§3 (11) — under $y=D^{\mathbf r}_{\lambda(t)}x$, (1) becomes $y'=(a D_{\mathbf r}+J_n)y+(bu+d)e_n$

Let $T>0$ and let $a$ be an admissible weight on $[0,T]$ (continuous, nonnegative, with $\int_t^T a>0$ on $[0,T)$), and let $\lambda(t)=1/\int_t^T a$. Let $b,d,u:\mathbb R\to\mathbb R$ and $x:\mathbb R\to\mathbb R^n$. Suppose that at some instant $t\in(0,T)$ the state $x$ is differentiable and satisfies the chain of integrators (1):

Then $y(\tau) = D^{\mathbf r}_{\lambda(\tau)} x(\tau)$ is differentiable at $t$ and

Since $ds/dt = \lambda(t)$, this is equation (11) of the paper, $y' = (a D_{\mathbf r} + J_n) y + (b u + d) e_n$, where $y'$ is the derivative with respect to the new time $s$. It reduces the prescribed-time problem for (1) to a stabilization problem on $[0,\infty)$ for a system with bounded time-varying coefficient $a$.

Formalization Note The identity is stated pointwise in the original time $t$, as $\dot y = \lambda\, y'$ (equation (8) of the paper), so that the inverse of the time change $s$ is not needed. The standing assumption $n\ge 1$ is not needed for this identity and is omitted.