Invertibility conditions for J_φ ensuring existence of the surrogate system

Investigate conditions under which the Jacobian matrix J_φ(x) = [∂(P_N φ_{λ_i})/∂x_j](x) is invertible on a domain D, thereby ensuring that the surrogate vector field \widetilde{F}(x) = J_φ(x)^{-1} [λ_1 P_N φ_{λ_1}(x), …, λ_n P_N φ_{λ_n}(x)]^T is well-defined and can be used to provide stability guarantees based on the truncated Koopman eigenfunctions.

Background

To strengthen stability guarantees, the paper constructs a surrogate system with vector field \widetilde{F}(x) defined via the truncated principal eigenfunctions P_N φ{λ_i}. This construction requires the invertibility of the Jacobian Jφ(x) of the mapping formed by these eigenfunctions.

The authors explicitly caution that the existence of the surrogate system can be hindered by invertibility issues of J_φ and state that this aspect requires further investigation, which they leave for future research.

References

However, the existence of the surrogate system might potentially suffer from invertibility issues related to the matrix \mathbf{J}_{\phi}. This requires further investigation that is left for future research.

— Error bounds on analytic Koopman-based Lyapunov functions  (2603.29351 - Bierwart et al., 31 Mar 2026) in End of Subsection 'Alternative result with a surrogate system', Section 4 (Error bounds on the Lyapunov function and application to stability)

Future work will focus on distributed implementations, adaptive selection of $\varepsilon$, extensions to constrained dynamics and collision avoidance, and a theoretical characterization of the conditions ensuring the invertibility of $\mathbf M\varepsilon$.

— Entropy-Regularized Optimal Transport for Time-Varying Multi-Agent Coverage Control  (2609.09829 - Napolitano et al., 9 Sep 2026) in Conclusion