  This study addresses a distributed state feedback controller design problem
for continuous-time linear time-invariant systems by means of linear matrix
inequalities (LMI). As the exact convexification is still open, the
block-diagonal relaxation of Lyapunov functions has been prevalent despite its
conservatism. In this work, we target a class of non-block-diagonal Lyapunov
functions that has the same sparsity as distributed controllers. By leveraging
a block-diagonal factorization of sparse matrices and Finsler's lemma, we first
present a (nonlinear) matrix inequality for stabilizing distributed controllers
with such Lyapunov functions, which boils down to a necessary and sufficient
condition for such controllers if the sparsity pattern is chordal. As a
relaxation of the inequality, we derive an LMI that completely covers the
conventional relaxation and then provide analogous results for $H_\infty$
control. Lastly, numerical examples underscore the efficacy of our results.
