Regularity of Optimal Solutions and the Optimal Cost for Hybrid Dynamical Systems via Reachability Analysis

The paper’s Theorem 13 states continuity of the optimal cost/value map under outer well-posedness, pre-forward completeness, inner well-posed perturbations dominated by a ρ-perturbation, and the non-jump/non-terminal-time assumption at the target hybrid time, together with continuity of the cost in its parameter; see the formal statement of Theorem 13 and its surrounding discussion . The candidate solution reproduces the same claim and proves it by (i) establishing limsup/liminf inclusions for the reachable sets via outer/inner well-posedness and (ii) passing to the limit using continuity of the cost—exactly the strategy the paper adopts (via Theorems 10 and 11, Berge’s maximum theorem, and graphical convergence) . Minor differences are stylistic: the model gives an explicit “evaluation along converging graphs” lemma (the paper references a similar fact as [38, Lemma 2]) and proceeds directly with limsup/liminf arguments, whereas the paper explicitly invokes Berge’s theorem. No missing hypotheses materially affecting correctness were found. Hence, both are correct and essentially the same proof.