Extend eigenvalue-cone characterization beyond 2×2 and p ∈ {1, ∞}

Establish necessary and sufficient spectral conditions for when a linear time-invariant system ẋ = A x is weakly infinitesimally contracting with respect to a weighted ℓp norm in dimensions n ≥ 3 and for p values outside {1, ∞}, thereby generalizing the 2×2 case for p ∈ {1, ∞} where contraction is characterized by the eigenvalues of A lying in the cone {α + iβ : α ≤ 0 and |β| ≤ −α}.

Background

The paper proves an eigenvalue-cone characterization for 2×2 diagonalizable matrices: a real 2×2 matrix is weakly infinitesimally contracting (WIC) in a weighted ℓp norm for p ∈ {1, ∞} if and only if its eigenvalues lie in the cone {α + iβ : α ≤ 0, |β| ≤ −α}. This shows that, in dimension two, non-Euclidean contraction imposes stricter spectral constraints than marginal stability in the ℓ2 case.

The authors note that this result quantifies a trade-off between computational tractability and expressiveness, and explicitly point out that determining analogous characterizations in higher dimensions and for other p remains unresolved.