Indices of equilibrium points of linear control systems with saturated state feedback

The paper proves, for the single‑input case, that generically there is one equilibrium when n is even and three when n is odd by (i) noting the origin’s index is (−1)^n and each other equilibrium has index +1, and (ii) using a bounded‑perturbation/homotopy argument to show the index on a large ball equals ind(Ax)=1, so the sum of local indices is 1 (Theorem 3.1 and its proof sketch; Proposition 2.2; Theorem 2.3) . The candidate solution reproduces the same degree argument more explicitly: it constructs the large‑sphere homotopy to Ax, computes local Jacobians in saturated/unsaturated regions, and identifies the two saturated equilibria x±=±A^{-1}bM with the consistency condition kA^{-1}b<−1; substituting indices then forces exactly 1 or 3 equilibria by parity, matching the paper’s result. Minor wording ambiguities in the paper (e.g., “off the saturated region”) do not affect correctness.