Asymptotic achievability for linear time invariant state space systems

We consider here the problem of finding a controller such that when interconnected to the plant, we obtain a system which is asymptotically equivalent to a desired system. Here `asymptotic equivalence´ is formalized as `asymptotic bisimilarity´. Intuitively speaking, two systems are asymptotically bisimilar if the difference between their outputs decays to zero with time. We give necessary and sufficient conditions for the existence of such a controller. These conditions can be verified computationally using standard algorithms in linear geometric control. The systems we consider are linear time invariant input-state-output systems.