Resolvent-Based Stability Criterion for Non-Normal Systems via the Kaupp Number

Abstract We study stability of linear dynamical systems generated by non-normal operators. Classical spectral criteria fail to predict transient amplification that may exceed operational limits. We introduce the Kaupp number K = \frac{\|X_0\| G_{\max}}{S}, \quad G_{\max} = \sup_{t \ge 0} \|e^{At}\|, where S denotes system capacity. We establish a resolvent-based lower bound G_{\max} \ge \sup_{\operatorname{Re} z > 0} \operatorname{Re}(z)\,\|(zI - A)^{-1}\|, and derive a frequency-domain formulation of the instability condition. A system fails when K \ge 1, even if all eigenvalues lie in the left half-plane. Constructive worst-case perturbations are obtained via singular value decomposition. Numerical examples demonstrate a fundamental gap between spectral stability and bounded response.