This paper develops a structured optimal-control framework to design coupling gains for synchronization of weakly nonlinear oscillator circuits connected in resistive networks with arbitrary topologies. The oscillators are modeled as weakly nonlinear Liénard-type circuits, and the coupling gain amounts to the current gain which scales the output current of the oscillator. The structured optimal-control problem allows us to seek a decentralized control strategy (equivalently, a diagonal feedback matrix) that precludes communications between oscillators. To this end, a sparsity-promoting optimal control algorithm is developed to tune the optimal diagonal feedback-gain matrix with minimal performance sacrifice. This involves solving an ?2 optimal control problem with ℓ1 regularization by applying the alternating direction method of multipliers (ADMM). Simulation studies with application to voltage regulation in islanded networks composed of power-electronic inverters are provided to validate the approach.