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We consider the problem of steering the state of a linear system to an affine target when the admissible controls are required to satisfy magnitude constraints. A necessary and sufficient condition for the existence of an admissible control which steers the system to the target from a specified initial condition is presented, as well as a necessary condition and a sufficient condition for global controllability to the target. The conditions are similar to those available in the literature in that they involve a finite dimensional search. Here, however, the affine nature of the target is exploited to reduce the dimensionality of the resulting optimization problem. Hence, the results are more easily applied than those developed in .