  We consider the problem of designing a stabilizing and optimal static
controller with a pre-specified sparsity pattern. Since this problem is NP-hard
in general, it is necessary to resort to approximation approaches. In this
paper, we characterize a class of convex restrictions of this problem that are
based on designing a separable quadratic Lyapunov function for the closed-loop
system. This approach generalizes previous results based on optimizing over
diagonal Lyapunov functions, thus allowing for improved feasibility and
performance. Moreover, we suggest a simple procedure to compute favourable
structures for the Lyapunov function yielding high-performance distributed
controllers. Numerical examples validate our results.
