Towards cyber-physical systems robust to communication delays: A differential game approach

The paper’s Theorem 2 establishes a min–max “sandwich” between the true time-delay system (TDS) value and those from the control‑affine approximation (8)–(9), and shows the gap is O(d_max) (its inequalities (15)–(16)) , building on the TDS model (1) , the approximation (8)–(9) , and the small modeling‑error Lemma 1 (10) . The Appendix B proof constructs an exact‑realization error signal and then applies a Lipschitz/Grönwall bound to compare trajectories, yielding the O(d_max) estimate . The candidate solution repeats essentially these ideas: (i) builds the exact‑realization strategy Γ* to get the sandwich, and (ii) derives a uniform trajectory and cost perturbation bound via Grönwall, concluding |V_b^− − V_b^+| = O(d_max). Minor differences are present (the model adds a smallness condition that can be absorbed into constants and assumes DAE well‑posedness for (8)–(9)), but the logical core and conclusions match the paper.