Quantitative Prediction of Monotone Decay Envelopes in Nonlinear Fractional Delay Systems via Asymptotically Accurate Comparators

Abstract

The transient behavior of nonlinear fractional-order systems with time delays is critical for safety-critical applications, yet classical stability theory provides only asymptotic decay rates without specifying the time at which a system response begins to decrease monotonically. We develop a rigorous framework of asymptotically accurate comparators for a class of forced nonlinear Caputo fractional delay equations. An upper-bound comparator V(t) is constructed satisfying |u(t)| ≤ V(t) and lim_{t→+∞} V(t)/|u(t)| = 1, so that V captures the exact leading-order asymptotic behavior of the true solution; under a quantitative rate condition on the forcing, V is dominated by a fully explicit comparator computable from the known kernel and forcing alone. Exploiting the complete monotonicity of the fractional resolvent kernel and a monotone-density argument, we prove that V is strictly decreasing beyond the onset time of monotonicity of an explicitly computable convolution G = K * w, and we derive an explicit sufficient bound for this threshold in terms of the system parameters. The threshold estimator T* depends only on the known kernel and forcing, hence is computable a priori by numerical quadrature; no knowledge of the solution is required. The comparator error is of strictly smaller order than the envelope itself, V(t) - |u(t)| = o(t^{-β}). The theory is applied to the axial vibration of a shield machine cutter head subject to a decaying rock–soil load, and is further tested numerically on a memory-damped oscillator and a fractional Hopfield neural network; in all examples and parameter scans the predicted decay-onset time matches the numerically observed one with relative errors within 17%.