  In this work, we revisit the Linear Quadratic Gaussian (LQG) optimal control
problem from a behavioral perspective. Motivated by the suitability of
behavioral models for data-driven control, we begin with a reformulation of the
LQG problem in the space of input-output behaviors and obtain a complete
characterization of the optimal solutions. In particular, we show that the
optimal LQG controller can be expressed as a static behavioral-feedback gain,
thereby eliminating the need for dynamic state estimation characteristic of
state space methods. The static form of the optimal LQG gain also makes it
amenable to its computation by gradient descent, which we investigate via
numerical experiments. Furthermore, we highlight the advantage of this approach
in the data-driven control setting of learning the optimal LQG controller from
expert demonstrations.
