Critical resetting and a maximum-resistance regime in long-memory fractional stochastic systems

Systems with anomalous dynamics, driven by fractional evolution 1<ρ<2 and persistent long-memory noise (H>1/2), challenge standard stabilization strategies because past history can be amplified rather than dissipated. Here we develop an analytical setting for such long-memory fractional stochastic systems under impulsive resetting interpreted as stochastic resetting. Using sectorial-operator-based evolution families together with fixed-point arguments, we derive sufficient conditions for well-posedness and mean-square exponential stability, yielding an explicit stability criterion that defines a critical resetting frequency fc(ρ,H). Numerical simulations reveal that fc depends non-monotonically on the fractional order ρ, exhibiting a robust peak near an intermediate value (ρ≈1.5). A reduced spectral criterion further links this peak to a mode-wise amplification boundary and gives a conditional analytical mechanism for the interior maximum. This “maximum-resistance” regime constitutes a new control paradigm: it marks the strongest competition between memory-driven amplification and resetting-induced contraction, requiring the highest resetting rate for stabilization. The results demonstrate that no single global power law governs multi-memory systems; instead, stabilization is controlled by competing memory mechanisms. In addition, numerical experiments reveal a discrepancy between ensemble and time-averaged observables, indicating nonergodic behavior in the weak-resetting regime, while stronger resetting reduces this discrepancy. Our framework provides a mechanistic basis for designing impulsive stabilization in long-memory media, with potential implications for viscoelastic materials and resetting-based neural control such as Deep Brain Stimulation (DBS).