Output-Feedback Control of the Semilinear Heat Equation via the L2 Residue Separation and Harmonic Inequality

The paper’s main theorem (stabilization via the L2 residue separation with the harmonic inequality and two Riccati equations) is internally consistent and its proof steps check out, including the precise treatment of the residue via ζ, the optimal choice of γ from a harmonic series, and the cancellation of interconnection terms u and ζ between the low-mode and residue estimates. By contrast, the candidate solution contains a critical error in its Lyapunov functional: it weights the infinite-dimensional residue by v_n = 1/(λ_n − q − σ), then claims a norm equivalence V ≳ ∥z∥^2; this is false because v_n decreases to 0 as n→∞, so (1/γ)∑ v_n z_n^2 does not control ∑ z_n^2. The candidate also tacitly assumes ∑ v_n f_n^2 ≤ ∑ f_n^2 (i.e., 1/(λ_N − q − σ) ≤ 1) without justification. Hence the model’s final L2-stability conclusion is not supported by its chosen Lyapunov functional, whereas the paper avoids this pitfall by using V∞ = γ^{-1}∑_{n≥N} z_n^2 and the harmonic inequality to close the estimates.