History-space triggered impulses for delay differential equations: well-posedness, anti-Zeno bounds, and stability under expansive jumps

We study impulsive delay differential equations whose event times are generated by threshold crossing of a continuous functional on the recent history segment. The paper analyses when an already exponentially stable delay flow retains stability under trigger-generated, possibly expansive, jumps. We formulate the hybrid problem on an admissible phase set, prove existence and uniqueness of global solutions for admissible initial histories, and show that strict post-jump reset together with a verifiable upper bound on trigger growth yields a uniform lower bound on all post-reset inter-event times and an initial-event lower bound determined by the initial trigger margin. This implies Zeno-freeness and, combined with average dwell-time and Lyapunov–Krasovskii decay, gives an explicit global exponential estimate for stable delay flows subjected to expansive impulses. A detector-augmentation theorem shows that a broad class of finite-dimensional dynamic triggers can be represented within the abstract framework. In addition, we derive a direct differentiation formula for weighted-window history triggers and show that, under a global Lipschitz condition on the observation map, this directly history-dependent class also satisfies the trigger regularity required by the abstract framework. The theory yields both state-space and full history-norm stability on the admissible set, and it is illustrated by detector-based examples with trigger-generated updates.