In this paper we study the optimizability of infinite-dimensional systems with admissible control operators. We show that under a weak condition such a system is optimizable if and only if the system can be split into an exponentially stable subsystem and an unstable subsystem that is exactly controllable in finite time. The state space of the unstable subsystem equals the span of all unstable (generalized) eigenvectors of the original system. This subsystem can be infinite-dimensional. Furthermore, the unstable poles satisfy a summability condition. The state space of the exponentially stable subsystem is given by all vectors for which the action of the original C-0-semigroup is stable.