  We consider a discrete-time linear-quadratic Gaussian control problem in
which we minimize a weighted sum of the directed information from the state of
the system to the control input and the control cost. The optimal control and
sensing policies can be synthesized jointly by solving a semidefinite
programming problem. However, the existing solutions typically scale cubic with
the horizon length. We leverage the structure in the problem to develop a
distributed algorithm that decomposes the synthesis problem into a set of
smaller problems, one for each time step. We prove that the algorithm runs in
time linear in the horizon length. As an application of the algorithm, we
consider a path-planning problem in a state space with obstacles under the
presence of stochastic disturbances. The algorithm computes a locally optimal
solution that jointly minimizes the perception and control cost while ensuring
the safety of the path. The numerical examples show that the algorithm can
scale to thousands of horizon length and compute locally optimal solutions.
