A time-delay system may or may not be stable for different lengths of delay, and further, may or may not be stabilized via feedback. When will then a delay system be stable or unstable, and for what intervals of delay? What will be the largest range of delay that a feedback system can tolerate? Fundamental questions of this kind have long eluded engineers and mathematicians alike, yet ceaselessly invite new thoughts and solutions. In this talk I shall present an analytical tool that answers to these questions, seeking to provide exact and efficient computational solutions to stability and stabilization problems of time-delay systems. The approach consists of the development of eigenvalue perturbation series and intrinsic delay bounds for stabilization. The former seeks to characterize the analytical and asymptotic properties of eigenvalues of matrix functions or operators. When applied to stability problems, the essential issue 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. The latter characterizes analytically the largest range of delay for which a system can be stabilized by a feedback controller. These results depart from the currently pervasive, typically LMI conditions, and yet are conceptually appealing and computationally efficient, requiring only the solution of a generalized eigenvalue problem.
