This paper addresses the design of controllers, subject to sparsity constraints, for linear and timeinvariant
plants. Prior results have shown that a class of stabilizing controllers, satisfying a given sparsity
constraint, admits a convex representation of the Youla–type, provided that the sparsity constraints
imposed on the controller are quadratically invariant with respect to the plant and that the plant is strongly
stabilizable. Another important aspect of the aforementioned results is that the sparsity constraints on
the controller can be recast as convex constraints on the Youla parameter, which makes this approach
suitable for optimization using norm-based costs. In this paper, we extend these previous results to
non-strongly stabilizable plants. Our extension also leads to a Youla-type representation for the class
of controllers, under quadratically invariant sparsity constraints. In our extension, the controller class
also admits a representation of the Youla–type, where the Youla parameter is subject to only convex
constraints
