The Set of Controls with Values in a Prescribed Subset of Euclidean Space

Let $T>0$ be a real number and let $m$ be a natural number. Adopt the notation of the Lebesgue space $L^{2}([0,T];\mathbb{R}^{d})$ in the case $d=m$: the set $\mathcal{L}^{2}([0,T];\mathbb{R}^{m})$ of square-integrable maps, the class $[u]$ of such a map, the space $L^{2}([0,T];\mathbb{R}^{m})$, and the measure space $([0,T],\mathcal{B}_{[0,T]},\lambda_{[0,T]})$. Let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^{m}$.

The set of $\mathcal{A}$-valued controls is

$$\mathcal{U}_{\mathcal{A}}=\bigl\{[u]\;:\;u\in\mathcal{L}^{2}([0,T];\mathbb{R}^{m})\text{ and }u(t)\in\mathcal{A}\text{ for every }t\in[0,T]\setminus N\text{ for some }N\in\mathcal{B}_{[0,T]}\text{ with }\lambda_{[0,T]}(N)=0\bigr\},$$

a subset of $L^{2}([0,T];\mathbb{R}^{m})$. An element of $\mathcal{U}_{\mathcal{A}}$ is called an $\mathcal{A}$-valued control.