Let Q be a convex subset of a linear space, $\mathcal{U} \subset Q$, $\mathcal{Z}$ a topological vector space, $C \subset \mathcal{Z}$ convex with a nonempty interior, $\bar q \in Q$, $\varphi = (\varphi _1 ,\varphi _2 ):Q \to \mathbb{R}^m \times \mathcal{Z}$, and $\varphi _2 (\bar q) \in C$. Also, let $\mathcal{A}: = \{ u \in \mathcal{U}|\varphi _2 (u) \in C\} $. This framework can be used to model many optimization problems including control problems with unilateral (state inequality) constraints. Sufficient conditions are derived for $\varphi _1 (\mathcal{A})$ to cover the interior of a convex set containing $\varphi _1 (\bar q)$ and an open neighborhood of a point $\varphi _1 (\bar q) + w$ for $w \in \mathbb{R}^m $. These sufficient conditions can be used to provide information about the attainable sets of control problems and, in some cases, to show that $\varphi _1 (\mathcal{A})$ contains a neighborhood of $\varphi _1 (\bar q)$. Several illustrative examples are also discussed.