We study controlled homogeneous dynamical systems and derive conditions under which the system is perspective controllable. We also derive conditions under which the system is observable in the presence of a control over the complex base field. In the absence of any control input, we derive a necessary and sufficient condition for observability of a homogeneous dynamical system over the real base field. The observability criterion obtained generalizes a well known Popov-Belevitch-Hautus rank criterion to check the observability of a linear dynamical system. Finally, we introduce rational, exponential, interpolation problems as an important step toward solving the problem of realizing homogeneous dynamical systems with minimum state dimensions.