The paper proposes a method for structured state-feedback controllers design for linear time-invariant systems. A necessary and sufficient condition for structured state-feedback stabilizability of linear systems, making an appeal to the linear-quadratic (LQ) regulator theory, is first proposed. The latter is presented in the form of a nonlinear matrix equation. Then, it is recast as a non-smooth unconstrained equation using projection onto the positive semi-definite matrices cone. Thereby, a non-smooth Newton's iterative algorithm, based on the Clarke generalized Jacobian of said projection, is proposed. This method has a guaranteed local convergence. Finally, numerical examples illustrate the effectiveness of the proposed method.