  We propose a convex distributed optimization algorithm for synthesizing
robust controllers for large-scale continuous time systems subject to exogenous
disturbances. Given a large scale system, instead of solving the larger
centralized synthesis task, we decompose the problem into a set of smaller
synthesis problems for the local subsystems with a given interconnection
topology. Hence, the synthesis problem is constrained to the sparsity pattern
dictated by the interconnection topology. To this end, for each subsystem, we
solve a local dissipation inequality and then check a small-gain like condition
for the overall system. To minimize the effect of disturbances, we consider the
$\mathrm{H}_\infty$ synthesis problems. We instantiate the distributed
synthesis method using accelerated alternating direction method of multipliers
(ADMM) with convergence rate $O(\frac{1}{k^2})$ with $k$ being the number of
iterations.
