I'm currently reading the paper "Reachability and minimal times for state constrained nonlinear problems without any controllability assumption" from Bokanowski, Forcadel, and Zidani.
We consider a controlled system \begin{align} \label{system} &\dot{y}(s)=f(y(s),\alpha(s))\quad \text{for a.e. }s\in[0,T], &y(0)=x, \end{align} where $y(s)\in\mathbb{R}^d$ represents the position at time $s\in[0,T]$ with initial position $x\in\mathbb{R}^d$. $\alpha:\,[0,T]\rightarrow\mathcal{A}$ is a measurable function which maps the time $s$ into the compact and non-empty set of admissible controls $\mathcal{A}\subset\mathbb{R}^m$. The right hand side $f:\,\mathbb{R}^d\times\mathcal{A}\rightarrow\mathbb{R}^d$ is a continuous function which models the dynamics. We have to make the following two assumption on $f$:
Assumption1: The function $f$ fulfills the following lipschitz condition and is bounded: There exists an $L_f\geq 0$ such that for every $x,\tilde{x}\in\mathbb{R}^d$ and $a\in\mathcal{A}$, \begin{align*} &\vert f(x,a)-f(\tilde{x},a)\vert\leq L_f\vert x-\tilde{x}\vert\quad\text{and}\\ &\vert f(x,a)\vert\leq L_f. \end{align*}
Assumption2: For every $x\in\mathbb{R}^d$, the set $f(x,\mathcal{A})\subset\mathbb{R}^d$ is convex.
I understand that Assumption 1 implies that the system has a unique (absolutely continuous) solution on $[0,T]$. Now, we define the Capture Basin.
Definition (Capture Basin): Let $\mathcal{K}\subset\mathbb{R}^d$ be a non-empty and closed set of state contrains and let the target $\mathcal{C}\subset\mathcal{K}$ be non-empty and closed, too. Then, for a given time $t\in[0,T]$, the Capture Basin is defined by \begin{align*} CapB(t):= \left\{x\in\mathbb{R}^d\mid\exists\alpha :[0,T]\rightarrow\mathcal{A} \text{ measureable},\,y_x^\alpha(t)\in\mathcal{C}\,\text{and}\,y_x^\alpha(\theta)\in\mathcal{K}\,\forall\theta\in[0,t]\right\}, \end{align*} where $y_x^\alpha$ denotes the solution of our System depending on the intial position $x$ and control $\alpha$.
In the paper it is said that Assumption 2 implies that $CapB(t)$ is closed (Remark 1). This is what I don't understand. Does anybody have an idea?