Online Algorithms and Policies Using Adaptive and Machine Learning Approaches

The paper’s Theorem 1 proves boundedness, asymptotic tracking (e→0), and constant regret R=O(1) for the AC‑RL controller under Assumptions A1–A3 and A4′ using a Lyapunov function that weights parameter errors by Λ*; its appendix derives ė = A_H e + B_r Λ* Θ̃ ω and shows V̇ = −e^T Q e, then invokes Barbalat’s lemma. The candidate solution reproduces the same structure: introduces K* and Λ*-weighted errors, matches the error model, selects the same Lyapunov function up to notation, obtains exact cross‑term cancellation to V̇ = −e^T Q e, concludes e ∈ L2∩L∞ and e→0 by Barbalat, and deduces R(T) ≤ V(0). Minor notational differences aside, the arguments coincide with the paper’s proof sketch and appendix details (Assumptions A1–A3, A4′; controller (36)–(39); proof steps (83)–(86); regret (13)) .