Consider the deterministic controlled system:
$$\dot x(t) = Ax(t) + Bu(t), \ t \in [0, T]$$ $$x(0) = x_0$$
where $x: [0, T] \to \mathbb R^n$ is the controlled state process, $A \in \mathbb R^{n \times n}$, $B \in \mathbb R^{n \times k}$ are real valued matrices, $u: [0, T] \to \mathbb R^k$ is the control process, and $x_0 \in \mathbb R^n$ is an arbitrary fixed initial point.
For each $U \subset \mathbb R^k$, we define the reachable set $\mathcal R_U$ as the set of all possible final states of the system using controls with values in $U$ - that is, the set
$$\mathcal R_U := \{x(T, u(.)) \ | \ u(t) \in U, \forall t \in [0, T]\}$$
In the book Stochastic Controls by Yong and Zhao, the following proposition is stated as Proposition 6.1 on page 77:
Proposition: Let $U \subset \mathbb R^k$ be compact. Then $\mathcal R_U$ is convex and compact in $\mathbb R^n$, and further it is equal to $\mathcal R_{\overline{\text{co } U}}$, where $\overline{\text{co }U}$ denotes the closure of the convex hull of $U$.
The proposition is said in turn to be a consequence of the following Lyapunov’s theorem, which is Theorem 6.3 on the same page:
Theorem: Suppose $f \in L^1 ([0, T], \mathbb R^n)$. Then the set $\mathcal H := \{\int_S f(x) dx \ | \ S \in \mathcal B [0, T]\}$ is a convex subset of $\mathbb R^n$.
where here $\mathcal B[0, T]$ is the set of Borel subsets of $[0, T]$.
On the same page, they refer to the book Functional Analysis and Time Optimal Controls by Hermes and LaSalle for proofs of both these statements. However, I was only able to find a proof of the Theorem above and not the Proposition.
Question:
Does anyone know of an alternative reference for this result?
