  In decentralized control systems with linear dynamics, quadratic cost, and
Gaussian disturbance (also called decentralized LQG systems) linear control
strategies are not always optimal. Nonetheless, linear control strategies are
appealing due to analytic and implementation simplicity. In this paper, we
investigate decentralized LQG systems with partial history sharing information
structure and identify finite dimensional sufficient statistics for such
systems. Unlike prior work on decentralized LQG systems, we do not assume
partially nestedness or quadratic invariance. Our approach is based on the
common information approach of Nayyar \emph{et al}, 2013 and exploits the
linearity of the system dynamics and control strategies. To illustrate our
methodology, we identify sufficient statistics for linear strategies in
decentralized systems where controllers communicate over a strongly connected
graph with finite delays, and for decentralized systems consisting of coupled
subsystems with control sharing or one-sided one step delay sharing information
structures.
