  In this paper, we present a structured solver based on the preconditioned
conjugate gradient method (PCGM) for solving the linear quadratic (LQ) optimal
control problem for $K \times N$ sub-systems connected in a two-dimensional
(2D) grid structure. Our main contribution is the development of a structured
preconditioner based on a fixed number of inner-outer iterations of the nested
block Jacobi method. We establish that the proposed preconditioner is
positive-definite. Moreover, the proposed approach retains structure in both
spatial dimensions as well as in the temporal dimension of the problem. The
arithmetic complexity of each PCGM step scales as $O(KNT)$, where $T$ is the
length of the time horizon. The computations involved at each step of the
proposed PCGM are decomposable and amenable to distributed implementation on
parallel processors connected in a 2D grid structure with localized data
exchange. We also provide results of numerical experiments performed on two
example systems.
