We present new methods for proving stability of time-varying linear systems with delays. Our main tools include positive systems and linear Lyapunov functionals. Our work applies to key classes of systems that arise in numerous engineering applications, including neutral systems, and systems that are not necessarily periodic in time and not necessarily positive. We prove stability by comparing the trajectories of the original systems with trajectories of higher dimensional positive systems. One of our key results requires an upper bound on the delay, but the delay can be unknown. Our work also provides robustness of the stability with respect to uncertainties in the coefficient matrices of the system. We illustrate our work in three examples, which show how our methods can sometimes be used with backstepping and linearization to cover even more general classes of systems.

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