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In this paper we present an analytical tool that has the promise to provide rather efficient computational solutions to a wide variety of control problems, in which the system under consideration depends on a continuously varying parameter. Notable examples in this category include time-delay and polynomially dependent systems. The approach, which appears to be conceptually appealing and computationally efficient and is referred to as eigenvalue perturbation approach, seeks to characterize the analytical and asymptotic properties of eigenvalues of matrix functions or operators. When applied to time-delay and polynomially dependent systems, the essential problem dwells on the asymptotic behavior of the critical eigenvalues on the imaginary axis, that is, on how the imaginary eigenvalues may vary with respect to the varying parameter. This behavior determines whether the imaginary eigenvalues cross from one half plane into another, and hence plays a critical role in determining the stability of such systems. Our results reveal that the eigenvalue asymptotic behavior can be characterized by solving a simple eigenvalue problem, which together with the existing matrix pencil approach in computing critical eigenvalues, leads to a numerically efficient stability analysis approach.