Motivated by different applications in circuits, electromagnetics, and optics, this paper is concerned with the synthesis of a particular type of linear circuit, where the circuit is associated with a control unit. The objective is to design a controller for this control unit such that certain specifications on the parameters of the circuit are satisfied. It is shown that designing a control unit in the form of a switching network is an NP-complete problem that can be formulated as a rank-minimization problem. It is then proven that the underlying design problem can be cast as a semidefinite optimization if a passive network is designed instead of a switching network. Since the implementation of a passive network may need too many components, the design of a decoupled (sparse) passive network is subsequently studied. This paper introduces a tradeoff between design simplicity and implementation complexity for an important class of linear circuits. The superiority of the developed techniques is demonstrated by different simulations. In particular, for the first time in the literature, a wavelength-size passive antenna is designed, which has an excellent beamforming capability and which can concurrently make a null in at least eight directions.