Extremum Seeking with High-Order Lie Bracket Approximations: Achieving Exponential Decay Rate

The paper’s Theorem 1 establishes semi-global practical exponential stability (SGPES) for the two-input ES system (8) when N = m under Assumption 2, using a Chen–Fliess one-period expansion in which all words of length ≤ N except I_N are cancelled and g_{I_N}(J) = −c_N J^{(N−1)}; crucially, the remainder is shown to be of order ε^{1+1/m} (not ε^{m+1}) and the dithers scale as u^ε_i(t) = ε^{1/N−1} v_i(t/ε) (so their sup-norm is not uniform in ε) . The candidate solution asserts the same high-level conclusion but relies on two incorrect technical claims: (i) a remainder bound O(ε^{N+1}) with N = m, contradicting the paper’s ε^{1+1/m} estimate; and (ii) uniform L∞ bounds on the dithers to obtain an inter-sample O(ε) bound, which is incompatible with the required amplitude scaling u^ε_i = ε^{1/N−1} v_i(t/ε). These errors lead to an unrealistically small ultimate bound ρ = O(ε^m) and an incorrect inter-sample estimate. The paper’s argument (backed by the precise remainder estimate and correct scaling) yields SGPES with a residual radius that decays with ε at the rate implied by ε^{1+1/m}, consistent with the referenced construction and bounds .