A Stability Condition for Online Feedback Optimization without Timescale Separation

The paper proves stability and optimality for the OFO closed loop by introducing a max-type Lyapunov function V = max{ξ Vx, Vu} and a Dini-derivative argument (Lemma 3), which yields exponential decay V(t) ≤ e^{-τ t} V(0) under Assumption 3’s block-dominance inequalities, for any α > 0; this establishes global exponential stability and links equilibria to critical points of Φ̃ via the chain rule ∇Φ̃(u) = H̃(u)∇Φ(u, h(u)) and u̇ = −α H̃(u)∇Φ(u, y) (equations (8)–(11), Theorem 1) . The candidate solution uses a different Lyapunov construction—a weighted sum W(x,u) = Vx(x) + (ζ/α) Vu(u)—to cancel α in the cross terms and obtain Ẇ ≤ −λ W under the same Assumption 3(i)–(iii), and then argues uniqueness and optimality as in the paper. The equilibria/critical-point correspondence and existence arguments match the paper’s chain-rule characterization and Assumption 2; uniqueness follows from global exponential stability in both approaches . Net: the paper’s result is correct, and the model’s proof is also correct but uses a different (sum-type) Lyapunov function than the paper’s max-type Lyapunov approach.