In this paper, the optimal H² control problem for spatially invariant systems is considered. The framework adopted is borrowed from the work pioneered by Voulgaris et al. In particular, the H² optimal control problem is solved via the computation of an orthogonal projection of a tensor Hilbert space onto a particular subspace. Next, the optimal H² decentralized control problem is solved by computing an orthogonal projection on a class of Youla parameters with a decentralized structure. The latter uses Riesz projections after invoking a particular L²-basis.