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In this paper we study an imitation and transfer learning setting for Linear
Quadratic Gaussian (LQG) control, where (i) the system dynamics, noise
statistics and cost function are unknown and expert data is provided (that is,
sequences of optimal inputs and outputs) to learn the LQG controller, and (ii)
multiple control tasks are performed for the same system but with different LQG
costs. We show that the LQG controller can be learned from a set of expert
trajectories of length $n(l+2)-1$, with $n$ and $l$ the dimension of the system
state and output, respectively. Further, the controller can be decomposed as
the product of an estimation matrix, which depends only on the system dynamics,
and a control matrix, which depends on the LQG cost. This data-based separation
principle allows us to transfer the estimation matrix across different LQG
tasks, and to reduce the length of the expert trajectories needed to learn the
LQG controller to~$2n+m-1$ with $m$ the dimension of the inputs (for
single-input systems with $l=2$, this yields approximately a $50\%$ reduction
of the required expert data).