  Consider that a linear time-invariant (LTI) plant is given and that we wish
to design a stabilizing controller for it. Admissible controllers are LTI and
must comply with a pre-selected sparsity pattern. The sparsity pattern is
assumed to be quadratically invariant (QI) with respect to the plant, which,
from prior results, guarantees that there is a convex parametrization of all
admissible stabilizing controllers provided that an initial admissible stable
stabilizing controller is provided. This paper addresses the previously
unsolved problem of determining necessary and sufficient conditions for the
existence of an admissible stabilizing controller. The main idea is to cast the
existence of such a controller as the feasibility of an exact model-matching
problem with stability restrictions, which can be tackled using existing
methods. Furthermore, we show that, when it exists, the solution of the
model-matching problem can be used to compute an admissible stabilizing
controller. This method also leads to a convex parametrization that may be
viewed as an extension of Youla's classical approach so as to incorporate
sparsity constraints. Applications of this parametrization on the design of
norm-optimal controllers via convex methods are also explored. An illustrative
example is provided, and a special case is discussed for which the exact model
matching problem has a unique and easily computable solution.
