Definition (Safe Set)

The safe set $\mathcal{C}$ is the Superlevel Set of a Smooth function $h:D\subset\mathbb{R}^{n}\to \mathbb{R}$, such that $$\begin{align*}\\ \mathcal{C}&= \{ x\in D\subset\mathbb{R}^{n}:h(x)\ge 0 \}\\ \partial \mathcal{C}&= \{ x\in D\subset \mathbb{R}^{n}:h(x)=0 \}\\ \text{Int}(\mathcal{C})&= \{ x\in D\subset \mathbb{R}^{n}:h(x)>0 \} \end{align*}$$where $\forall x\in \mathbb{R}^{n}$ s.t. $h(x)=0$ we have that $$\frac{ \partial h }{ \partial x } (x)\not=0.$$

Definition (Safe)

The system $$\dot{x}=f_{\text{cl}(x)}:=f(x)+g(x)k(x)$$where $k(x)=u$ is the feedback controller, is safe with respect to the set $\mathcal{C}$ if the set $\mathcal{C}$ is Forward Invariant.