is well-defined and constraint-admissible, meaning that $x_k \in X$ and $\kappa_N(x_k) \in U$ for every $k \ge 0$. Moreover, for every $x \in \mathcal X_N$ and $x^+=F(x,\kappa_N(x))$,
Finally, assume that there is a neighbourhood $D \subset \mathcal X_N$ of $0$ relative to $\mathcal X_N$ such that $V_N|_D$ is continuous and positive definite, in the sense that $V_N(0)=0$ and $V_N(x)>0$ for every $x \in D \setminus \{0\}$. Assume also that $V_N|_D$ is proper on $D$ in the following sense: whenever $c \ge 0$ and the sublevel set
is a neighbourhood of $0$ relative to $\mathcal X_N$ and satisfies $\Omega_c \subset D$, then $\Omega_c$ is positively invariant for the closed loop, and the origin is asymptotically stable for the closed-loop system restricted to $\Omega_c$.