We consider the problem of finding a controller such that, when interconnected to the plant, we obtain a system that is equivalent to a desired system. Here, "equivalence" is formalized as "bisimilarity." We give necessary and sufficient conditions for the existence of such a controller. The systems we consider are linear input-state-output systems. A comparison is made to previously obtained results about achievable/implementable behaviors in the behavioral approach to systems theory. Among the advantages of using the notion of bisimilarity is the fact that it directly applies to state-space systems, while the computations involved are operations on constant matrices.