Forward Euler Solutions and Weakly Invariant Time-Delayed Systems

This paper presents a necessary and sufficient condition for the weakinvariance property of a time-delayed system parametrized by a differential inclusion. The aforementioned condition generalizes the well-known Hamilton-Jacobiinequality that characterizes weakly invariant systems in the nondelay setting. Theforward Euler approximation scheme used in the theory of discontinuous differentialequations is extended to the time-delayed context by incorporating the delay andtail functions featuring the dynamics. Accordingly, an existence theorem of weaklyinvariant trajectories is established under the extended forward Euler approach