Robust Bounds on the Eigenvalues of Linear Systems With Delays and Their Applications in Stability Analysis

ABSTRACT We review some known bounds on the eigenvalues of a matrix and use similar techniques to derive bounds for the class of nonlinear eigenproblems and, as a special case, the eigenvalues for LTI systems with delays. We present two classes of results. The first class is based on the Hermitian decomposition, and the second on Gershgorin's theorem. The bounds are easily computable and provide regions in that include or exclude eigenvalues. We reflect on their implications for the stability theory of LTI systems. In particular, for delay systems, the new bounds provide an alternative to the use of the LambertW function and the known Riccati‐type stability criteria for stability based on the Lyapunov‐Krasovskii theory.