In this paper, we derive new readily testable criteria for exponential stabilizability, approximate controllability, and exact controllability for multiplicative systems arising from linear partial differential equations on an infinite domain. These multiplicative systems have an unbounded semigroup generator, but bounded input and output operators. The theoretical results are illustrated by several examples. In particular, explicit, easily verifiable conditions for exponential stabilizability, approximate and exact controllability are given for second-order P.D.E. systems. Dual results for exponential detectability, approximate and exact observability are also included.