Beyond partial control: Controlling chaotic transients with the safety function

The paper correctly defines the safety function U, the Bellman-style fixed-point recursion used to compute it, and the greedy control step that minimizes max(|u|, U(next)) after observing the realized disturbance, and it informally argues that U(q_n) is nonincreasing and that trajectories “eventually” enter the safe set, illustrating this numerically on the tent map. However, it provides no general proof or conditions guaranteeing finite-time entry for all disturbance sequences; it simply asserts eventual entry and that the safe set becomes a global attractor (outside of figures), without ruling out equal-value cycles or tie-induced stalling outside the safe set. The candidate solution supplies the missing fixed-point–based one-step Lyapunov argument (nonincrease and forward invariance) and makes explicit an additional structural hypothesis (or a strict tie-breaking condition) needed to guarantee finite-time entry for all disturbance sequences. Hence, the paper’s claims are incomplete, while the model’s solution is correct and appropriately qualified. See the paper’s statements of the safety function and control policy, and the claim of eventual convergence and “global attractor” language in the conclusions, which lack a general proof (e.g., “Eventually, the orbit enters into the safe set to remain forever in it” and “the safe set is an attractor of any initial condition in Q”) ; and the fixed-point operator defining U in the appendix, matching the model’s Bellman operator .