Stability of delay equations via Lyapunov functions

The importance of Lyapunov functions is well known. In the general setting of nonautonomous linear delay equations v ′ = L ( t ) v t , we show how to characterize completely the existence of a nonuniform exponential contraction or of a nonuniform exponential dichotomy in terms of Lyapunov functions. This includes uniform exponential behavior as a very special case, and it provides an alternative (usually simpler and particularly more direct) approach to verify the existence of exponential behavior or to obtain the robustness of the dynamics under sufficiently small perturbations.