Stabilization of solutions of the controlled non-local continuity equation

The paper proves s-stabilization for the controlled non-local continuity equation under Assumption 2.7 and the Control–Lyapunov Pair (CLP) of Definition 4.4 by a careful inf-convolution regularization, a constructed proximal ε-subgradient at a nearby pseudo-minimizer, and an extremal-shift feedback; it also supplies the needed intersample and truncation estimates via Propositions 2.11–2.12, and assembles a global, piecewise-defined feedback achieving Definition 6.1 s-stabilization (Sections 5–6). These ingredients are explicitly present in the text and are internally consistent . The candidate model relies on choosing, at each sampled measure m, a proximal ε-subgradient α ∈ ∂^P_ε φ(m), but the paper’s CLP does not guarantee nonemptiness of ∂^P_ε φ(m) at arbitrary m; indeed, the paper circumvents this by constructing γε_κ(m) ∈ ∂^P_ε φ(µ^ε_κ(m)) at a different point produced via inf-convolution and Ekeland’s principle, and then uses an extremal-shift feedback. Because the model’s feedback is undefined where ∂^P_ε φ(m) = ∅, its argument does not go through in the paper’s generality. Hence the paper’s proof is correct, while the model’s solution is flawed by a missing key hypothesis (global nonemptiness of the proximal subdifferential).