors note that these inter-model differences are unlikely to be resolved quickly and that controlled SAI, to achieve specific temperature goals and with well-quantified risks of unexpected consequences, is likely to remain out of reach for many years.