  We consider a class of $\ell_0$-regularized linear-quadratic (LQ) optimal
control problems. This class of problems is obtained by augmenting a penalizing
sparsity measure to the cost objective of the standard linear-quadratic
regulator (LQR) problem in order to promote sparsity pattern of the state
feedback controller. This class of problems is generally NP hard and
computationally intractable. First, we apply a $\ell_1$-relaxation and consider
the $\ell_1$-regularized LQ version of this class of problems, which is still
nonconvex. Then, we convexify the resulting $\ell_1$-regularized LQ problem by
applying affine approximation techniques. An iterative algorithm is proposed to
solve the $\ell_1$-regularized LQ problem using a series of convexified
$\ell_1$-regularized LQ problems. By means of several numerical experiments, we
show that our proposed algorithm is comparable to the existing algorithms in
the literature, and in some cases it even returns solutions with superior
performance and sparsity pattern.
