Singularly Perturbed Averaging with Application to Bio-Inspired 3D Source Seeking

The paper’s main theorem (Theorem 2.1) and its supporting trajectory approximation (Proposition B.1) are consistent and technically justified under Assumption 2.1, combining a refined fast-variable coordinate shift (including ϕ1, ϕ2) with second-order averaging to produce the averaged vector field f̄ and a finite-horizon closeness result sufficient to establish the paper’s nonstandard stability notion; see system (2), assumptions, and the averaged vector field (9)-(10) and the formal statement of SSGPUAS in Definition A.1 . By contrast, the model’s Step (3) asserts a uniform-in-time bound dist(u(t),S) ≤ β(·) + O(1/√ω) for the perturbed system u̇ = f̄(u) + d(t) using a simple Grönwall/Lipschitz argument; this is not generally valid without an ISS-type robustness property, and the claimed time-uniform (global) additive-disturbance bound does not follow from GUAS of the averaged system alone. The paper avoids this pitfall by proving finite-horizon closeness to the averaged dynamics with exponentially fast fast-variable decay (37)-(38), parameterized by any tf and D, then invoking the SSGPUAS definition to conclude stability by choosing ω* depending on tf and D (i.e., a sliding-window argument) . The model’s Step (1)-(2) are aligned in spirit with the paper’s construction, but Step (3) is a substantive logical gap needed to claim Item 1 of Definition A.1 for all t ≥ 0.