Structural results and explicit solution for two-player LQG systems on a finite time horizon

It is well-known that linear dynamical systems with Gaussian noise and quadratic cost (LQG) satisfy a sepa-ration principle. Finding the optimal controller amounts to solving separate dual problems; one for control and one for estimation. For the discrete-time finite-horizon case, each problem is a simple forward or backward re-cursion. In this paper, we consider a generalization of the LQG problem with two controllers and a partially nested information structure. Each controller is responsible for one of two system inputs, but has access to different sub-sets of the available measurements. Our paper has three main contributions. First, we prove a fundamental struc-tural result: sufficient statistics for the controllers can be expressed as conditional means of the global state. Sec-ond, we give explicit state-space formulae for the optimal controller. These formulae are reminiscent of the classi-cal LQG solution with dual forward and backward re-cursions, but with the important difference that they are intricately coupled. Lastly, we show how these recursions can be solved efficiently, with computational complexity comparable to that of the centralized problem.