QUASI-CONVERGENCE OF AN IMPLEMENTATION OF OPTIMAL BALANCE BY BACKWARD-FORWARD NUDGING

The paper proves quasi-convergence of the backward–forward nudging scheme by establishing a uniform one-step contraction with a small forcing term (Propositions 4 and 5), and then combining this with the endpoint matching identity Fn(q*,T)=Gn(q*) implied by the order conditions, yielding Theorems 6 (algebraic rate) and 7 (exponential-in-ε^{-1/3} rate) via a geometric-series argument. This is explicitly visible in the statement and proof of Theorem 6, which uses the identity Gn(q*)=Fn(q*,T) and the estimate ‖w_{m+1}‖≤θ‖w_m‖+c(ε/T)^n to obtain limsup bounds, and in Theorem 7, which relies on the exponential one-step forcing de^{-c√(T/ε)} and the inequality √(T/ε)≥T^{1/6}ε^{-1/6}≥T^{1/3}ε^{-1/3} for ε≤T to reach an ε^{-1/3}-type exponent in the final bound . The order conditions on ρ (algebraic or Gevrey-2) and the nudging scheme’s definitions are clearly stated, including the definition of w_m and the role of Fn and Gn . The candidate solution reproduces exactly this structure: assumes the one-step recursion, invokes the matching Fn=Gn identity under the endpoint conditions, applies the standard affine-contraction lemma to sum the geometric series, and passes to limsup, concluding Theorems 6–7 with the same dependence of constants and exponents. Minor omissions in the candidate’s write-up (e.g., explicit smallness of T0, bounds on w0 within a ball, and analytic radii for V) are precisely the hypotheses under which Propositions 4 and 5 hold in the paper and do not affect the core logic or conclusion. Hence, both are correct and the proofs are substantially the same .