We study optimal stabilization of submanifolds. We show that the adjoint equation of the Hamiltonian formalism associated with such optimal control problems has structured solutions. As a consequence, optimal controls for submanifold stabilization are necessarily structured state feedbacks. In particular, we show that optimal controllers depend linearly on a function mapping to the normal spaces of the submanifold. We connect our findings to the classical linear quadratic regulator via classes of Riccati differential equations and algebraic Riccati equations. We illustrate our findings on the synchronization problem, on the broadcasting problem, and on the pattern generator problem.