Let $n,m,N \in \mathbb N$ with $N \geq 1$. Let $X \subset \mathbb R^n$, let $U \subset \mathbb R^m$, let $X_f \subset X$, and let $F: X \times U \to X$ be a discrete-time system map. For $z \in X$, say that the horizon-$N$ terminal-constrained feasibility problem at $z$ is feasible if there exist an input tuple $(u_0,\dots,u_{N-1}) \in U^N$ and a state tuple $(x_0,\dots,x_N) \in X^{N+1}$ such that $x_0=z$, $x_{i+1}=F(x_i,u_i)$ for every $i \in \{0,\dots,N-1\}$, and $x_N \in X_f$. Define

\begin{align*} \mathcal X_N := \{z \in X : \text{the horizon-}N\text{ terminal-constrained feasibility problem at }z\text{ is feasible}\}. \end{align*}

Suppose there exists a terminal controller $\kappa_f: X_f \to U$ such that

\begin{align*} F(y,\kappa_f(y)) \in X_f \end{align*}

for every $y \in X_f$. If $x \in X$ and $(u_0^*,\dots,u_{N-1}^*) \in U^N$ with predicted state tuple $(x_0^*,\dots,x_N^*) \in X^{N+1}$ is any feasible solution of the horizon-$N$ terminal-constrained feasibility problem at $x$, then the exact closed-loop successor

\begin{align*} x^+ := F(x,u_0^*) \end{align*}

belongs to $\mathcal X_N$.