Krasovski\u{i} Stability Theorem for FDEs in the Extended Sense

The analysis of the stability of systems' equilibria plays a central role in the study of dynamical systems and control theory. This note establishes an extension of the celebrated Krasovski\u{\i} stability theorem for functional differential equations (FDEs) in the extended sense. Namely, the FDEs hold for $t \geq t_0$ almost everywhere with respect to the Lebesgue measure. The existence and uniqueness of such FDEs were briefly discussed in J.K Hale's classical treatise on FDEs, yet a corresponding stability theorem was not provided. A key step in proving the proposed stability theorem was to utilize an alternative strategy instead of relying on the mean value theorem of differentiable functions. The proposed theorem can be useful in the stability analysis of cybernetic systems, which are often subject to noise and glitches that have a countably infinite number of jumps. To demonstrate the usefulness of the proposed theorem, we provide examples of linear systems with time-varying delays in which the FDEs cannot be defined in the conventional sense.