Let $n \in \mathbb N$ with $n \ge 2$. For $1 \le i \le n$, write $x_{1:i}:=(x_1,\dots,x_i)$, and let $\pi_i:\mathbb R^n \to \mathbb R^i$ be the projection $\pi_i(x)=x_{1:i}$. Let $D \subseteq \mathbb R^n$ be an open neighbourhood of $0$, and set $D_i:=\pi_i(D)$ for $1 \le i \le n$.

For each $1 \le i \le n$, let $f_i,g_i \in C^\infty(D_i;\mathbb R)$ satisfy $f_i(0)=0$ and $g_i(\xi)\ne 0$ for every $\xi \in D_i$. Consider the scalar strict-feedback system on $D$ with state $x=(x_1,\dots,x_n)\in D$ and scalar input $u\in\mathbb R$ given by

\begin{align*} \dot{x}_i = f_i(x_{1:i})+g_i(x_{1:i})x_{i+1}, \qquad 1\le i\le n-1, \end{align*}

and

\begin{align*} \dot{x}_n = f_n(x_{1:n})+g_n(x_{1:n})u. \end{align*}

Suppose that there exist an open neighbourhood $U_1\subseteq D_1$ of $0$, a function $\alpha_1\in C^\infty(U_1;\mathbb R)$ with $\alpha_1(0)=0$, and a function $V_1\in C^\infty(U_1;\mathbb R)$ such that $V_1(0)=0$, $V_1(x_1)>0$ for $x_1\ne 0$, and the function

\begin{align*} x_1\mapsto \frac{dV_1}{dx_1}(x_1)\bigl(f_1(x_1)+g_1(x_1)\alpha_1(x_1)\bigr) \end{align*}

is negative definite on $U_1$.

Then there exist open neighbourhoods $U_i\subseteq D_i$ of $0$ for $1\le i\le n$, chosen so that $x_{1:i}\in U_i$ whenever $x_{1:i+1}\in U_{i+1}$ and $1\le i\le n-1$, smooth virtual controls $\alpha_i\in C^\infty(U_i;\mathbb R)$ for $1\le i\le n-1$ with the given initial $\alpha_1$ restricted to the chosen $U_1$ and satisfying $\alpha_i(0)=0$, and a smooth feedback $k\in C^\infty(U_n;\mathbb R)$ satisfying $k(0)=0$, such that the origin is a locally asymptotically stable equilibrium of the closed-loop vector field $F_k:U_n\to\mathbb R^n$ obtained by setting $u=k(x)$ in the displayed system.

Moreover, suppose $D=\mathbb R^n$, each $f_i,g_i$ is defined on all of $\mathbb R^i$, each $g_i$ is nowhere zero on $\mathbb R^i$, and the initial data $\alpha_1\in C^\infty(\mathbb R;\mathbb R)$ and $V_1\in C^\infty(\mathbb R;\mathbb R)$ are global, with $V_1$ proper and with the displayed derivative negative definite on $\mathbb R$. If the recursive choices of virtual controls can be made so that each recursively defined Lyapunov function $V_i:\mathbb R^i\to\mathbb R$ is proper, then the virtual controls $\alpha_i$ and the feedback $k$ may be chosen globally, and the closed-loop origin is globally asymptotically stable.