Transient Growth versus Asymptotic Stability: Limitations of Classical Criteria in Non-Normal Systems

Abstract Classical stability theory is primarily based on spectral properties of system operators, asserting stability when all eigenvalues lie in the left half-plane. However, this criterion fails to capture transient amplification in non-normal systems. We demonstrate that asymptotic stability does not imply bounded system response and introduce a framework in which transient growth dominates system behavior. By combining resolvent estimates, pseudospectral analysis, and the K-functional, we establish conditions under which spectrally stable systems exhibit large finite-time amplification leading to operational instability.