The Set of Controls with Values in a Closed Bounded Set is Nonempty, Bounded, Convex and Closed

Let $T>0$, $m$ and $\mathcal{A}$ be as in the definition of the set $\mathcal{U}_{\mathcal{A}}$ of $\mathcal{A}$-valued controls, write $H=L^{2}([0,T];\mathbb{R}^{m})$, and adopt the norm $\lVert\cdot\rVert_{L^{2}}$ and the metric $d_{L^{2}}$ of the Lebesgue space in the case $d=m$. Let $|\cdot|$ be the Euclidean norm on $\mathbb{R}^{m}$ and let $R\ge0$ be a real number with $|a|\le R$ for every $a\in\mathcal{A}$.

Then the following hold.

1. (Nonempty.) $\mathcal{U}_{\mathcal{A}}$ contains the class of the constant map with value any prescribed element of $\mathcal{A}$; in particular it is nonempty.

2. (Bounded.) $\lVert\xi\rVert_{L^{2}}\le R\,T^{1/2}$ for every $\xi\in\mathcal{U}_{\mathcal{A}}$, the square root being the nonnegative square root.

3. (Convex.) If $sa+(1-s)a'\in\mathcal{A}$ for all $a,a'\in\mathcal{A}$ and every real $s$ with $0\le s\le1$, then $\mathcal{U}_{\mathcal{A}}$ is a convex subset of $H$.

4. (Closed.) If $\mathcal{A}$ is closed for the topology of metric open subsets of $\mathbb{R}^{m}$ determined by the Euclidean distance, then $\mathcal{U}_{\mathcal{A}}$ is closed for the topology of metric open subsets of $H$ determined by $d_{L^{2}}$.