Extension of the projected closed-loop ISS result to the full tangent bundle

Let $\mathcal{T}\checkmathcalq$ and $\mathcal{T}\mathcalq$ denote the learned submanifold and the full tangent bundle, respectively, and let $\Delta_N$ and $\Delta_{\perp,N}$ be bounded disturbances caused by unmodeled nullspace dynamics and cross-couplings in a compact neighborhood $\mathcal{N}\subseteq\mathcal{T}\mathcalq$ of the training data distribution. Write

$$\Delta=\tilde\Delta_\theta+\tilde\Delta_\perp+\Delta_N+\Delta_{\perp,N}.$$

Extension of the projected closed-loop ISS result. Under standard model-order-reduction assumptions, the closed-loop dynamics on the submanifold $\varphi(\mathcal{T}\checkmathcalq)$ extend to the manifold $\mathcal{T}\mathcalq$, and the resulting closed-loop dynamics under the disturbance $\Delta$ are locally exponentially input-to-state stable.

This extends the stability conclusion for the projected dynamics to the full system in the presence of nullspace disturbances and cross-couplings. The source presents this as a conjectural consequence of the preceding projected ISS theorem; no resolution is supplied.