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.