  In this paper, our goal is to study fundamental foundations of linear
quadratic Gaussian (LQG) control problems for stochastic linear time-invariant
systems via Lagrangian duality of semidefinite programming (SDP) problems. In
particular, we derive an SDP formulation of the finite-horizon LQG problem, and
its Lagrangian duality. Moreover, we prove that Riccati equation for LQG can be
derived the KKT optimality condition of the corresponding SDP problem. Besides,
the proposed primal problem efficiently decouples the system matrices and the
gain matrix. This allows us to develop new convex relaxations of non-convex
structured control design problems such as the decentralized control problem.
We expect that this work would provide new insights on the LQG problem and may
potentially facilitate developments of new formulations of various optimal
control problems. Numerical examples are given to demonstrate the effectiveness
of the proposed methods.
