Competitive Control with Delayed Imperfect Information

The model’s solution reproduces the paper’s main bound (Theorem 3) and follows essentially the same proof structure: (i) a cost-difference characterization showing Alg−Opt equals a weighted quadratic in the deviation from the offline-optimal action (Lemma 10; Alg−Opt = ∑ η_t^T H η_t) , (ii) representing the offline optimum via the co-state/preview sum ψ_t = ∑ F^{>i}P w_{t+i} and lower-bounding Opt by λ_min(P^{-1}−FP^{-1}F^>−H)∑||ψ_t||^2 , and (iii) bounding the deviation sequence (η_t or, equivalently, δy_t = Hη_t) through a stable convolution whose ℓ1 norm yields the exact coefficients shown in the theorem for both k ≥ d and k ≤ d cases (see bounds on η_t in Eqs. (13) and (16) and the matrix/Young-type inequality in Appendix C) . The handling of terminal terms and the O(1) additive constant matches Lemmas 5–6 and Theorem 3’s statement (O(1)=0 when Q_f=P) . Minor notation differences (the model uses s_t while the paper uses ψ_t, and writes Alg−Opt as ∑ δy_t^T H^{-1} δy_t with H^{-1} understood on Range(B)) are benign and map directly to the paper’s η_t/H formulation.